Collected Papers: Volume 2 (1951-1964)

ANDRÉ WEIL ŒVRES SCIENTIFIQUES COLLECTED PAPERS

Paris, 1953

André Weil Œuvres Scientifiques Collected Papers Volume II (1951–1964)

 

Soft cover reprint of the 1979 edition published with the ISBN 0-387-90330-5 Springer-Verlag New York 3-540-90330-5 Springer-Verlag Berlin Heidelberg Library of Congress Control Number: 2008940272 Mathematics Subject Classification (2000): 01A75, 11-03, 14-03, 22-03, 46-03, 53-03, 57-03, 58-03

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Table des Mati6res Volume II [1951-1964]

[1951c] [1952a] [1952b] [1952d] [1952e] [1953] [ 1954a] [ 1954b] [1954c] [1954d] [ 1954e] [ 1954f] [ 1954g] [1954h] [ 1954i] [1955a] [1955b] [1955c] [1955d] [1955e] [19561 [1957a] [1957b] [1957c]

Bibliographie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R e v i e w o f " I n t r o d u c t i o n to the t h e o r y o f algebraic functions, b y C. C h e v a l l e y " . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sur les th6or&mes de de R h a m . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lettre b. Henri C a r t a n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sur les "formules explicites" de la th6orie des n o m b r e s premiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jacobi sums as "Gr/Sssencharaktere" . . . . . . . . . . . . . . . . . . . . . . O n Picard varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Th~orie des points p r o c h e s sur les vari&6s differentiables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R e m a r q u e s sur un m6moire d ' H e r m i t e . . . . . . . . . . . . . . . . . . . . Mathematical teaching in Universities . . . . . . . . . . . . . . . . . . . . T h e m a t h e m a t i c s curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sur les crit~res d'6quivalence en g6om6trie alg6brique . . . . . . F o o t n o t e to a recent paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N u m b e r o f points o f varieties in finite fields . . . . . . . . . . . . . . . O n the projective embedding o f abelian varieties . . . . . . . . . . . A b s t r a c t versus classical algebraic g e o m e t r y . . . . . . . . . . . . . . . Poincar6 et l'arithm&ique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O n algebraic groups o f transformations . . . . . . . . . . . . . . . . . . . . O n algebraic groups and h o m o g e n e o u s spaces . . . . . . . . . . . . . O n a certain type o f characters o f the id61e-class group o f an algebraic number-field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O n the t h e o r y of c o m p l e x multiplication . . . . . . . . . . . . . . . . . . Science Fran~aise? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T h e field of definition o f a variety . . . . . . . . . . . . . . . . . . . . . . . . Z u m Beweis des Torellischen Satzes . . . . . . . . . . . . . . . . . . . . . Hermann Weyl (1885-1955) ............................ 1) R6duction des formes quadratiques; 2) G r o u p e s des formes quadratiques ind6finies et des formes bilin6aires altern6es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii 2 17 .45 48 63 73 103 111 118 121 127 161 165 175 180 189 197 235 255 263 277 291 307 329

360

vi [1958a] [1958b] [1958c] [1958d] [ 1959a] [1959b] [ 1960a] [1960b] [1960c] [1961b] [ 1962a] [1962b] [1962c] [ 1964a]

Tabledes Matieres Introduction ~ l'6tude des vari6t6s k~ihl6riennes (Pr6face)... 379 On the moduli of Riemann surfaces (to Emil Artin) . . . . . . . . 381 Final report on contract A F 18(603)-57 . . . . . . . . . . . . . . . . . . 391 Discontinuous subgroups of classical groups (Introduction) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 Ad/~les et groupes alg6briques . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 Y. Taniyama . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 De la m6taphysique aux math6matiques . . . . . . . . . . . . . . . . . . 408 Algebras with involutions and the classical groups . . . . . . . . . 413 On discrete subgroups of Lie groups . . . . . . . . . . . . . . . . . . . . . 449 Organisation et d6sorganisation en math6matique . . . . . . . . . . 465 Sur la th6orie des formes quadratiques . . . . . . . . . . . . . . . . . . . 471 On discrete subgroups of Lie groups (ll) . . . . . . . . . . . . . . . . . 486 Algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 Remarks on the cohomology of groups . . . . . . . . . . . . . . . . . . . 517 Commentaire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 Appendix I: Correspondence, by X X X (Am. 555 J. of Math. 79 (1957), pp. 951-952) . . . . . . . . . . . . . . . . . . . Appendix II: Correspondence, by R. Lipschitz (Ann. 557 of Math. 69 (1959), pp. 247-251) . . . . . . . . . . . . . . . . . . . . .

Bibliographie

[1926] [1927a] [1927b] [1927c] [1928] [1929] [1932a] [1932b] [1932c] [1934a]

[1934b] [1934c] [1935a] [1935b] [1935c]

(Les caract{res gras dksignent les livres et notes de tours; C.R. = Comptes Rendus de l'Acadkmie des Sciences). Sur les surfaces ~t courbure n6gative, C.R. 182, pp. 1069-1071. Sur les espaces fonctionnels, C.R. 184, pp. 67-69. Sul calcolo funzionale lineare, Rend. Linc. (VI) 5, pp. 773-777. L'arithm&ique sur une courbe alg6brique, C.R. 185, pp.

1426-1428. L'arithm6tique sur les courbes alg6briques, A c t a Math. 52, pp. 281-315. Sur un th6or6me de Mordell, Bull. Sc. Math. (II) 54, pp. 182-191. On systems of curves on a ring-shaped surface, J. Ind. Math. Soe. 19, pp. 109 114. Sur les s6ries de polynomes de deux variables complexes, C.R. 194, pp. 1304-1305. (avec C. Chevalley) Un th4orhme d'arithm6tique sur les courbes alg6briques, C.R. 195, pp. 570-572. (avec C. Chevalley) Uber das Verhalten der Integrale erster Gattung bei Automorphismen des Funktionenk6rpers, H a m b . Abh. 10, pp. 358 361. Une propri6t4 caract6ristique des groupes de substitutions lin6aires finis, C.R. 198, pp. 1739-1742. Une propri6t6 caract6ristique des groupes finis de substitutions, C.R. 199, pp. 180-182. Uber Matrizenringe auf Riemannschen Fl~ichen und den Riemann-Rochschen Satz, H a m b . Abh. 11, pp. 110-115. Arithm6tique et G6om&rie sur les vari4t6s alg6briques, Act. Sc. et Ind. no. 206, Hermann, Paris, pp. 3-16. Sur les fonctions presque p6riodiques de von Neumann, C.R. 200, pp. 38-40.

vii

,viii [1935d]

Bibliographie

L'int6grale de Cauchy et les fonctions de plusieurs variables, Math. Ann. 111, pp. 178-182.

[1935e] [1936a] [1936b] [1936c] [1936d] [1936e] [1936f] [1936g] [ 1936h] [1936i] [1937] [1938a] [1938b] [1938c] [1939a] [1939b] [ 1940a] [1940b] [1940c]

[1940d3 [1941] [1942] [ 1943a] [1943b] [1945]

D6monstration topologique d'un th6or6me fondamental de Cartan, C.R. 200, pp. 518-520. Les familles de courbes sur le tore, Mat. Sbornik (N.S.) 1, pp. 779 781. Arifmetika algebrai6eskykh mnogoobrazii (Arithmetic on algebraic varieties), Uspekhi Mat. N a u k 3, pp. 101-112. Matematika v Indii (Mathematics in India), Uspekhi Mat. N a u k 3, pp. 286 288. La mesure invariante darts les espaces de groupes et les espaces homog6nes, Enseign. Math. 35, p. 241. La th6orie des enveloppes en Math6matiques Sp6ciales, Enseign. Scient. 9 e ann6e, pp. 163-169. Les recouvrements des espaces topologiques; espaces complets, espaces bicompacts, C.R. 202, pp. 1002-1005. Sur les groupes topologiques et les groupes mesur6s, C.R. 202, pp. 1147-1149. Sur les fonctions elliptiques p-adiques, C.R. 203, pp. 22-24. Remarques sur des r6sultats r6cents de C. Chevalley, C.R. 203, pp. 1208-1210. Sur les espaces h structure uniforme et sur la topologie g6n6rale, Act. Sc. et Ind. no. 551, Hermann, Paris, pp. 3-40. G6n6ralisation des fonctions ab61iennes, J. de Math. P. et App., (IX) 17, pp. 47-87. Zur algebraischen Theorie der algebraischen Funktionen, CrellesJ. 179, pp. 129-133. "Science Fran~aise" (in6dit). Sur l'analogie entre les corps de hombres alg6briques et les corps de fonctions alg6briques, R e v u e Scient. 77, pp. 104-106. Les groupes 5- pn 616ments, R e v u e Scient. 77, pp. 321-322. Une lettre et un extrait de lettre 5- Simone Weil (in6dit). Sur les fonctions alg6briques 5. corps de constantes fini, C.R. 210, pp. 592 594. Calcul des probabilit6s, m6thode axiomatique, int6gration, Revue Scient. 78, pp. 201-208. L'intkgration dans les groupes topologiques et ses applications,

Hermann, Paris (2 e edition 1953). On the Riemann hypothesis in function-fields, Proc. Nat. Ac. Sci. 27, pp. 345-347. Lettre 5- Artin (in6dit). (jointly with C. Allendoerfer) The Gauss-Bonnet theorem for Riemannian polyhedra, Trans. A. M. S. 53, pp. 101-129. Differentiation in algebraic number-fields, Bull. A.M.S. 49, p. 41. A correction to my book on topological groups, Bull. A. M. S. 51, pp. 272-273.

Bibliographie

[1946a] [1946b3 [1947a]

[1947b]

[1948a,b]

[ 1948c] [1949a]

[ 1949b] [ 1949c] [1949d] [ 1949e] [ 1950a] [1950b] [1951a] [1951b]

ix

Foundations of algebraic geometry, Am. Math. Soc. Coll., vol. XXIX, New York (2rid edition 1962). Sur quelques r6sultats de Siegel, S u m m a Brasil. Math. 1, pp. 21-39. L'avenir des math6matiques, "Les Grands Courants de la Pensbe Mathbmatique", 6d. F. Le Lionnais, Cahiers du Sud, Paris, pp. 307 320 (2 e 6d., A. Blanchard, Paris 1962). Sur la th6orie des formes differentielles attach6es ~t une vari6t6 analytique complexe, Comm. Math.. Heir. 20, pp. 110-116. (a) Sur les courbes algkbriques et les varibtbs qui s'en dbduisent, Hermann, Paris; (b) Variet~s abbliennes et courbes algbbriques, ibid; [2 e 6dition de (a) et (b), sous le titre collectif "Courbes algkbriques et varibtOs abbliennes", ibid., 1971 ]. On some exponential sums, Proc. Nat. Ac. Sc. 34, pp. 204-207. Sur l'6tude alg6brique de certains types de lois de mariage (Syst~me Murngin), Appendice 5, la I ~ partie de: C. L6viStrauss, Les structures klementaires de la parentb, P. U. F. Paris 1949, pp. 278-285. Numbers of solutions of equations in finite fields, Bull. Am. Math. Soc. 55, pp. 497-508. Fibre-spaces in algebraic geometry, in Algebraic Geometry Conference, U. of Chicago (mimeographed), pp. 55 59. Th6orbmes fondamentaux de la th6orie des fonctions th~ta, Sbminaire Bourbaki no. 16, mai 1949, 10 pp. G6om6trie differentielle des espaces fibr6s (in6dit). Vari6t6s ab61iennes, in Colloque d'AIgbbre et Theorie des Nombres, C.N.R.S., Paris, pp. 125-127. Number-theory and algebraic geometry, Proc. Intern. Math. Congress, Cambridge, Mass., vol. II, pp. 90-100. Arithmetic on algebraic varieties, Ann. o f Math. 53, pp. 412-444. Sur la th6orie du corps de classes, J. Math. Soc. Japan 3, pp.

1-35. [1951c]

[1952a]

Review of "Introduction to the theory of algebraic functions of one variable, by C. Chevalley", Bull. Am. Math. Soc. 57, pp. 384-398. Sur les th6or6mes de de Rham, Comm. Math. Heir. 26, pp.

119-145. [1952b]

[1952e] [1952d3

[1952e] [ 1952f]

Sur les "formules explicites" de la th6orie des nombres premiers, Comm. Lund (vol. d6di6 5, Marcel Riesz), p. 252. Fibre-spaces in algebraic geometry (Notes by A. Wallace). U. of Chicago, (mimeographed) 48 pp. Jacobi sums as "Gr~issencharaktere", Trans. Am. Math. Soc. 73, pp. 487-495. On Picard varieties, Am. J. of Math. 74, pp. 865-894. Criteria for linear equivalence, Proc. Nat. Ac. So. 38, pp. 258-260.

X [1953]

[1954a] [1954b] [1954c] [1954d] [1954e] [1954f] [19548]

[1954h] [1954i] [1955a] [1955b] [1955c]

[1955d] [1955e] [1956] [1957a] [1957b] [1957c]

[1958a] [1958b] [1958c]

[1958d]

Bibliographie Thdorie des points proches sur les vari&~s differentiables, in Colloque de G bombtrie Differentielle (Strasbourg 1953), C.N.R.S., pp. l l l - l l 7 . Remarques sur un m6moire d'Hermite, Arch. d. Math. 5, pp. 197-202. Mathematical Teaching in Universities, Am. Math. Monthly 61, pp. 34-36. The mathematical curriculum (a guide for students) (inddit). Sur les crit6res d'6quivalence en g6om~trie alg~brique, Math. Ann. 128, pp. 95 127. Footnote to a recent paper, Am. J. of Math. 76, pp. 347-350. (jointly with S. Lang) Number of points of varieties in finite fields, Am. J. of Math. 76, pp. 819 827. On the projective embedding of abelian varieties, in Algebraic geometry and Topology, A Symposium in honor orS. Lefschetz, Princeton U. Press, pp. 177-181. Abstract versus classical algebraic geometry, Proc. Intern. Math. Congr. Amsterdam, vol. III, pp. 550-558. Poincar6 et l'arithm&ique, in Livre du Centenaire de Henri Poincarb, Gauthier-Villars, Paris, 1955, pp. 206-212. On algebraic groups of transformations, Am. J. of Math. 77, pp. 355-391. On algebraic groups and homogeneous spaces, Am. J. of Math. 77, pp. 493-512. On a certain type of characters of the idble-class group of an algebraic number-field, in Proc. Intern. Symp. on Algebraic Number Theory, Tokyo-Nikko. pp. 1-7. On the theory of complex multiplication, ibid., pp. 9-22. Science Fran~aise?, La Nouvelle N.R.F., Paris, 3e ann6e, n~ pp. 97 109. The field of definition of a variety, Am. J. of Math. 78, pp. 509 524. Zum Beweis des Torellischen Satzes, Gi)tt. Nachr. 1957, no. 2, pp. 33-53. (avec C. Chevalley) Hermann Weyl (1885 1955), Enseign. Math. I11, pp. 157-187. (1) R6duction des formes quadratiques, 9 pp.; (2) Groupes des formes quadratiques inddfinies et des formes bilindaires altern6es, 14 pp., Seminaire H. Cartan, 10e ann6e, novembre 1957. Introduction b l'i'tude des varibt~s k~lhlbriennes, Hermann, Paris. On the moduli of Riemann surfaces (to Emil Artin), (inddit). Final Report on contract AF 18(603)-57 (in6dit). Discontinous subgroups of" classical groups (Notes by A. Wallace), U. of Chicago (mimeographed).

Bibliographie [1959a] [1959b] [1960a] [1960b] [1960c]

[1961a] [1961b] [1962a]

[1962b] [1962c] [1964a] [1964b] [1965] [1966] [1967a] [1967b] [1967e] [1968a] [1968b] [1970]

[1971a] [1971b] [1971c]

xi

Addles et groupes alg~briques, S~minaire Bourbaki, mai 1959, n ~ 186, 9 pp. Y. Taniyama (lettre d'Andr6 Weil), Sugaku-no Ayumi, vol. 6. no. 4, pp. 21-22. De la m~taphysique aux math6matiques, Sciences, pp. 52-56. Algebras with involutions and the classical groups, J. Ind. Math. Soc. 24, pp. 589-623. On discrete subgroups of Lie groups, Ann. of Math. 72, pp. 369-384. Adeles and algebraic groups, I.A.S., Princeton. Organisation et d6sorganisation en math6matique, Bull. Soc. Franco-Jap. des Sc. 3, pp. 25-35. Sur la th6orie des formes quadratiques, in Colloque sur la Thkorie des Groupes Algkbriques, C.B.R.M., Bruxelles, pp. 9-22. On discrete subgroups of Lie groups (II), Ann. of Math. 75, pp. 578-602. Algebraic geometry, in Encyclopedia Americana, New York, pp. 455-457. Remarks on the cohomology of groups, Ann. of Math. 80, pp. 149-157. Sur certains groupes d'op6rateurs unitaires, Acta Math. 11 l, pp. 143-211. Sur la formule de Siegel dans la th6orie des groupes classiques, Acta Math. 113, pp. 1-87. Fonction z~ta et distributions, Skminaire Bourbaki no. 312, juin 1966. Uber die Bestimmung Dirichletscher Reihen dutch Funktionalgleichungen, Math. Ann. 168, pp. 149-156. Review: "The Collected papers of Emil Artin", Scripta Math. 28, pp. 237-238. Basic Number Theory (Grundl. Math. Wiss. Bd. 144), Springer (3 rd edition, 1974). Zeta-functions and Mellin transforms, in Proc. of the Bombay Coll. on Algebraic Geometry, T.I.F.R., Bombay, pp. 409-426. Sur une formule classique, J. Math. Soc. Japan 20, pp. 400-402. On the analogue of the modular group in characteristic p, in "Functional Analysis, etc.", Proc. Conf. in honor of M. Stone, Springer, pp. 211-223. Automorphic forms and Dirichlet series, Lecture-Notes no. 189, Springer. Notice biographique, in t~Euvres de J. Delsarte, C.N.R.S., Paris 1971, t.I, pp. 17-28. L'ceuvre math~matique de Delsarte, ibid., pp. 29-47.

xii [1972]

Bibliographie

Sur les formules explicites de la th6orie des nombres, lzv. Mat. N a u k (Ser. Mat.) 36, pp. 3-18.

[1973] [1974a] [1974b] [1974c] [1974d] [1974e] [1975a]

[1975b]

[1976a] [1976b] [1976c] [1977a]

[1977b] [1977c] [1978a] [1978b]

Review of "The mathematical career of Pierre de Fermat, by M. S. Mahoney", Bull. A m . Math. Soc. 79, pp. 1138-1149. Two lectures on number theory, past and present, Enseign. Math. XX, pp. 87-110. Sur les sommes de trois et quatre carr6s, Enseign. Math. XX, pp. 215-222. La cyclotomie jadis et nagu6re, Enseign. Math. XX, pp. 247-263. Sommes de Jacobi et caract6res de Hecke, Gi)tt. Nachr. 1974, Nr. 1, 14 pp. Exercices dyadiques, Invent. math. 27, pp. 1-22. Review of "Leibniz in Paris 1672-1676, his growth to mathematical maturity, by Joseph E. Hofmann", Bull. A m . Math. Soc. 81, pp. 676-688. Introduction to E.E. Kummer. Collected Papers vol. I, pp. 1-11. Elliptic Functions according to Eisenstein and Kronecker, (Ergebnisse d. Mathematik. Bd. 88), Springer. Sur les p6riodes des int6grales ab61iennes, Comm. on Pure and Appl. Math. XXIX, pp. 813-819. Review of "Mathematische Werke, by Gotthold Eisenstein", Bull. A m . Math. Soc. 82, pp. 658-663. Remarks on Hecke's lemma and its use, in Algebraic N u m b e r Theory, Intern. Symposium Kyoto 1976, S. lyanaga (ed.), Jap. Soc. for the Promotion of Science 1977, pp. 267-274. Fermat et l'6quation de Pell, HPI~MATA (W. Hartner Festschrift), Fr. Steiner Verlag, Wiesbaden 1977, pp. 441-448. Abelian varieties and the Hodge ring (in6dit). Who betrayed Euclid?, Arch. Hist. Exact Sci. 19, pp. 91-93. History of mathematics: Why and how, Proc. Intern. Math. Con~4ress, Helsinki.

ANDRt~ WEIL (EUVRES SCIENTIFIQUES COLLECTED PAPERS

[1951c] Review of "introduction to the theory of algebraic functions, by C. Chevalley" Introduction to the theory of algebraic functions of one variable. B y C. C h e v a l l e y . ( M a t h e m a t i c a l S u r v e y s , no. 6.) N e w Y o r k , A m e r i c a n M a t h e m a t i c a l Society, 1951. 1 2 + 188 pp. $4.00. Here is a l g e b r a with a v e n g e a n c e ; algebraic a u s t e r i t y could go no Reprinted from Bulletinof the AmericanMathematicalSociety, V1. 5"7,pp. 384-398 by permissionof the American MathematicalSociety.9 1951 by the AmericanMathematical Society.

[1951c1 x95~]

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385

further. "We have not tried to hide (says the author) our partiality to the algebraic a t t i t u d e . . . "; he has not indeed; and, if it were not for a few hints in the introduction and one casual r e m a r k at the end of C h a p t e r IV, one might never suspect him of having ever heard of algebraic curves or of taking a n y interest in them. Fields and only fields are the object of his study. A field is given, or rather two fields: one, the function-field R; the other, the field K of constants; K is algebraically closed in R; and R is finitely generated and of degree of transcendency 1 over K. E v e r y t h i n g m u s t be "intrinsic," i.e. m u s t be born from these by some standard operations. Later on the family circle is enlarged b y the a p p e a r a n c e of another function-field S containing R, with a field of constants 25 containing K, and a large portion of the book is devoted to the mutual relations between R and S; b u t nowhere except in one or two lemmas is a n y element allowed to a p p e a r unless it is contained in those fields or canonically generated from them. T h e contents of the book are as follows. Valuations are introduced and the basic existence theorem on valuations is proved in the standard m a n n e r (th. 1, p. 6), b y the use of Zorn's lemma: t h i s is the theorem according to which every "specialization" of a subring 0 of a field R (i.e. every h o m o m o r p h i c m a p p i n g of 0 into a field) can be extended to a "valuation" of R (i.e. a specialization of a subring K) of R such t h a t R = 9 the theorem, however, is not stated in its full generality. One might observe here that, in a functlon-field of dimension l, every valuation-ring is finitely generated over the field of constants, and therefore, if a slightly different a r r a n g e m e n t had been adopted, the use of Zorn's l e m m a (or of Zermelo's axiom) could have been avoided altogether; since T h e o r e m 1 is formulated only for such fields, this t r e a t m e n t would have been more consistent, and the distinctive features of dimension 1 would have appeared more clearly. Places are defined as being in one-to-one correspondence with the non-trivial valuation-rings of R, i.e. with those proper subrings 0 of R which contain the field of constants K of R and satisfy R = 0k-J0-1. In Zariski's terminology, on the other hand, a place is a homomorphic m a p p i n g of a valuation-ring o into a field; in consequence, if 0 is a non-trivial valuation-ring of R, and l~ the ideal of non-units in o (the "place" in Chevalley's sense), there will be as m a n y "places" belonging to 0 and I~, with values in a given "universal domain" t2, as there are isomorphisms of the "residue-field" E = 0/~ into f~; their n u m b e r is equal to the degree de over K of the maximal separable extension ~, of K contained in ~. According to Chevalley's self-

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[1951c] 386

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[September

imposed taboos, however, only the field 0/!0 is allowed to exist, and the "place" determined b y 0 and l) (or b y either of them) m u s t be unique. This has far-reaching consequences: while otherwise sums (e.g. the sum of the residues of a differential) could be extended over all the d, places belonging to 0 and l), here the d8 t e r m s belonging to such a sum can never be separated from each other. I t is true t h a t traces (of elem e n t s of 21, over K) are adequate substitutes for such sums; b u t it m a y well be doubted whether the constant use of traces is not an unnecessary complication, and whether it helps a beginner to understand the subject. C h a p t e r I then brings, as usual, proofs for the existence of a uniformizing variable at a place (i.e. a t e l ) such t h a t l) =to), for the independence of valuations (or of "places"), and for the existence of the divisor of a function; it ends up with the theorem t h a t the degree of the divisor of zeros of xCR is equal to [R:K(x)]. Chapter I I follows, with the definition of differentials and the proof of the RiemannRoch theorem due to A. Weil. T h e genus is defined by means of R i e m a n n ' s theorem. A "repartition" is defined as a function assigning to each place l) an element x(l)) of R (or, later, an element x(l)) of the la-adic completion R~ of R at !a), so t h a t those places !0 for which x(l)) has a pole at l0 are in finite n u m b e r ; then a differential is a linear function on the space of repartitions, continuous in a suitable sense, which vanishes on the subspace of "principal" repartitions (those for which x ( l ) ) = x for all !0, with xCR). This rather a b s t r a c t concept of differential is of course w h a t makes possible such a brief proof of the R i e m a n n - R o c h theorem; while this is v e r y convenient for m a n y purposes, one should not forget t h a t eventually (in the case where R is separably generated over K) differentials h a v e to be identified with the expressions ydx, or, w h a t a m o u n t s roughly to the same thing, it m u s t be shown t h a t the sum of the residues of ydx is 0; for this, in the present volume, one has to wait until p. 117. C h a p t e r I I I introduces the local or !~-adic completions of the function-field K by means of the usual definitions a n d of Hensel's l e m m a ; E~ being defined as before, it is shown t h a t E, can be canonically identified with a subfield of the completion R of R at l~, and that, if E =E,, this completion is essentially the ring E((t)) of power-series in t with coefficients in E, where t is a n y uniformizing variable at !0; the structure of R when ~ is not separable over K is not further discussed. T h e last w of C h a p t e r I I I brings the concept of residue of a differential, in terms of the values of the differential at certain "repartitions"; it then becomes trivial t h a t the sum, not of the residues, b u t of the traces of the residues of a differential is 0.

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So far only one function-field has been considered. Now another one, S, is introduced, with the field of constants L, such t h a t S ~ R and K=L(AR; the next three chapters (nearly half the book) are d e v o t e d to the simultaneous s t u d y of the two fields R, S, under various assumptions. Some of the questions raised here b y the author had never been treated before; unfortunately, as he treads new ground, his footsteps become more uncertain, and to follow in t h e m is at times no easy task. In the language of algebraic geometry, the passage from R to S consists p a r t l y in enlarging the field of constants of a given curve, p a r t l y in considering the mutual relations of two curves in a (1, m)-correspondence with each other; the a u t h o r tries to t r e a t b o t h problems b y the same methods ; however t e m p t i n g this idea m a y a p p e a r to the algebraist, it is not altogether successful, and m a y well h a v e caused some blurring of the picture. C h a p t e r I V is chiefly devoted to the case where S is of finite degree over R, and to the behavior of the places of R and S with respect to one another; these questions are fairly familiar, at least in the parallel case of number-fields, and no surprises are to be expected here. Because of the too special formulation which has been given of the theorem on the extension of specializations in C h a p t e r I, the existence of a place of S lying over a given place p of R is made to depend, strangely enough, upon R i e m a n n ' s theorem. T h e ramification indexes ex and the relative degrees d• of the places ~x of S lying over p are defined, and it is proved t h a t ~_~xdxex=[S:R]; this m i g h t well have been postponed until it is shown t h a t dxex is the degree of the ~3x-adic completion Sx of S over the p-adic completion of R, and a basis is explicitly given for the former over the latter (Theorems 4 and 5, p. 60-61); in between those results are inserted some r e m a r k s on the case of normal extensions, the proof for the existence of a base of S, integral at p, and auxiliary definitions and results on the Kronecker product of fields or c o m m u t a t i v e algebras, the latter being necessary in order to show t h a t the direct sum of the Sx is no other t h a n the algebra over R obtained b y considering S as an algebra over R and extending its ground-field to R. T h e n norms and conorms, traces and cotraces are defined for divisors and repartitions in S and in R ; norms and traces are defined as usual; the conorm and cotrace are the dual operations, i.e. consist in "lifting" divisors, repartitions, etc., from R to S; the consistent use of these terms (rather t h a n the more usual identification of divisors, etc., in R with the corresponding ones in S) is perhaps cumbersome, b u t is v e r y helpful in keeping a p a r t essentially distinct concepts while their main properties are being developed. T h e different is then de-

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fined, and its basic properties are given. C h a p t e r V discusses the extension of the field of constants; in the absence of a universal domain, such extensions have to be generated b y the clumsy device of the tensor-product of fields; in the inseparable case, one has then to face the disagreeable appearance of radicals, whose rather a r b i t r a r y dismissal follows at once, not without the intervention of minimal ideals. More t h a n half of 'the chapter is spent in such a w k w a r d discussions, beginning with the definition of separable (not necessarily algebraic) extensions b y the non-exlstence of nilpotent elements in certain tensor-products and leading up to T h e o r e m 3 (p. 92) which expresses, again in t e r m s of such products, the effect on a place of R of an extension of the field of constants. T h e r e is hardly a n y connection between the foregoing and the basic T h e o r e m 4; according to the latter, if the field of constants K of R is extended to a field L, separable over K, and if a is a divisor of R, every element y of the extended function-field which is a multiple of ~I (more accurately, of the " c o n o r m " of ~I) is a linear combination, with coefficients in L, of elements of R which are multiples of a (cf. A. Well's Foundations, C h a p t e r V I I I , th. 10); from this it is deduced t h a t the genus is not altered b y the extension of the field of constants from K to L if L is separable over K, and t h a t it can only decrease b y an a r b i t r a r y extension. C h a p t e r VI takes up the behavior of differentials under an extension of the function-field; one of its main objectives is to identify the differentials in R with the symbols ydx, provided R is separable (i.e., separably generated) over K. This is done b y means of a general theory for the "lifting" of a differential from a field R to a field S under suitable conditions; this operation is called the "cotrace." An explicit definition being given for a certain differential, called dx, in a purely transcendental extension K(x) of the field of constants, dx is then "lifted" from K(x) to R for every n o n - c o n s t a n t x in R. A d r a w b a c k of this m e t h o d is, of course, that, as dx and dy are lifted from different fields, there is no obvious connection between them, and d ( x + y ) = d x + d y becomes a deep t h e o r e m ; perhaps a more satisfactory a r r a n g e m e n t would have been provided by a definition similar to t h a t adopted for m e r o m o r p h i c differentials in C h a p t e r V I I . H o w e v e r t h a t m a y be, after a preliminary discussion of the field K(x), the fields K, R, L, S are again considered; the trace of a differential of S is defined, in the case where S is of finite degree over R; and the cotrace of a differential of R is defined, b u t merely for the case K = L; the behavior of residues under the operations of trace and cotrace, and other e l e m e n t a r y properties, are established. T h e dif-

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ferent (which had disappeared during the whole of C h a p t e r V) turns up again, and it is shown that, when a differential is lifted from R to S, its divisor is multiplied b y the different of S over R; it is a p i t y t h a t this w is separated from the w on the different, as both could just as easily have been p u t together, either here or, even better, in C h a p t e r IV. One then comes back to dx, which can now be lifted from K(x) to R; a m o n g other results, its divisor is calculated; and it is shown that, if x is a uniformizing variable at a place ~ of degree 1, the residue of ydx a t p is the coefficient of x -1 in the powerseries expressing y in t e r m s of x in the completion of R at ~. T h e investigation is again interrupted, this time in order to introduce the general concept of derivation in fields, algebraic function-fields and power-series fields; it is resumed for the proof of the decisive T h e o r e m 9, according to which, for a given x, dy/dx is the derivation D,y of R which vanishes on K and has the value 1 at x; this is ingeniously proved by showing t h a t the differential dy-(D,y)dx has infinitely m a n y zeros. T h e concept of cotrace is then extended to the case K ~ L , provided R is separable; since in t h a t case the differentials of R can be written as ydx, with x, y in R, these same expressions can be used to lift t h e m into S; in particular, under an extension of the field of constants of R, it is shown t h a t the residues of a differential remain the same, t h a t the divisor of a differential is unchanged if the genus is unchanged, t h a t otherwise it is divided b y an integral divisor. A further section (the purpose of which remains unexplained) discusses the effect on differentials and their residues of a derivation of the field of constants; and the c h a p t e r ends up, rather disappointingly, with a theory of differentials of the second kind confined to characteristic 0, in which case it is an easy application of the R i e m a n n - R o c h theorem. M a y b e the a p p e a r a n c e of characteristic 0 at the end of C h a p t e r VI was m e a n t as a transition to the extensive C h a p t e r V I I (more t h a n 50 pages), which treats the "classical" case, i.e. the case where the field of constants is the complex number-field, with its topology; this is almost a different book. I t is hard to say w h a t knowledge is ass u m e d of the reader in this chapter; while it is tediously proved t h a t m e r o m o r p h i c functions in an open set form a field, and one full page is devoted to the calculation of fdx/x on a circle surrounding the origin in the complex plane (the value being found to be _ 27r(- 1)1/3), Schwarz's l e m m a suddenly turns up from nowhere (p. 152) in order to prove t h a t holomorphic mappings preserve the orientation, a fact for which, fortunately, a more reasonable justification is given later (p. 181). T h e reader is further required to take for granted the

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validity of all the "axioms" of Eilenberg and Steenrod for the singular homology theory in arbitrary topological spaces, a statement of which is given in w ; thanks to this, says the author, "we have avoided the cumbersome decomposition of the Riemann surface into triangles." T h e truth is that this triangulation is a quite trivial matter; and, while the reduction of a triangulation to standard form (as done e.g. in Seifert-Threlfall) is a somewhat clumsy process, the canonical dissection of the Riemann surface which is so obtained has immense advantages over a purely homological theory; it shows t h a t all Riemann surfaces of a given genus are homeomorphic ; it gives the structure of the fundamental group, which, even to the pure algebraist, is of prime importance in determining the nature of the nonabelian extensions of the given function-field; such advantages seem to be more than enough to outweigh those of the more algebraic (and "intrinsic") procedure adopted by the author. T h e chapter begins with the definition of the Riemann surface, i.e. of the set of places of the given field, as a topological space; unfortunately, its definition as an analytic manifold is given only much later, so t h a t orientation is defined twice, and various special cases of Stokes' formula have to be proved separately. Meromorphic functions and differentials on open subsets of the Riemann surface are defined ; it is shown in the usual manner t h a t the meromorphic functions on the Riemann surface are the elements of the function-field. Periods of differentials are defined, essentially by analytic continuation (not by integration, since the 1-chains are not assumed to be differentiable), so t h a t their definition virtually depends upon the concept of fundamental group, which however is carefully avoided. We come now to one of the most interesting and original features of the whole book. With the author, let us denote by S the Riemann surface, b y P and Q two mutually disjoint finite subsets of S. Then Hi(S--P, Q) is the "relative" homology group of the open set S - P modulo Q; in other words, it is the group of classes of 1-chains lying in S - P , with boundary in Q (the "relative cycles" in S - P mod. Q), such a chain being homologous to 0 if it bounds in S - P . If 3' is such a relative cycle, and 3,' is a relative cycle in S - Q mod. _P, then, as the boundary of each cycle is disjoint from the other, the intersectionnumber or Kronecker index I(% 3") is defined; it depends only upon the homology classes of % 3"; and it determines a duality between H ~ ( S z P , Q) and H i ( S - Q , .P), in the sense t h a t I(3", 3") cannot be 0 for all 3" unless 3" is homologous to 0, and t h a t there is a 3" such t h a t I(3", 3") is equal to an arbitrarily given integral-valued linear function on H i ( S - P , Q). These groups, and the duality between them, can

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now be translated into algebraic terms by means of the following concepts. Let E(P, Q) be the set of differentials on S with no poles at the points of Q and no residue ~ 0 at any point outside P. Take a canonical dissection of S by means of curves, not going through the points of P and Q; S is then represented as a canonical polygon of 4g sides (where g is the genus, which we assume to be ~ 0 , the case g = 0 being similar but simpler), all the vertices corresponding to one and the same point of S, and the sides occuring in the order albla~lbrl 999 aoboa-~lb~; join the origin 0 of al (the extremity of b~-1) to the points P . of P, and to the points Q. of Q, by mutually disjoint simple arcs p. resp. q., interior (except for their common origin O) to the fundamental polygon. In the polygon, cut along the arcs q., an element ~o of E(Q, P) is the differential w = d r of a one-valued function r similarly, in the polygon, cut along the arcs p., an element ~0' of E(P, Q) is the differential c0'=d4b' of a one-valued function ~'; we m a y assume that, at the vertex O, q ~ = r T h e integral of Cd4b', or t h a t of --r along the contour of the canonical polygon is equal to

= fod,p

=

(1) ~-

~ X

-f4,'d, COp - ;~

60 d

bx

60 p d

ax

. /

Apply now Cauchy's theorem, either to the differential Cw'=r and to the polygon cut along the arcs q., or to the differential - r = - r 1 6 2 and to the polygon cut along the arcs p.. We get 1 - - I(~0, cop) -k E qY(Q.)'Rese. ~0 -t- E Resq, { [(~' -- 4d(Q.)lw } 27ri . .. = ~ Resp, (4~op) + ~ Res~o (0~o') where the Ro are all the poles of co or of ~o', other than the Pu and Q,, and therefore: I"(~o,r (2)

---- ~ R e s e .

{ [q~- ,(P.)]o0'} -- ~ Resq. { [q~'- q~P(Q.)]w}

+ Y'~ Rese, (q~op) o

1

= 2~ri I(o~, o/) + ~_~ 4d(Q,) Resq, o~ - ~ q~(Pu) Respu o/. Here j(oa, oJ) is an alternating bilinear form, defined for ooE.E(Q, P),

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to'CE(P, Q). Of the two expressions for it given by (2), the first one depends only upon the power-~eries expansions of o0, oa' at the points Pu, Q., Rp; in fact, r 1 6 2 is the function vanishing at P . with the differential to; cb'-cb'(@) is the function vanishing at Q. with the differential to'; and at a point R. we may, in calculating ResR. (r take for ~ any function with the differential co (this being meromorphic tlaere since ResRp to = 0), the residue of Cw' being independent of the choice of the additive constant in q~ since ResR0 to' is 0; and we have ResRp (qSto')=--ResR ~ (r Put j(to, to')=j~(~0'); then to--+j~ maps E(Q, P) into the space of linear functions on E(P, Q) ; the second expression forj(to, to') in (2) shows at once t h a t the kernel of this mapping consists of the differentials w=df of the meromorphic functions f on S which are 0 at the P , ; if F(P) is this kernel, then that same expression shows t h a t E(Q, P)/F(P), E(P, Q)/F(Q) are two vector-spaces of finite dimension equal to 2 g q - ( p - 1 ) + q- ( q - 1) + (where p, q are the numbers of points in P, Q respectively, and a + = max (a, 0)), and that j(to, to') establishes a duality between them. Now take to'EE(P, Q), and c C H t ( S - P , Q); let y be a relative cycle in S - P mod. Q belonging to the homology class c and disjoint from the poles of to'; it is easy to see that f.~to' depends only upon c and upon the class of to' mod. F(Q); therefore there is an tocEE(Q, .P) such t h a t f~0' =j(to~, to') for all to', and the class of toc mod. F(P) depends only upon c. Similarly, if c'EHa(S- Q, P), one will attach to it an element oa.'., of E(P, Q), well defined mod. F(Q). It is easy to determine, in terms of the canonical dissection, the structure of Ha(S-P, Q); this is generated by the ax, bx, by small circles % surrounding the P . and positively oriented, and b y the linear combinations ~--~.~m.q.of the arcs q., with integral coefficients m~ satisfying ~ . m . = 0; the only relation between these generators is ~--~..'l,.eo0; H I ( S - P , Q) is therefore a free abelian group of rank 2 g - k - ( p - 1 ) + - i - ( q - 1 ) +. Also the intersection-numbers of cycles in H i ( S - P , Q) with cycles in Ha(S-Q, P) are then obvious. On the other hand, if one uses the second expression in (2) for j(to, to'), one obtains the conditions which to=toe has to satisfy, for a given c, in terms of the periods of to and of the r ; proceeding similarly for to',, one finds t h a t j(to~, to~,) is equal to the intersection-number of c, c' (i.e. of two cycles belonging to these homology classes). While these are some of the main results of the author, he proceeds in an entirely different way. He first gives the algebraic definition of j(to, r and shows in a purely algebraic manner that this is a bilinear function on the spaces E(Q, .P)/F(P), E(P, Q)/F(Q) and establishes

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a duality between them. He next defines r ~0~,as above; and he calls j(r to~,) the intersection-number of c, c'! He then (without condescending to say so) goes on to show t h a t this intersection-number has the properties which characterize it from the topological point of view, viz., t h a t it is an integer, t h a t / ( 3 " , 3") = 0 if 3", 3"P are disjoint, and t h a t I(3", 3") = 1 if 3', 3" are arcs inside a small circle, with extremities on t h a t circle, the cyclic order of these extremities being suitably related to the orientation; the proof for the second one of these facts (pp. 158-161) is a singularly difficult and tortuous one, appeals to a theorem (attributed to Montel) on so-called "normal families," and also (without a n y reference) to the fact t h a t a continuous function of two complex variables, separately holomorphiC in each, is holomorphic in both. All this could easily have been shown by means of the canonical dissection, even if one did not w a n t merely to verify it a posteriori in the m a n n e r sketched above. T h e author now proceeds to the determination of the homology groups, which is far from easy and requires all the combined resources of topology and of algebraic function-theory. As this gives him the technical equivalent of the tools ordinarily provided b y the canonical polygon and integration along its contour, he can then prove Abel's theorem and Riem a n n ' s billnear inequalities; even at t h a t stage, he needs two pages to prove Stokes' formula for the c o m p l e m e n t of the union of finitely m a n y small circles on S, and two more pages, involving the use of differentials of the second kind, for the proof of the bilinear inequalities for periods of differentials of the first kind. On the other hand, a v e r y simple and direct proof is given for the fact t h a t the group of divisor-classes of degree 0 is isomorphic to the torus-group of real dimension 2g. We have not yet mentioned some illustrative sections in this and the earlier chapters, on fields of genus 0 and 1, on fields of elliptic functions, and one (Chapter IV, w on hyperelliptic fields. Except for the latter which is a kind of tour de force (hyperelliptic fields over an a r b i t r a r y field of constants had p r o b a b l y never been discussed before), these are e l e m e n t a r y and could well have been given in the form of exercises or series of exercises; and it is greatly to be regretted t h a t the author has not added m a n y more in t h a t form, out of his rich stock of knowledge on such subjects; he could thus have greatly enhanced the usefulness of the book at the cost of a v e r y m o d e r a t e increase in size. As it is, one will not even find in it a calculation of the genus of a field k(x, y) defined b y y2=P(x), where P(x) is a polynomial. T h e book is also without a n y bibliography beyond a small n u m b e r of references in the brief introduction, a few lines of which

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comprise all t h a t the reader is told a b o u t the history of the subject; the n a m e of R i e m a n n occurs only as a label for some theorems and in " R i e m a n n surface." Chapters I - V I are without a single reference; it is true t h a t they are almost entirely self-sufficient, but even at places where a reference could help the reader (as e.g. in the section on elliptic functions) none is given; of the four references in Chapter V I I , two (those to Montel and to Bourbaki) are irrelevant to the a u t h o r ' s main purposes. T h e r e is nothing to indicate to the reader which results should be considered important, which ones could be further extended. As to the style, it is t h a t of the modern algebraic or formalistic school ; the resources of the English v o c a b u l a r y and syntax could not be cut down a n y further. Definitions are either not motivated, or else the patronizing tone in which this is done indicates t h a t it is mere condescension to h u m a n weaknesses of which the author does not approve. At the same time, as one could expect of him, he achieves everywhere the u t m o s t precision; there is never one vague word to mislead the unwary, perplex the novice, or let loose the fancy of the imaginative. T h e reader is not to look forward to a conducted tour through a picturesque countryside; he is on a bus which runs to a schedule. W h y should he w a n t to look out of the window? Enough has been said to indicate that, in spite of some shortcomings which it was our d u t y to point out, this is a valuable and useful book, and also a timely one. While it is not as a t t r a c t i v e l y written as the classical paper of Dedekind and Weber, or as H. Weyl's Idee der Riemannschen Fldche, it covers far more ground t h a n the former, and, even in its final chapter, has little in c o m m o n with the latter. I t was highly desirable t h a t the principles of the theory of algebraic functions should be treated at least once in their full generality b y purely algebraic methods; this is w h a t the author has done as perhaps no one but he could do it, and for this he has a right to expect the gratitude of the m a t h e m a t i c a l c o m m u n i t y . His a t t i t u d e towards his subject has been professedly one-sided; b u t his work should be of value, not only to those who will always prefer the algebraic methods for their own sake, but also to those who wish to ascertain both their scope and their limitations. Indeed some conclusions already seem to emerge from it, and will now briefly be set forth. This branch of m a t h e m a t i c s , says the author (p. v), has an "algebraico-arithmetical" and a geometric aspect; it is surprising t h a t he should not even mention the function-theoretic method, which was t h a t of Riemann, and is still in m a n y ways the m o s t powerful one of all; alone it supplies the proof for R i e m a n n ' s existence theorems,

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and is therefore the only source of our present knowledge a b o u t the structure of the non-abelian extensions of a function-field; in higherdimensional problems, it leads directly to harmonic integrals and the theory of Kghler manifolds, which has achieved such striking successes in the last 20 years. Even now its advantages are such t h a t one who is chiefly interested in characteristic p will frequently begin by investigating the "classical" case and will do so by using functiontheory, topology and harmonic integrals. However, as the author points out, there are valid reasons for considering other fields of constants than the complex numbers; if the characteristic is 0, it is still possible, by "Lefschetz' principle," to apply to them m a n y of the results obtained in the classical case b y function-theory; but finite fields of constants are becoming increasingly important, both for their own sake and because of possible applications to analytic number-theory; in particular, the so-called "singular series" depend upon numbers of solutions of equations over finite fields. Therefore, if for no other reason, one must be able to deal at any rate with fairly general fields of constants; and while in substance the means for doing so must be of a purely algebraic nature, one has to choose here between the language and technique of algebraic geometry and another language, originating in the theory of fields of algebraic numbers, which takes the function-field as the primary object of its study. "Whichever method is adopted," says the author, "the main results to be established are the same"; his book is good evidence to the contrary. On the one hand, the algebraic method which he follows emphasizes the analogies with algebraic numbers; one of its main advantages, in fact, is that m a n y questions concerning algebraic numbers and functions of one variable can be so treated simultaneously, as has been shown by Artin and Whaples, and more recently by Hasse in his Zahlentheorie; in the book we are reviewing, the greater part of Chapters I, III and IV applies with little change to numberfields, and it is perhaps unfortunate t h a t this is not pointed out there. For the same reasons, class-field theory can best be treated, at least at present, from that point of view; it is true t h a t a t r e a t m e n t of class-field theory over function-fields of dimension 1 by the methods of algebraic geometry is greatly to be desired, for its own sake and as a preparation for the same theory in higher dimensions; but this would not, at least for some time to come, deprive the algebraic method of the advantages which it now seems to possess. It also appears that the algebraic method is as yet the only one to deal with certain phenomena connected with inseparability. T h e algebraic geometer, in order to study a function-field, must assume,

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in the case of dimension 1, t h a t it possesses a model without multiple points (a curve whose points are all absolutely simple in the sense of Zariski). But perhaps not much is lost by t h a t assumption. I t is always fulfilled in the case of a finite ground-field, since such fields are perfect; and the curves which arise from the application of the Picard method to varieties of higher dimension are without singular points, provided one starts from a "normal" variety. T h e fields which have no non-singular model m a y therefore, a t the present m o m e n t , be regarded as pathological beings; these are the fields whose genus decreases under a suitable extension of the ground-field, and they are necessarily inseparable. If the author had excluded them, he could have spared himself some of the complications inherent in C h a p t e r s IV, V and V I ; and the reader would not be left with the uncomfortable feeling of being told (sometimes without proof) t h a t certain unpleasant things can happen, but not how and when they m a y happen. Thus, b y imposing upon himself the task of working with an entirely unrestricted field of constants, the author, perhaps unintentionally, has overemphasized certain more or less pathological features connected with characteristic p, because he had to devise procedures which do not exclude t h e m ; on the other hand, far more interesting facts on characteristic p, such as W i t t ' s residue-theorem for "Witt's vectors" (Crelles J., 176 (1936), p. 140), are not included, perhaps because the author considers them as pertaining to class-field theory. T h e algebraic method begins to show its weakness when it comes to dealing with extensions of the field of constants. Here also a new language and new techniques had to be invented b y the author, chiefly in order to show the invariance of the most i m p o r t a n t properties of a function-field under such extensions; in his introduction, he acknowledges the considerable effort which this has cost him, and, strangely enough, finds no better justification for it t h a n a reference to a n o t a b l y unsuccessful paper of Deuring on correspondencetheory, where the latter rediscovered rather clumsily a few of Severi's more e l e m e n t a r y results on the same subject. Undoubtedly, w h a t e v e r method is adopted, there are some crucial proofs (e.g. t h a t of T h e o r e m 4, C h a p t e r V, p. 96) which cannot be avoided ; b u t most readers of this book will feel the need for a language b y which those properties and results which are invariant under an extension of the ground-field can be expressed and proved in a m a n n e r independent of t h a t field; this is w h a t algebra does not do, and w h a t algebraic g e o m e t r y does without a n y effort. All this would not be decisive; as long as the geometer is exploring curves and nothing else t h a n curves, the algebraist can keep pace with

[1951c]

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him ; he will sometimes be in front, and a t worst not far behind. W h a t is decisive is t h a t algebra stops short of higher-dimensional problems; and, even in the theory of curves, these cannot be avoided. T o begin with, there are times when curves have to be embedded into projective spaces; even the a u t h o r could not refrain at least once (at the end of C h a p t e r IV) from interpreting in this m a n n e r a s t a t e m e n t on the differentials of a non-hyperelliptic field. But the crucial test is supplied b y the theory of correspondences, which is the theory of the product of two curves, and b y t h a t of the jacobian v a r i e t y of a curve; there it would be impossible to take function-fields as the p r i m a r y object, since one has to deal with properties of surfaces and varieties which depend upon the use of a particular model, and are not inv a r i a n t under a r b i t r a r y birational transformations. I t is therefore no accident t h a t in the present book the group of divisor-classes of degree 0, which is nothing else t h a n the jacobian, is discussed only in the "classical case" and b y topological methods (v. T h e o r e m 16 of C h a p t e r V I I , p. 176, and its corollary) ; and it is no accident t h a t the algebraists who a t t a c k e d those problems b y their own methods failed to obtain a n y significant results. T h u s it appears t h a t the author has s o m e w h a t overstated his claims, and has been too partial to the method dearest to his algebraic heart. W h o would throw the first stone at him? I t is rather with relief t h a t one observes such signs of h u m a n frailty in this severely dehumanized book. And it would only remain for us to congratulate him on the service he has rendered to the m a t h e m a t i c a l public, if it were not necessary to devote some of our attention to typographical matters. T h e book is generally well printed, and fairly free from misprints, much more so indeed than most previous publications of the same author; here are a few which might e m b a r r a s s the reader: pp. 98-99, the references to T h e o r e m 5 are really to T h e o r e m 4; p. 111, 1. 6-7, instead of "for the elements . . . to all be" read "for all the e l e m e n t s . . . to be"; p. 121, 1. 6 from below, for "ramification index" read "differential e x p o n e n t " ; p. 123, 1. 14, for " T h e o r e m 5, V, w read " T h e o r e m 4, V, w p. 132, last line, for "such t h a t " read "such t h a t r (mod. a -2) and t h a t " ; p. 165, 1. 22, for 2-chain, read 2-cycle. I t is regrettable t h a t the running title at the top of each page does not include the indication of the C h a p t e r and w as this makes references unnecessarily hard to find. But the point upon which we wish to draw a t t e n t i o n is a far more serious one, and one which affects not merely this volume, but all modern m a t h e m a t i c a l printing in America. No t y p e s e t t e r would separate the word "and" into "a-" and

16

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BOOK REVIEWS

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"nd" at the end of a line. Yet on p. 169 of this book, H I ( S - P , Q) is broken into H I ( S - and P, Q) ; on p. 149-150, a similar formula is similarly broken between one page and the next; on p. 121, the two factors of a product occur, one on line 22 and the other on line 23; several dozens of such instances could easily be given. I t is difficult enough to follow such a text in detail without having c o n s t a n t l y to reconstruct in one's mind w h a t has been separated on paper; and, a p a r t from all aesthetical considerations, such practices, which in this country are fast becoming the rule rather than the exception, m a y soon m a k e m a n y of our m a t h e m a t i c a l texts intolerably hard to read. I t is high time t h a t a reaction should set in against the tendency to cram as much text as possible into each page at the lowest possible cost, regardless of the effect on the reader; this will require a coordinated effort on the p a r t of authors, editors and the printingpresses. T h e authors, who u n d o u b t e d l y bear some responsibility for the present situation, should be more mindful of such m a t t e r s in the preparation of their manuscripts; editors and editorial assistants should cooperate with t h e m to a greater extent than sometimes happens now. As to the typesetters, who are doing an extraordinarily good job of setting the most complicated formulas, they could v e r y easily be trained to avoid broken formulas, if their attention were drawn to it by the presses; they could well be trusted to use their j u d g m e n t in displaying some long formulas, even in the absence of an indication from the a u t h o r or editor; as to short formulas, all t h a t is mostly required is some a d j u s t m e n t in the spacing of words; this might sometimes take more time t h a n mechanically running along, but would still be far less expensive t h a n later corrections which m a y affect a whole p a r a g r a p h of type. Possibly, at least in the transitional period until typesetters acquire experience in such matters, the average cost of the printed page in m a t h e matical texts would increase slightly; possibly the n u m b e r of pages to be printed every year by m a t h e m a t i c a l journals would h a v e to be somewhat cut down. M a y b e the gain would be greater t h a n the loss. A. WEIL

[ 1952a] Sur les th6or5mes de de Rham L a ddmonstration actuellement la plus satisfaisante des cdl6bres thdor6mes de de R h a m est celle qni rdsulte de la th6orie de l'homologie de H. Cartan, qui les renferme, ainsi que le thdor~me de dualitd de Poincard, c o m m e cas particuliers. Mais cette thdorie o ' a fair l'objet que de publications partielles sous forme de notes de cours mimdographidesl). A son origine se t r o u v e n t d'ailleurs, d ' u n e p a r t u n mdmoire de Leray, et d ' a u t r e p a r t j u s t e m e n t une d6monstration des thdor6mes de de R h a m que je c o m m u n i q u a i ~ C a r t a n en 1947. A ddfaut d ' a u t r e utilit6, celle-ci p e u t encore servir d ' i n t r o d u c t i o n a u x m6thodes de C a r t a n ; et c'est a v a n t t o u t ~ ce titre que je la prdsente ici, avec des amdliorations d e n t je dois quelques-unes ~ G. de R h a m et s N. H a m i l t o n ; j ' y joins une d d m o n s t r a t i o n ( d a t a n t aussi de 1947) du m i t que t o u t espace poss6dant un r e c o u v r e m e n t d ' u n certain t y p e (dit ~4opologiquement simple>>) a m~me t y p e d ' h o m o t o p i e que le nerf de ce r e c o u v r e m e n t .

w 1. Construction d'un recouvrement simple Soit (X~)iE I une famille de parties d ' u n espace E , ~ ensemble d'indices I quelconque ; on dit, c o m m e on salt, que cette famille est localement finie si t o u t point de E a un voisinage qui ne rencontre q u ' u n n o m b r e fini des X i ; si E ess localement compact, il revient au m 6 m e de dire que route partie c o m p a c t e de E ne rencontre q u ' u n n o m b r e fini des X~. Nous conviendrons une fois p o u r routes, si (Xi)i~ 1 est une famille localement finie et si J c I , de poser Xj :i~jX~; l'ensemble N des parties non rides J de I telles que X j ne soit pas vide s'appelle le n e f f de la famille (X~) ; si J E N , J est finie. L ' o b j e t de notre ~tude sera une vari~t~ diff~rentiable V de dimension n, ((paracompacte>) c'est-~-dire d e n t t o u t e c o m p o s a n t e eonnexe est d~nombrable h l'infini ; il revient au rhyme de dire que V a d m e t un rec o u v r e m e n t localement fini p a r des ((cartes)~, c'est-s p a r des parties ouvertes munies chacune d ' u n isomorphisme diff~rentiable sur une p a t t i e a) Cours de H a r v a r d , 1948; S6minaire do I'E. N. S., P a r i s 1948--1949 et 1950--1951.

119 Reprinted by permission of the editors of Comm. Math. Heir.

17

18

[1952a] ouverte de R n. Le m o t sera toujours pris au sens > (ou (<de classe C~>>). Cela n'est pas v r a i m e n t une restriction si on tient compte du thdor6me de W h i t n e y d'apr~s lequel toute varidtd de classe C n, pour n >~ 1, a d m e t un homdomorphisme de classe C ~ sur une varidtd de classe C ~ ; d'ailleurs la mdthode qui va 6tre expos@ s'applique aussi aux varidtds de classe C ~ pour n ~> 2. Notre outil principal sera un recouvrement lI = (Ui)~EI localement f i n i de V par des ensembles ouverts U i relativement compacts, qui devra avoir de plus la propridt~ suivante : chaque ensemble non vide Uj : i~: Ui

poss~de une (> c'est-s une application diffdrentiable ~Vg de Uj X R dans Uj telle que ~g (x, t) = x chaque fois que x E Uj et que t / > 1, et que ~v: soit constante sur Uz x ] - - ~ , 0]. Un tel recouvrement, muni de la donnde des rdtractions ~z, sera dit diff6rentiablement simple. P o u r construire un tel recouw'ement, on peut, comme le fair de R h a m 2), se servir d ' u n ds ~, mais il est peut-6tre plus dldmentaire de procdder comme suit. P a r t o n s d ' u n recouvrement localement fini de V par des cartes ouvertes relativement compactes V~; s V~ sera donc attachd un isomorphisme diffdrentiable de Vi sur une partie ouverte de R n au m o y e n de t~~ . . . , t~~). On peut alors, pour chaque i, ddfinir des ouverts Wi, W~ et une fonction fi diffdrentiable sur V de mani~re qne les W~ forment encore un recouvrement de V, que l'on air W i c W ~ et W i c V i , et q u e f i a i t l a v a l e u r l s u r W i e t 0 e n d e h o r s d e W+i. Posons f~o : f ~ et ddsignons p a r f i j la fonction dgale ~ f.tff) dans Vi et s 0 en dehors de Vi; l'ensemble des fonctions fij pour 0 ~ ~" ~< n, et pour toutes les valeurs de i, ddtermine une application de V dans l'espace R (A), oh A est l'ensemble des couples (i, j) ; on salt q u ' o n ddsigne ainsi l'espace vectoriel des applications de A dans R qui prennent la valeur 0 p a r t o u t sauf en un hombre fini d'dl6ments de A . De plus, l'application (fis) de V dans tR(~) ddtermine sur route partie ouverte relativement compacte Z de V un isomorphisme diffdrentiable de Z sur une sous-varidtd d ' u n sous-espace vectoriel de dimension finie de R (~). On pourra donc simplifier le langage en identifiant V avec son image dans R (~). Sur R (a), nous mettrons une structure d'espace m6trique ()) au m o y e n de la distance d ( x , y) = [~. (xi~ --yi~)~]~; i,i

elle

~) CL G. de Rham, Complexes ~ automorphismes et hom6omorphie diff6rentiable, Ann. Gren. 2 (1950) p. 51. Ce dernier expos6, comme ma d6monstration de 1947, resto limit6 au eas compact; mais e'est de Rham qui m'a indiqu6 la possibilit6 d'6tendre l'une et l'autre m6thode aux vari6t6s non eompactes. 120

[1952a]

19

fair de t o u t sous-espace de dimension finie de R (~) un espace euclidien. D ' a p r ~ s ce qui precede, la distance de -Wi ~ V - - W if e s t ~ 1, puisque la coordonnde xi 0 a la valeur 1 sur le premier ensemble et 0 sur le second. P o u r t o u t x E V, ddsignons p a r T~ la vuridtd linduire t a n g e n t e ~ V en x, et p a r P~ la projection orthogonale de R (~) sur T~, consid~r~e c o m m e application lindaire de R (~) sur T~ ; et ddsignons p a r U(x, r) l'intersection de V uvec la boule ouverte de centre x et de r a y o n r ; si x E W~ e t r < 1, on a u r a U (x, r) c W'i ; donc U (x, r) est r e l a t i v e m e n t c o m p a c t p o u r v u que r < 1. Soit x E W~ ; soit E un espace vectoriel de dimension finie c o n t e n a n t - - I W~. E n p r e n a n t duns E des coordonndes orthogonales d'origine x, les n premiers vecteurs coordonn~s dtant choisis dans T~, on volt que x poss~de un voisinage ouvert U contenu duns W~ et a y a n t les propri~t~s suivantes : (a) quel que soit yE U, P~ induit sur U u n isomorphisme diffdrentiable (c'est-k-dire une application biunivoque, p a r t o u t de r a n g n) de U sur son image U~ = P~(U) duns Tu ; (b) quels que soien~ y, zl, z 2 duns U , on a d(zl, z2)r/2 en v e r t u de (b). ~r maintenant que, si x e K , 0 < r ~ r ( K ) / 4 , et y e U ( x , r ) , P~ induit sur U ( y , r ) nn i s o m o r p h i s m e diffdrentiable de U(y, r) sur une partie eonvexe de T~. C o m m e on a U(y, r) ~ U(x, 2r), le seul point s d d m o n t r e r est la eonvexitg de P~ [ U ( y , r)]. Or c'est l~ l'ensemble des points z' --~ P~(z) 121

20

[1952a] zE U ( x , r ( K ) ) et d(y, z)2 O pour i E J ; si J ~ ( i } , Z j se r~duit au s o m m e t e i de N, et S t j , q u ' o n 6crira S t i , est dire l'~toile ouverte de e~; p o u r J E N , on a S t j : ~ N j S t , . Si J a m 616ments, donc si ~ j est de dimension m - 1, le centre de gravitd (ou barycentre) de z~j s e r a l e p o i n t e j = ( x i ) , avec x i : l / m pour iEJ, x~=O pour ~'n'app a r t e n a n t pas s J . Si la fonction 2(t) est ddfinie c o m m e plus haut, (x, t)-->eg ~- 2 ( t ) ( x - - e j ) est une rdtraction de S t j ; les S t i f o r m e n t donc bien un r e c o u v r e m e n t simple de N . w 2. Les formes dii~rentielles P a r une forme diffgrentielle, on e n t e n d r a toujours une telle forme d o n t les coefficients, lorsqu'on e x p r i m e localement la forme au m o y e n de coordonn~es locales, soient des fonctions de classe C ~ de ces coordonn~es. Une forme co est dire ferm6e si d~o ----0 ; elle est dite homologue s 0, sur la vari~t~ oh elle est d6finie, s'il existe sur cette vari~t~ une f o r m e telle que oJ = d~. Soit U une partie ouverte d ' u n e varidt6 diffdrentiable V, munie d ' u n e rdtraction ~0 ; soit eo une f o r m e de degrd m sur U ; considdrons sur U • R la f o r m e co[~0(x, t)], image rdciproque de oJ p a r q. Si, au voisinage d ' u n point de U, x 1. . . . . x,, sont des coordonndes locales, on p o u r r a dcrire :

123

22

[1952a] r

(x, t)] = • f ( i ) ( x , t ) d x i ~ s (i)

Adx~,~q- ~ g(~)(x,t)dtAdxjl A . . . Adxj,~_, , (t)

oh A d6signe le produit extdrieur. Dans le m4me voisinage, considdrons la forme I (o de degrd m - - 1 dr par 1

I co = s (S g(i)(x, t) dr) dxj A . . . A dx m_, (i)

o

On v6rifie imm6diatement que cet op6rateur est compatible avec les changements de coordonn6es locales et peut donc 4tre consid6r6 eomme d6fini globalement dans U ; si m = 0 , on a I c o = 0 . Au m o y e n de l'expression locale de I , on v6rifie aussitSt que l'on a r = Ida) q- dla) si m ~ 0 ; si m ~ - 0 , o n a t o = I d a ) q-co(a) s i a e s t l a v a l e u r constante de q~(x,t) pour t H 0 . I1 s'ensuit que, si m ~ O , do~ = 0 entraine

(o = d l o~. Supposons m a i n t e n a n t donn6, une fois pour toutes, un recouvrement diff6rentiablement simple lI = (Ui)iE I de V; soit N le neff de R. Si H = (io, i l , . . . , i T) est une suite quetconque d'616ments (distincts ou non) de I , on d6signera par ] H I l'ensemble des i v distincts. P a r un codldment diffdrentiel de bidegrd (m, p), on entendra un syst4me /2 = (con) = (~oioi~...ip) de formes de degr6 m, respectivement attach6es aux suites H = ( i 0 i l . . . i ~ ) de p Jr 1 616ments de I telles que I H I E N , n U, v . Le coo n 6tant pour t o u t H une forme d6finie dans UI~ I = o 0 ) soit c o n t e n u dans U~ p o u r t o u t i E 1 . Cela pos~, soient JEN, i E I et J' = J u ( i ) ; si o9 e s t u n e f o r m e d~finie d a n s Uj,, on c o n v i e n d r a de ddsigner p a r fi o9 la f o r m e ddfinie dans Uj qni est 6gale 's fie9 d a n s Uj, et h 0 d a n s Uj fl C ( U j , ) ; il est i m m d d i a t en effet que c ' e s t bien l~ u n e f o r m e (~ coefficients diff~rentiables) d a n s Uj ; si J ' n ' a p . p a r t i e n t pas h N , c'est-h-dire si Uj, = o, cette d~finition e n t r a l n e que fie9 = 0. A v e c cette c o n v e n t i o n , si 12 = (~on) est u n co~16ment de bidegr6 ( m , p ) avec p > 0 , nous poserons a) K12 = ($io...i~_), avec ~i . . . . ~_1 =

XAog~0...~_~

,

kEI

oh les t e r m e s d u s e c o n d m e m b r e d o i v e n t ~tre e n t e n d u s c o m m e fl v i e n t d ' e t r e dit. D e m~me, si 12 = (o9~) est u n codl6ment de bidegr6 (m, 0), on ddsignera p a r K12 la f o r m e o) = z~ f~ o9~, oh on dolt e n t e n d r e p a r ~ez

f~ w~ la f o r m e ddfinie sur V, dgale h f~ o9~ dans U~ et ~ 0 en d e h o r s de s) $e dois l'op6rateur K h N. H a m i l t o n . Ma d~monstration primitive se sorvait, au lieu de K, du th6or~mo de prolongoment de W h i t n e y

125

24

[1952a] U k ; co est d o n c u n e f o r m e dgfinie s u r V. Si Y2 e s t d e b i d e g r d (m, p ) e t est fini, K t 9 e s t fini si p > 0, e t e s t u n e f o r m e s s u p p o r t c o m p a c t si p = 0 ; si Y2 e s t a l t e r n d e t p ~ 0, K~9 est a l t e r n 6 . On v6rifie i m m d d i a t e m e n t q u ' o n a z9 = K~Y2 + SKY2, d o n c q u e (5f2 = 0 e n t r a i n e $2 = 5KD, pour p~0; si w est u n e f o r m e , on a ~o = K & o , e t 6~o = 0 e n t r a l n e d o n c co = 0. D a n s ces c o n d i t i o n s , c o n s i d 6 r o n s t o u t e s les s u i t e s (~o, ~ 0 , ~r . . . . , D.,_a, ~ ) oh oJ est u n e f o r m e d e degr6 m > 0 s u r V, -Qh u n co616ment d e b i d e g r 6 (m - - h - - l , h) p o u r 0 ~ h ~m--1, e t ~ m l co61dment. d e b i d e g r 6 (0, m ) , s a t i s f a i s a n t a u x r e l a t i o n s

&o = dD o ;

~12h = dDh+ 1

(0 ~ h ~ m - - 2) ;

5Y2m_a = 3 .

(I)

S'fl en e s t ainsi, on a dSY2 h = 0 pour 0 ~h~m--1, dS =0 et (~S = 0, e t ~dco = 0 d ' o h do9 = K(~d~o = 0. D o n c ~o a p p a r t i e n t l ' e s p a c e v e c t o r i e l ~m (sur R) des f o r m e s f e r m d e s slur V ; D h a p p a r t i e n t l ' e s p a c e v e c t o r i e l ~m, h d e s c o d l 6 m e n t s d e b i d e g r 6 (m - - h - - 1, h) q u i s a t i s f o n t ~ d5~9 = 0 ; q u a n t ~ ~ , p u i s q u ' o n a d Z = 0, on p e u t , c o m m e on a v u , le c o n s i d 6 r e r c o m m e u n e c o c h a l n e d e N ; c o m m e ~ = 0, c ' e s t u n c o c y c l e ; d o n c ~" a p p a r t i e n t s l ' e s p a c e v e c t o r i e l des c o c y c l e s d e d i m e n s i o n m d e N (h coefficients r6els). S u p p o s o n s Y2h d o n n d d a n s ~ , a, e t h < m - - 1 ; a l o r s la r e l a t i o n 5D~ = dDh+~ est s a t i s f a i t e p o u r D~+~ = I ~tg~. S u p p o s o n s que ~ga s o i t d a n s l a s o m m e ~m, a d e s s o u s - e s p a c e s d e ~ , ~ r e s p e c t i v e m e n t d d t e r m i n d s p a r les c o n d i t i o n s dY2 = 0 et 5D = 0 ; on a u r a d o n c D ~ = X + Y, d X = 0 , ~Y =0; comme X est de b i d e g r 6 (m - - h - - l , h), e t q u ' o n a m--h--l>0, dX =0 entralne X =dIX; on a u r a d o n c ~t9~ = ( ~ d ( I X ) , &off d Z = O en p o s a n t Z=Da+~--MX; commeona Y2~+~=(~(IX)+Z, dZ=O, ~9~+aest d a n s . ~ , a+a. E x a c t e m e n t d e m ~ m e , on v o l t que, si D~+I e s t d o n n 6 d a n s ~ , ~+~, l a r e l a t i o n ~D~ = dY2h+~ e s t s a t i s f a i t e p a r D~ = K d Q h + l , p u i s que $2~+~ e ~m, ~+~ e n t r a l n e ~2~ e ~ , ~. i1 s ' e n s u i t que la r e l a t i o n (~Y2~ ~ dD;~+a d 6 t e r m i n e u n i s o m o r p h i s m e e n t r e les e s p a c e s v e c t o r i e l s

~m, hi'm, h e t ~m,hU-il~m,h-l-l* D e m~me, si t9 o est d o n n ~ d a n s ~m, o, o n s a t i s f e r a ~ ~ o = d ~ o en prenant ~ =KdDo; si 9 0 e s t d a n s ~ , 0 , on a u r a t9 o = X + Y, dX =0, 5Y =0, d'oh Y =5(KY), et 5o~ = d Y = ( ~ d ( K Y ) , d ' o h , e n p o s a n t ~ = K Y , ~(o~ - - d ~ ) ~ O, d o n c ~o = d ~ . R d c i p r o q u e m e n t , si ~o est d o n n 6 e d a n s ~m, on s a t i s f e r a h ~ o = dDo en p r e n a n t ~o = I ( ~ o ; si ~o = d ~ , o n a u r a (~d~ = d D o, d o n c , en p o s a n t X = ~ 2 o - - 5 ~ , ~9o = X + (~, d X = 0, d o n c t90 ~ ~ , o . En ddsignant par ~ l'espace v e c t o r i e l des f o r m e s d e degr6 m h o m o l o g u e s h 0 s u r V, o n v o i t d o n c que 126

[1952a]

25

la r e l a t i o n (~(o -----dr0o d6termine u n i s o m o r p h i s m e entre le ~ / ~ et ~ , 0 / ~ , 0 . Enfin, si t 2 m _ ~ = X + Y , d X = O , 6Y = 0 , on a S = S X , et X est u n e c o c h a i n e d e N , d o n c ~ e s t u n c o b o r d de N ; r @ i p r o q u e m e n t , si S est d o n n d et (~3 = 0, on satisfait &Qm_1 = ~ en p r e n a n t D~_~ = K ~ ; si ~ = ( ~ X , oh X est une c o c h a i n e c'est-~-dire ml co616ment satisfaisant ~ d X = 0, on a u r a D~_~ = X + Y, dX = 0 , bY =0. D o n c la relation (~Y2~_~ = ~ d d t e r m i n e u n i s o m o r p h i s m e entre ~ . . . . ~/~ . . . . ~ et le g r o u p e de c o h o m o logic Hm(N) de d i m e n s i o n m de N ~ coefficients rdels. E n ddfinitive,

(~) dtablit donc un isomorphisme entre le groupe de de Rham ~ / ~ de V, et le groupe H m (N) ; et cet isomorphisme est canoniquement ddtermind par la seule donnde du recouvrement simple lI. On v o i t de plus que, si on se d o n n e la f o r m e fermde ~o, on p e u t p r e n d r e D h = (I 6)h+~ ~o, ~ = (~(I (~)m~o ; r d e i p r o q u e m e n t , si on se d o n n e le cocycle -- ( ~ .... i~), on p o u r r a p r e n d r e ~2~ = K ( d K ) m - h - ~ , ~o = K ( d K ) m ~ , c'est-s : re(m-l)

~o =- ( - - 1)

2

v

$io...imf~mdf~oA...Adfi,~_~

io, i l . . . . . i m

P o u r m = 0, on s u b s t i t u e r a a u x relations (I) l ' u n i q u e r e l a t i o n (5o9 = ~ , d ' o h on d d d u i t t r i v i a l e m e n t les m4mes rdsultats. I1 n ' y a rien ~ c h a n g e r ~ ce qui prdcgde si on ddsire considdrer exclusiv e m e n t des codl6ments et cochaines alternds. I1 n ' y a rien ~ y c h a n g e r si, au lieu des formes, on ddsire eonsiddrer les () (ce s o n t les formes d o n t les coefficients, q u a n d on les e x p r i m e a u m o y e n de c o o r d o n ndes locales, s o n t des distributions au lieu d ' e t r e des fonetions diffdrentiables). Enfin, il n ' y a rie~ s y c h a n g e r n o n plus si on ddsiz'e considdrer e x c l u s i v e m e n t les codldments et cochalnes finis et les formes h s u p p o r t c o m p a c t ; en ce cas, bien e n t e n d u , on n ' a b o u t i t pas en gdn6ral a u x mSmes g r o u p e s que p r d c d d e m m e n t , mais on o b t i e n t u n i s o m o r p h i s m e e n t r e les g r o u p e s de de R h a m s s u p p o r t c o m p a c t et les g r o u p e s de cohomologie de N rclatifs a u x cochaines finies. Enfin, supposons q u ' o n se soit d o n n d d e u x f o r m e s fermdes ~o, co' de degrds respectifs m , r, et q u ' o n air form6 d e u x suites (w, D o . . . . , ~9m_1, S) ! f ! ~[ et (co, f20, . . . , ) satisfaisant ~ (I). On p e u t alors former, sans if S " ) satisfaisant nouvelle intdgration, u n e suite ((o", Q o , . . . , Din+r-l, (I) et c o m m e n g a n t p a r le p r o d u i t ext6rieur r = ~oA~o'. P o s o n s en effet t9 h = (~Oih0...ih), ~ - = - ( ~ i .... ira), et de m ~ m e p o u r ~9~, ~ ' ; on p o u r r a alors p r e n d r e : 127

26

[1952a]

f2hii

= (cob i0...ih A co')

H

(0 ~ 0,' 6t deg (b t) = 0 en t o u t a u t r e cas. Soit s u n simplexe singulier ddfini p a r u n e a p p l i c a t i o n diffdrentiable f dans V d ' u n voisinage W de ~y,m ; si o~ est une f o r m e de degrd m d a n s V, son i m a g e rdciproque o) [f(x)] p a r f est une f o r m e de degrd m d a n s W, d o n t l'intdgrale sur Z "* est p a r ddfinition l'intdgrale ~(o de o~ sur s ; $

cette ddfinition s ' d t e n d p a r lindarit6 a u x chalnes finies h coefficients r@ls, et m ~ m e ~ t o u t e s los chalnes h coefficients rdels si co est ~ s u p p o r t c o m p a c t . O n a la f o r m u l e de Stokes ~deo = j'r valable c h a q u e lois t

bt

que t e s t une chalne finie ou que co est s s u p p o r t c o m p a c t . A u m o y e n de ~r qui est n n e f o r m e bilindaire en t et ~o, les chalnes finies sont t

nfises en d u a h t 6 avec les formes, et les chalnes avec les f o r m e s ~ s u p p o r t c o m p a c t , ce qui p e r m e t de t r a n s p o s e r a u x chalnes, p a r dualit6, los o p d r a t i o n s et les rdsultats du w 2 ; mais nous allons en d o n n e r u n exposd i n @ p e n d a n t , de mani6re h ne p a s a v o i r ~ s u p p o s e r G = R . Soit d ' a b o r d U une p a r t i e o u v e r t e de V m u n i e d ' u n e r d t r a c t i o n diffdrentiable q ; soit p la v a l e u r c o n s t a n t e de q ( x , t) p o u r t ~< 0. On d6signera p a r s~ le simplexe ddgdndr6 [ f ; a a . . . a ] de d i m e n s i o n m, oh 9

Commentarii Mathematici ttelvettci

129

28

[1952a]

ou, ce qui revient au m~me, le simplexe d6fini par la restriction s Z m de l'applieation constante de R m+l sur p ; on a b s m = Sm_l si m est pair et > 0 , b s m = 0 s i m est impair ou 0. Consid6rons un simplexe singulier s = [ f ; a 0 . . . am] dans U , les au dtant des points o aul les points (au, 0) et (au, 1) d ' u n espace affine E ; ddsignons p a r au, de E • R . P a r ddfinition, f est une application diff6rentiable dans U d ' u n voisinage du plus petit ensemble convexe K contenant les au; alors, si on pose f ' (x, t) = ~ [ f ( x ) , t], f ' est une application diff6rentiable dans U d ' u n voisinage de K x R . Posons, dans ces conditions:

f(a) = p,

m

Psi-

~ ( - - 1 ) ~ [ f ' ; a o .0. . a ~ a ~0 . . .1

a ~ ] + sin+ I ,

et dtendons cet opdrateur par lin~arit6 a u x chalnes finies dans U. U n calcul facile donne b P s ~- P b s = s pour m > 0 et b P s ~- P b s = s - - S o p o u r m = 0, donc, pour t o u t e chaine finie de dimension m , t = b P t ~P b t si m > 0 et t = b P t ~-(bot)So s i m ~ 0 . Donc bt = 0 entra~ne t = b P t si m > O , et bo t = 0 entra~ne t = b P t si m = 0 . P a r un 1Lsimplexe, on e n t e n d r a un simplexe singulier contenu dans Fun au moins des ensembles U~ du r e c o u v r e m e n t ~I; p a r une 1Lchalne, on e n t e n d r a une chalne dont t o u s l e s simplexes sont des 1Lsimplexes. L ' a p p l i c a t i o n de notre mdtbode exige q u ' o n se restreigne aux 1Lchalnes ; d'aprSs un thdor~me de S. Eilenberg 5), cela ne change rien a u x groupes d'homologie ; rappelons les points principaux de sa d6monstration. Soit 8 = [ f ; a 0 . . . a m ] un simplexe singulier. Posons I~ = {0, 1 , . . . , # } pour 0 ~/~ ~m; et, si I = { / ~ , . . . , / ~ k } est une partie quelconque k

de Ira, posons a x = Z (l/k)a~h. Alors on appelle subdivision barycenh=l

trique de s la cha~ne finie a s =- v

e,~ [ f ; a,(x0) . . . a~(im)] ,

7r

oh la s o m m e est 6tendue ~ toutes les p e r m u t a t i o n s ~ de Ira, et off e, 4- 1 suivant que z e s t paire ou impaire ; on ~tend l'op6rateur a u x chaines p a r lin6arit~; on vdrifie q u ' o n a ba = a b . D ' a u t r e p a r t , Eilenberg (loc. cir., note 5, p. 429) dd~nit un autre opdrateur Q, analogue mais d o n t l'expression explicite serait plus compliqu6e, tel que b ~ ~- ~b = a - - I ; ~s est une chaine finie de dimension m ~- 1, s o m m e de termes de la f o r m e -4- I f ; b 0 . . . bm+~] , oh chacun des b~ est Fun des az. Cela pos~, si 8 est un simplexe singulier, on p e u t t r o u v e r un entier v assez g r a n d p o u r que a ' s soit une lI-chaine; soit v(8) le plus p c t i t entier a y a n t cette propri6td ; soit ~ l'op6rateur ddfini sur les simplexes singuliers p a r 6) S. Eilenberg, Singular homology theory, Ann. Math. 45 (1944) p. 407. 130

[1952a]

29 zs =~(1

+a+...+a

v (~) 1) s ,

et 6 t e n d u a u x cha~nes p a r lindaritd. Alors, si on pose e o m m e plus h a u t bs = Z (--1)gsix, on vdrifie i m m 6 d i u t e m e n t q u ' o n a IX

v (s)--I

( b v 4 - zb) s = ( a v ( ~ ) - - l ) s - - ~ .

(--1)~ ix

x~

eaJsix ,

j = v (six)

ee qui m o n t r e que (1 4- b z 4- ~ b)s est une lI-ehaine finie, de s u p p o r t c o n t e n u duns celui de s. Done, si t est une chalne, t = (1 4- bz 4- zb)t est u n e ll-chalne, finie s i t est finie. S i t est u n cycle, on a t = t 4- bvt, d o n e t e s t u n cycle et t e s t h o m o l o g u e s t. D e plus, la f o r m u l e ci-dessus m o n t r e que (bz 4- zb)s = 0 si v(s) = 0, e'est-s s i s est u n lI-simplexe, d o n e t = t si t est une lI-chalne. S u p p o s o n s q u ' u n e l I - c h a l n e t' soit l e b o r d d ' u n e e h a i n e t ; on a u r a t' = b t , et b 3 - = b t 4 - b z b t = t ' = t', d o n e t' est aussi le b o r d d ' u n e ~I-ehaine. I1 s ' e n s u i t bien que la restriction a u x ~t-ehalnes ne c h a n g e rien a u x g r o u p e s d ' h o m o l o g i e ; ddsormais nous ne eonsiddrerons que eelles-ls et p o u r abrdger n o u s dirons ~chalne,, au lieu de 0 d u n s Uj, b t = 0 e n t r a i n e t = bPt. Alors, si T = (tz) est u n dldment de bidegr6 (m, p), on p o s e r a P T -= ( P l z l t B ) ; e ' e s t u n 61dment de bidegrd ( m 4 - 1 , p ) ; si m > 0 , bT = 0 entralne T =bPT; s i m = 0 , boT = 0 e n t r a l n e T = b P T ; engdndral, o n a T = b P T 4 - P b T si m > 0 ; P T e s t f i n i s i T e s t fini, alternd si T e s t alternd. D ' a u t r e part, si T = (ti .... ~v) est u n 61dment de bidcgr6 (m, p), et si p > 0 , nous ddfinirons n n dldment OT = (ui0...r de bidegr6 (m, p -- 1) au m o y e n de la formule 131

30

[1952a]

u~.... i~-1 = ~: ( - - 1)~t~0...i~_lki~...ip_l , oh la s o m m a t i o n doit ~tre dtendue a u x valeurs d e / ~ , k pour lesquelles I i0 999 i~_lki~ 999i~_1 I E N ; ces valeurs sont en h o m b r e fini, et t o u s l e s termes du second m e m b r e sont des chaines finies dans U I i .... ~p-11' donc ces formules ddfinissent bien un dldment a T , qui est fini si T est fini. De m~me, si T = (ti) est un dl~ment de bidegrd (m, 0), on posera OT = Z t ~ ; aTestalorsunechMne, finiesiTestfini. O n a a 2 = 0 , et k

a est p e r m u t a b l e avec b. D ' a u t r e part, on p e u t aussi, dans la formule qui ddfin]t OT, interprdter T c o m m e une chalne de N , les t~ dtant alors des dldments de G; cette formule, oh la s o m m a t i o n est dtendue a u x m~mes valeurs de #, k que t o u t ~ l'heure, ddfinit alors OT c o m m e chaine de N ; les groupes d'homologie de N sont ceux qui sont ddfinis au m o y e n des chalnes de N e t de l'op~rateur 0, ou encore au m o y e n des chaines finies de N e t de a. Dans ces conditions, a est p e r m u t a b l e avec b0. On v a d~finir un opdrateur L tel que a T = 0 entralne T ~ a L T . P o u r cela, convenons de choisir une lois p o u r routes, pour t o u t lI-simplexe s, Fun des U~ dans lesquels fl est contenu ; soit UI(,) cet ensemble. Soit T = (tH) un dldment de bidegr6 (re, p ) ; soit t~ = Z c ~ s q l'expression rdduite de t~. Si H = ( i 0 . . . i~), on posera i H = ( i i o . . . i ~ ) . Alors on ddfinira un ~l~ment L T = (vR,) de bidegrd ( m , p + 1) en p o s a n t v~H= ~, c~sq chaque lois que [ i H ] E N ; celaveut direque /(ae)=i

la s o m m e est ~tendue ~ ~outes les valeurs de ~ telles que f(s~) = i. Puisque t~ est une s o m m e finie, vi~ en est une aussi; et chaque simplexe sq figurant dans vi~ est eontenu dans U I~1 paree qu'il figure dans t~, et dans U~ parce que i = f ( % ) , donc aussi dans U l ~ l ; L T est donc bien un ~l~ment, fini si T e s t fini. De m~me, s i t = ~ cqsq est l'expression rdduite d ' u n e 1Lchalne de dimension m , on d~finit, a u m o y e n de v~ = ~ , cqsq, un ~l~ment L t ---- (vi) de bidegr~ (m, 0), fini /(#~)=i s i t est finie. On a T = a L T + L O T si T e s t un dldment, et t = a L t s i t est une chalne. Si donc T e s t un dl~ment tel que 0 T = 0 , on a T = aLT. I1 n ' e s t pas vrai que L T soit alternd chaque lois que T est alternd. Si on v e u t se servir exelusivement d%ldment altern~s, fl f a u t substituer 0, L l e s op~rateurs 0~, L ' qui, avee les m~mes notations que cLdessus, sont d~finis p a r les formules a ' T = ( ~ : t~0...~_~) , k

132

~952a]

31

oh la, somma,tion est dtendue a,ux k tels que I k i o . . 9i~_~ ] e N , et ~+1

L'T---- ( ~ t~=0

~

( - - 1 ) t ' e ~ 0 . . . ~ _ ~ , + , . . ~ + se) .

/(so)=it~

On v~rifie facflement qu'ils poss~dent des proprigtgs sembla,bles ~ celles de 0 et L lorsqu'on les a,pplique ~ des ~l~ments a,ltern~s et qu'fls transf o r m e n t cenx-ci en ~lSments a,ltern~s. Consid~rons ma,intenant routes les suites (t, T o , . . . , T,~,Z), oh t est une cha,lne de dimension m ~ 0 de V, T~ un ~l~ment de bidegr~ (m--h,h) pour 0 ~ h ~ m , e t Z u n e cha,ine de dimension m de AT, sa,tisfa,isa,nt a u x rela,tions

t = 0 T o;

bT~ =~Ta+~

(0 ~ h

~m--l);

b0 T ~ = Z

.

(II)

S ' i l e n e s t a , i n s i , ona, bOT h ~ 0 (O ~ h ~ m - - 1 ) , bo~T,~ = 0 , bt = 0 et 0Z = 0. Donc t a p p a r t i e n t a,u groupe ffm des cycles singuliers diff,rentia,bles ~ coefficients dans G sur V, et Z au groupe des cycles de N coefficients da,ns G ; T~ appa,rtient au groupe ffm,h des ~l~ments de bidegr5 ( m - - h , h ) q u i s a , t i s f o n t s b O T - ~ O pour h < m e t ~ b 0 a T - = 0 pour h = m. Soit ~Bm le groupe des bords da,ns V, e'est-s le groupe des 61~ments de ff~ de la, forme b t ' ; s o i t ~ m , h , pour 0 ~ h ~ m , le groupe des 61~ments de ff~, h de la forme b X ~ a Y, oh X , Y sont des 515ments de bidegrds respectifs (m - - h ~- 1, h) et (m - - h, h + 1). On satisfera, h la, rela,tion b T h OTh+1 en prena,nt T h = PaTh+ 1 si Th+l est donn6 dans ~ , h + l , et Th+ 1 = L b T h si T h e s t donn6 dans ~ , a ; on satisfera h t = O T o en prena,nt T O = L t si t est donn6 dans ~m; enfin il est clair q u ' o n p e u t former T ~ satisfa,isant s boTm ~ Z si Z e s t donnd. Si T h E ~ m , h , done si T h = b X - F - a Y , on a,ura,, en posa,nt U = Th+ ~ - b Y , a U = 0 , d'oh U = O ( L U ) et Th+ ~ = b Y + O(LU) e ~ m , h + l ; de m6me, si Th+ ~ = b Y + 0 V , on aura" b W = 0 e n p o s a n t W = T~ - - 0 Y , d'oh W = b P W puisque W est de bidegr6 ( m - - h , h) et que m - - h > 0 ; on a donc Ta = b ( P W) + 0 Y e ~ , a . Done la rela,tion b T~ =0T~+~ d6termine un isomorphisme entre ~m Y ~ m . ~ et ~ , ~ + ~ / ~ m , ~+~. De m6me, si t = 0 T o et T o = b X + 0 Y , on a, t = b ( O X ) e ~ , , ; si t=bt', etqu'onpose U=To--b(Lt'), ona, 0 U = 0 , donc U ~ - 0 L U , et T o = b ( L t ' ) - F O ( L U ) e ~ , o. Si b o T m = Z et T m - - - - b X + 0 Y , on a, Z = 0 ( b o Y ) , d o n e Z e s t h o m o l o g u e h 0 ; et, si Z = O Z ' et b o T ' = Z ' , on a,ura, en posa'nt X = T m - - O T ' , boX = 0 , donc X = b P X et T m ~ b ( P X ) + OT' e ~ m , m. E n d6finitive, on volt que les relations (II) dtablissent un isomorphisme entre le groupe d'homologie singuli~re diffd=

133

32

[1952a] rentiable ~ / ~ dimension m e t

de V e t le ffroupe d'homologie des chaines de N , pour la le groupe de coefficients G ; et cet isomorphisme est cano-

niquement d~termin~ par la donn~e du recouvrement simple lI. P o u r m = 0 , on p a r t i r a des relations t = ~ T o , boTo ~ Z ,

oh T 0 e s t un dldment de bidegrd (0, 0), et on arrive au m~me rdsultat p a r des raisonnements analogues mais plus simples. I1 n ' y a rien ~ changer ~ ce qui prdc~de si on v e u t considdrer exclusivem e n t les dldments et chalnes finis; on obtient ainsi un isomorphisme entre ]es groupes d'homologie de V e t de N obtenus au m o y e n de cha~nes finies. I1 n ' y a rien ~ y changer si on v e u t se servir de chaines de classe C k, c'est-A-dire d o n t les simplexes sont ddfinis par des applications k lois c o n t i n u m e n t diffdrentiables, k 6tant un entier quelconque ; en ce cas, il suffit que les rdtractions ~g soient elles-m~mes de classe Ck; p o u r k = 0, on voit q u ' o n obtient les m~mes rdsultats au moy~n de chaines singuli~res continues, les ~j d t a n t alors seulement assujetties /~ ~tre continues ; ce r~sultat s'applique en particulier au r e c o u v r e m e n t simple d ' u n complexe simplicial localement fini par les dtoiles ouvertes des s o m m e t s (voir w 1), et contient donc une ddmonstration de l'invariance topologique des groupes d'homologie combinatoires d ' u n tel complexe, qui d'ailleurs ne diff~re qu'en apparence de la ddmonstration classique. I1 n ' y a rien ~ changer non plus s ce qui prdc~de si l'on veut se servir exclusivement d'~ldments alternds, et de chalnes altern@s de N , sauf qu'il faut substituer a', L ' ~ a, L . Si on p r e n d G - - R , les op6rateurs q u ' o n a d6fini sur les ~16ments singuliers sont en dualit6 avee eeux q u ' o n a d6fini sur les co616ments diff6rentiels. Soient en~cffet ~9 = (~oB) et T = (tH) un eo616ment diff~rentiel et un 616ment singulier, tous deux de bidegr6 (m, p), d o n t Fun soit fini; on posera alors H

tH

et, s'ils sont tous deux altern~s : (T, ~9)' - - (p ~_ 1) !

,

~,

oh ~ ' indique q u ' o n prend une lois seulement chaque combinaison i0, 99 9 i~ de p ~- 1 414ments de I , rangds dans un ordre quelconque ; c'est de (T, ~2)' qu'il f a u t se servir dans la th4orie alternde. L a formule de Stokes donne (bT, D) = (T, d g ) ; et on v4rifie facilement q u ' o n a (OT, [2) = (T, ~f2), et de m6me (O'T, f2)' = (T, ~f2)' si T, f2 sont altern4s ; de m~me, si ~o est une forme de degr4 m sur V et T un 414ment de 134

[1952a]

33

bidegrd (m, 0), et que co soit s s u p p o r t c o m p a c t ou T fini, on a (T, (~) ~ co. Enfin, si T e s t un dldment de bidegr6 (0, p), et ~ un codl~ment ~T

de bidegrd (0, p) satisfaisant ~ d Z ~ 0 ou a u t r e m e n t dit une cocha~ne d e N , e t s i T o u ~ e s t fini, o n a ( T , ~ ) ~ ( b o T , S ) , ok d a n s l e s e c o n d m e m b r e figure le produit scalaire des chaines et cochalnes de N ddfini par ( Z , ~ ) = ~ Z ~ H pour Z = ( z ~ ) , ~ = ( ~ H ) ; ces f o r m u l e s s o n t H

modifier d ' u n e mani~re ~viden~e dans la th~orie alternde. Considdrons alors deux suites (eg,Y20,... , ~ - 1 , ~) et (t, T o. . . . . T~, Z), satisfaisant r e s p e c t i v e m e n t a u x relations (I) du w 2 et a u x relations (II) ci-dessus ; supposons co ~ support c o m p a c t et les ~9~ et ~ finis, ou t, les Ta et Z finis. Au m o y e n des formules ci-dessus, on obtient i m m d d i a t e m e n t : co = (To, d~0) . . . . .

(T~_ 1, dOra_l) = (Tin, (~Om_l) = (Z, .~) .

t

I1 s'ensuit que les groupes de de R h a m et les groupes d'homologie singuli~re ~ coefficients rdels de V ont entre eux les m~mes relations de dualit~ que les groupes de cohomologie et d'homologie de N . E n particulier, il cxiste toujours une forme fermde co sur V telle que y co soit une fonction t

lin~aire a r b i t r a i r e m e n t donnde sur le groupe d'homologie singufi~re finie de V, ou a u t r e m e n t dit soit dgale ~ une Ionction lindaire L(t) donn~e sur l'espace vectoriel des cycles finis de V, nulle sur les bords de chalnes finies. D ' a u t r e p a r t , si une forme ferrule co ~ support c o m p a c t sur V est tclle que ~co = 0 pour tout cycle t, f i n i o u non, de V, elle est d e l a t f o r m e co ~ d r , oh V e s t ~ s u p p o r t c o m p a c t ; de m~me, si une forme fermde e9 est telle que ~ co = 0 p o u r t o u t cycle fini t, elle est de la forme co ~ d~. E n effet, d'apr~s ce qui prdc~de, il suffit, pour obtenir ces rdsultats, de v~rifier les r~sultats analogues p o u r N , ce qui est immddiat. Les espaces vectoriels dont il s'agit ici sont en g~ndral de dimension infinie si V n ' e s t pas c o m p a c t e ; on ne p e u t donc espdrer dtablir entre eux de relations de dualit~ t o u t h fair satisfaisantes ~ moins d ' y introduire des topologies convenables ; c'est 1s un terrain sur lequel nous ne nous engagerons pas. E n revanche, si V e s t compacte, le r e c o u v r e m e n t lI est fini; ce qui prdc~de m o n t r e donc que t o u s l e s groupes d'homologie de V sont alors de t y p e fini, et s ' a n n u l e n t au-dessus d ' u n e certaine dimension ; sur R , en particnlier, tous ces groupes sont des espaces vectoriels de dimension finie. On conclut alors de ce qui precede que la fonc135

34

[1952a] tion bilingaire

S co m e t en dualit4 le g r o u p e de de R h a m de degr4 m et t

le g r o u p e d ' h o m o l o g i e diffdrentiable de d i m e n s i o n m s coefficients rdels. On p e u t compldter ces rdsultats au m o y e n des r e m a r q u e s suivantes, que nous b o r n e r o n s au cas c o m p a c t , oh r o u t e chaine est finie. I1 est imm d d i a t que r o u t e chalne t s coefficients rdels p e u t se m e t t r e sous la f o r m e t = z~ ~t~, oh les t~ sont des chalnes s coefficients entiers et les ~ s o n t des h o m b r e s rdels lindairement i n d d p e n d a n t s sur le corps Q des r a t i o n nels ; alors bt = 0 entralne b t~ = 0 p o u r t o u t i, d o n e t o u t cycle rdel est c o m b i n a i s o n lin4aire de cycles entiers ; et, si u n cycle entier t' est le b o r d bt d ' u n e chaine rdelle t, alors, en i n e r r a n t t sous la f o r m e ci-dessus, on v o i t que l ' u n des $i, p a r e x e m p l e ~1, doit 6tre r a t i o n n e l et q u ' a l o r s on a t' = b (~1 q ) , done q u ' u n multiple entier de t' est le b o r d d ' u n cycle entier. Le g r o u p e d ' h o m o l o g i e entibre de d i m e n s i o n m d t a n t de t y p e fini, il est s o m m e directe d ' u n g r o u p e fini et d ' u n g r o u p e abdlien libre engendrd p a r des classes d ' h o m o l o g i e enti6re en n o m b r e fini ; soient tl . . . . , G des cycles enticrs a p p u r t e n a n t r e s p e c t i v e m e n t s ces classes; d'apr~s ce qui prdc6de, les classes d ' h o m o l o g i e rdelle de t x , . . . , tr f o r m e n t alors une base du g r o u p e d ' h o m o l o g i e rdelle de dimension m consid4rd c o m m e espace vectoriel sur R ; et on p e u t identifier les formes lindaires sur ce dernier g r o u p e a v c c les h o m o m o r p h i s m e s dans R du g r o u p e d ' h o m o l o g i e enti~re, u n e telle f o r m e ou u n tel h o m o m o r p h i s m e d t a n t c o m p l 6 t e m e n t ddtermin4 p a r ses valeurs sur les classes des cycles ti. P a r u n e pdriode d ' u n e f o r m e co, on e n t e n d son intdgrale S co s u r u n t cycle entier t ; p o u r u n choix ddtermin4 des cycles t ~ , . . . , t r, on appelle s o u v e n t ~> de co les intdgrales de co sur les t~. On v o i t d o n e qu'il revient au m~me de se d o n n e r , soit la f o r m e lindaire co sur le g r o u p e d ' h o m o l o g i e rdelle de V, soit l ' h o m o m o r p h i s m e ~ co t

t

d u g r o u p e d ' h o m o l o g i e enti6re de V duns R , soit les pdriodes f o n d a m e n tales de co. On a done r e t r o u v 4 les ,~thdorbmes de de Rham>~ sous leur f o r m e classique :

Sur une varidtd difffrentiable compacte V, il existe des formes fermdes dont les pdriodes fondamentales so~t arbitrairement donndes; route forme fermde dont les pdriodes fondamentales sont nulles est homologue de 0 sur V. Q u a n t au ~(troisi~me t h d o r 6 m e de de Rham>), une p a r t i e en est cont e n u e dans le r d s u l t a t de la fin du w 2, d ' a p r 4 s lequel le (( c u p - p r o d u c t >) des cocycles de N c o r r e s p o n d au p r o d u i t extdrieur des formes sur V. P o u r passer de l~ s l'dnoncd classique du m 4 m e thdor5me, il f a u t se servir de la dualitd de P o i n c a r d dtablie p a r le n o m b r e d ' i n t e r s e c t i o n 136

[1952a]

35

entre les cycles rdels de dimensions m e t n - - m , ou encore (ce qui au fond revient a u mdme) passer au produit de la vari6t6 p a r elle-m~me, puis ~ la diagonalc duns ce produit. J e n'insisterai pus sur ees questions ddjA classiques ; mais il ne sera pus superftu de faire a p p a r a i t r e une consdquence i m p o r t a n t e de nos rdsultats, qui d ' h a b i t u d e se ddduit du troisi4me thdor4me de de l~ham. Bornons-nous toujours au cas c o m p a c t ; considdrons une forme ~ sur V dont toutes les pdriodes sont enti6res ; soit • un cocyele de N eorrespondant ~ (o, cocycle qui est bien ddtermind ~ un cobord arbitraire prds. Alors (Z, ~) est entier p o u r t o u t cycle entier Z . Mais le groupe des cycles entiers de N e s t le sous-groupe du groupe des ehaines enti4res ddtermin6 p a r les conditions 0Z = 0, done t o u t e ehalne entidre dont un multiple est un cycle est elle-m~me un cycle; d'aprds la thdoric des diviseurs 616mentaires, le groupe des chaines entidres est done s o m m e directe du groupe des cycles entiers et d ' u n a u t r e groupe, de sorte q u ' o n p e u t 6tendre au groupe des chaines t o u t h o m o m o r p h i s m e donn6 sur le groupe des cycles. C o m m e t o u t homom o r p h i s m e du groupe des ehalnes enti4res duns le groupe additif des entiers p e u t s'dcrire sous la forme, Z ~ (Z, ~0), oil Z0 est une coehalne enti4re, on voit qu'il existe une coehaine enti4re Bo telle que (Z, "~0) = (Z, s pour t o u t cycle entier Z , done aussi p o u r t o u t cycle rdel Z . I1 s'ensuit que ~ 0 - ~" est le cobord d ' u n e cochMne rdelle, done que H0 est, aussi bien que ~ , un cocycle correspondant ~ (o. P a r suite, pour qu'une forme ~o corresponde ~ un cocycle ~ 6 coefficients entiers,-il faut et il suffit que routes ses pdriodes soient des entiers. De 1s et du r6sultat final du w 2, on conclut que, si w et co' sont h pdriodes enti~res, il en est de m~me de leur produit extdrieur eo/~ ~o'. Bien entendu, on p e u t obtenir aussi ce m6me r6sultat en p a s s a n t au produit de V p a r elle-m~me et en se s e r v a n t du th6or4me de Kiinneth.

w 4. La dualit6 de Poincar6 T o u t ce que nous avons fait jusqu'ici repose en rdalit6 sur une seule propridt6 du r e e o u v r e m e n t l I : c'est que les Uj sont h o m o l o g i q u e m e n t triviaux, c'est-s ont l'homologie d ' u n espaee rdduit s un point. Nous nous s o m m e s servis, il est vrai, des rdtractions ~vj, mais seulement pour obtenir un exposd s la fois plus dldmentaire et plus 616gant grace la possibilit6 de ddfinir explicitement les opdrateurs I e t P. L'expos6 ci-dessus renferme done, du moins p o u r l'homologie singuli4re, une dgm o n s t r a t i o n du thdor4me de L e r a y d'apr4s lequel, si un r e c o u v r e m e n t lI d ' u n espace X est tel que les Us soient homologiquement triviaux, l'homologie de X est la m~me que eelle du nerf N de ll. 137

36

[1952a] E n revanche, puisque t o u t complexe simplicial a d m e t un recouvrem e n t simple, il est dvident que l'existenee d ' u n tel r e c o u v r e m e n t n'entraine pus le thdor4me de dualitd de Poincar6. P o u r obtenir ee thdorgme sur une varidtd au m o y e n du r e e o u v r e m e n t lI, il faut m e t t r e en oeuvre une propridtd des Uj qui n ' e s t pus encore intervenue, & savoir que leur homologie modulo leur fronti6re est triviale duns toutes les dimensions sauf la dimension n de V. Ce n ' e s t pas 1~ une propridt6 0 sur Fiet f/ = 0 en dehors de U~, p o u r t o u t i E I . Soit x = (xi) un point de N ; choisissons un i tel clue x~ = maxjE I (x~) ; alors x est dans un simplexe de la subdivision b a r y c e n t r i q u e de N a y a n t un s o m m e t en ei, et on a donc g (x) E F i, d'ofl f~ (g (x)) > 0. P o u r t o u t i E I et t o u t x = (xi) E IV, posons qJi(x) = min [xi, f / ( g ( x ) ) ] ; les ~v~ sont des fonctions continues >~ 0 sur N , et on a ~i (x) = 0 s i x i = O, e'est-s si x n ' a p p a r t i e n t p a s ~ St~ ; de plus, d ' a p r & ce q u ' o n vient de montrer, il y a p o u r t o u t x E N un i tel que ~vs(x)>0. I1 s'ensuit que ~ = Z ~vi est une fonction continue p a r t o u t > 0 sur N , et p a r suite que les h i = Wi/~ f o r m e n t sur N une partition de l'unit6 subordonnde au r e c o u v r e m e n t (Sti) ; si doric on pose h (x) = (h t (x)), h est une application de N dans N . Si, p o u r x E iV, J e s t l'ensemble des i E I tels que h i(x) > 0, on aura, p o u r t o u t iEJ, x ~ > 0 et f~(g(x))>O; alors h(x) est dans Z z, et x et f(g(x)) sont tous deux dans Stj, de sorte que les segments de droite qui joignent h (x) A x d'une p a r t et ~ f (g (x)) d ' a u t r e p a r t sont contenus dans N ; e o m m e t o u t h l'heure on conclut de 1s que h e s t h o m o t o p e l'application identique d'une part, et s f o g d ' a u t r e part. Enfin, soit p E E , et soit J l'ensemble des i E I tels que f~ (p) > 0 ; on a u r a donc p E U ~ pour t o u t i E J ; on a u r a f(p) E~gj, done f(p) a p p a r t i e n d r a h u n des simplexes de la subdivision b a r y c e n t r i q u e de ~ j ; mais ce sont l~, avec les notations employ6es plus haut, les simplexes ~Y"(J0 . . . . , J ~ ) avec J ~ c J ; si alors on p r e n d i E J o , on a u r a g (f(p)) EFt ; d o n c p et g (f(p)) sont tous deux dans V'i. Posons X~ = U~ ; soit N ' le nerf de la famille (Xi) ; on a u r a N ' c N . Si de plus on suppose m a i n t e n a n t que les Uj ont la propridtd &extension, c'est-h-dire que est t o p o l o g i q u e m e n t simple, on voit que les familles (Xi)ir x et (Uj)jr N, satisfont ~ routes les conditions d u l e m m e de t o u t ~ l'heure ; d'apr~s le corollaire de ce lemme, on p e u t donc affirmer que g o f est h o m o t o p e 144

[1952a]

43

l'application identique de E p o u r v u que E x E X [0, 1] soit normal. Le th6or~me annonc6 est done c o m p l 6 t e m e n t d~montr& Supposons en particulier que l'un des U i soit recouvert p a r la r~union des autres, done q u ' o n air U~ =iUUi~,~ et que U~ X U~ X [0, 1] soit normal. Alors les U~j non r i d e s f o r m e n t un r e c o u v r e m e n t topologiquem e n t simple de U~, d o n t le nerf a done m~me t y p e d ' h o m o t o p i e que Ui ; ce t y p e est trivial, puisque U i a la propridt6 d'extension et est done contractile. Si on o m e t U~ dans le r e c o u v r e m e n t II, ce qui reste est encore un r e c o u v r e m e n t ~i' de E en v e r t u de l'hypoth~se ; le nerf N ' dc 11' se ddduit de N en en r e t r a n c h a n t Sty; et la fronti~re de St~ n ' e s t a u t r e que le nerf du r e c o u v r e m e n t (U~j)i~i de Ui, donc est un complexe fini h o m o t o p i q u e m e n t trivial (c'est-s contractile); c o m m e on le volt facilement, cela 6quivaut ~ dire qu'il existe une rdtraction de l'adhdrence St~ de St~ sur sa fronti~re St~ D N ' , done une r6traction de N sur N ' , et m~me qu'il existe une telle rdtraction ddpendant continum e n t d ' u n param~tre, c'est-s une application continue F(x, t) de N X [ 0 , 1] dans N telle que F(x,O) = x et F(x, 1 ) E N ' p o u r t o u t x E N , F ( x , t ) = x quel que soit t p o u r t o u t x E N ' , et F ( x , t ) e S t ~ quel que soit t pour t o u t x E Sty. E n particulier, de R h a m a montr~ (loc. cir., note 6) que, si on se borne s consid6rer la famille des recouvrem e n t s simples qu'il appelle ((convexes)) d'une vari5t6 diff6rentiable compacte, on p e u t toujours passer de l'un s l ' a u t r e de ces recouvrements par insertions et omissions successives d'ensembles superflus ; le rdsultat que nous venons de d d m o n t r e r indique, d ' u n e mani~re un peu plus precise que ne le faisait de R h a m , l'effet de ces opdrations sur les nerfs des recouvrements correspondants.

(Regu le 22 n o v e m b r e 1951.)

DRUCK:

ART.

INSTITUT

ORELL

F~SSLI

AG., Z~RICH

R.Belgica lOV S.Paulo, 18 j a n v i e r . ~ Mon cher Caftan, Ayant commenc~ ~ r~fl~chir ~ un pro jet de rapport sur les espaces fibr4s et questions connexes, j'ai obtenu une d~monstration des th4or~mes de de Rham (pr~lim~naire indispensable ~ ces questions) que j e t e communique, dans l'espoir que ~a t'engage2a ~ te remettre toi-m~me au projet de topologie combinatoire dont tu nous as d4j~ fourni une premifire esquisse. T u n e manqueras pas de reeonnaltre dane cette d~monstration une idle qui appara~t d ~ dans ma lettre ~ de Rham. Le 3e th. de de Rham (sur le prodai% des formes) se d~dult ais4ment des 2 premiers, pourvu qu'ou comlaisse la base d'homologie dane le prodnit d'une vari~t4 par elie-mSme (cas simple: groupes d'homologie sur R) (et ~ ce propos~ on te o~ ; supplie humblement de d~brouiller au plus rite le th. de Kune%h~ i SVP). Quant aux deux premiers~ ils 4tablissent un isomorphieme (canonique) entre le p-i~me groups de de Rham d'une vari4t4 V (groups des formes diff~rentielles ferrules de degr~ p, i.e. des ~ ,,!~< ( ~ de degr4 p satisfaisant ~ d m = O~ modulo le sous-groupe des \ ~'~, :~ formes ~ = d v ~ ~ = forms quelconque de degr4 p-l)~ st le &q'oupe ~9 "~,, ~ de cohomologie" de dimension p sur R. Si je d4finis directement un ~\~ tel isomorphisme,.~'aurai l~essen~iel lie reste s'4tablit Le seul point un peu d~licat est de d~montrer la p o s s i b i l i ~ d'tu~ recouvrement de V par des ensembles A~?~ suffisamment petite pour que dans cnacun on puisse prendre des-coordonn4es locales (i.e. pour que chacun puisse 8ire diff4rentiablement appliqu4 sur une partie de R n )~ et tels que lee A i et routes leurs intersections Z ~ m 2 ~ non v~des soient hom~omorphes ~ des boules B ~. On peut y arriver par exsmple en plongeant diff4rentiablement V dane un espace euclidian R ~ (ce qui est 414mentaire) et en prenant pour lee Ai des intersections de V avec de petites boules euclidiennes ayant leurs centres sur V~ et ayant routes meme rayon (suffisamment petit). De toute mani@re, c'est l~ une question tech~nique infiniment plus facile que la triangulation. Cette quesii .....~ ~ tion 4rant r4solue, si une forme de degr~ p > 0, d4finie dane une : ,~ ~ ' "~jintersection ~ d'ensembles A i , satisfait A d ~ = O, on pourra i'~:~ ~9 / " %oujours l'~crire sous la forme ~ = d~[ ~ ~ 4rant d4finie dane \_i ...... la m~me intersection. Autrement dit~ si'A' d4signe une intersection T e ~ d'ensembles A i . nous avons attach4 ~ A' une suite G~(A') de groupes (p = O, i~ ~ . . . ) ~ ~ savoir lee groupes de fS~mes de degr4 p, d~finies dane A'~ pour toutes lee valeurs de p, ~cf~ " ~ --, et~ pour tout p, un homomorphisme ~ - ~ d ~ de GD(A') dane GD+I(A~ ~i ....... de tells sorte que le noyau de ~ - - - - x ~ I m ~ - ~ x ~ - ~ . -~-*-m--~-*e l'homomorphisme d de Gp+ 1 darts Gp+ 2 soit l'Ima~e par d de Gp aan_~ i~+i (cascade !)~ et cela quel que s01t p > O; quant au noyau de homomorphisme d de G o dane G 1 ~ il set form4 par lee constantes~ donc isomorphe ~ R. D4finir tune forme (de degr4 p donn4) globalement sur V~ c'es~ donner pour route valeur de i un 41~ment mz de Gp(Ai)~ de tells sorte que l'on alt, dane G D ( A i n A ~ ) , la relation ~)~- ~)~ = O, pour toutes lee valeurs de'i et ~"(si A i ~ Aj = ~ cette relation dolt ~tre consld4r4e comme automatiquement satisfai~e). J'utilise ici le falt trlvial~ male qui (en vue d'~ventuelles g~n4ralisatlons) m4rite d'Stre 4nonc4 explicitement~ que~ si A' et A" sont des intersections d'ensembles A i ~ tels que A' ~ A " ~ il y a u~ homomorphlsme canonique de GD(A') ~aqs GD(A"), a savoir celul qui, ~ route forme dane A ' , ~ a i t c o r ~ s p t n d r e la restriction de cette forms ~ A". ~ I

~

2. J'aurai de plus a me servir du principe de prolongement suivant. Soient A' 9 A~' ,...,A" des intersections d'ensembles Ai , toutee contenues dans la premiere A'; supposons donn~e~ dans chacun des A~ , une forme de degq'4 p~ ~ , de mani~re qu'on air &)~ - ~ - 0 dans~ A ~ A ~ quels. . que. solent, ~ ~ ,,~~ , alors ii exlste une forme 60 dans A qul lnElulse ~J~ sur A ~ pour Lout i~ . P o u r le voir~ il sui'fit de prendre darts A' des co ordonn4es locales, au moyen desquelles on pout exprimer los formes en question~ et d'appliquer le th@or~me de %Rs (prolonng.ement d'une fonction diff4rentiable donn4e dans une partie de Rl; a ee propos, il serait tr~s d4sirable d'avoir une d4monstration simple et naturelle de c6 th4or~me). Ce principe permet en particulier de r4soudre la question suivante: supposons donn4, pour chaque couple de valeurs de i~ j, un @14ment ~'i de G o ( A i ~ A j ) ; pout-on attacher ~ tout i un 414ment ~o. de G6(Ax) , de 1"a~on a avoir ~ = ~- ~ ? On a evlcemn~nt los conditions n4cessaLres ~ = O, V.~ + ~ i = O~dans A i ~ A j ) , ~ +~ +~ = 0 (~ans A i ~ A j ~ Aa). Ces c~onditions sont s~ffisautes: en effet~ supposant los ~ d~termin4es pour j < i, la d4termination de ~ clans A i est un ~robl~me de prolo~gement du type indiqu4 plus haut ( ~ est soumisea la condition d'induire des formes d4j~ connues sur t o u s l e s A i ~ Aj pour j ~ i). De m~me, on pourra se g~ma~ demander, 4rant dorn~4~ des ~ . ~ 6 ~,]~Gp(Ai,qAjnAk), s'il existe des ~i~ggGp(AioAj) tels que ~ =0~ 9[~ + ~'~ ~ O, et V,~ + ~'~ +~'~ = ~ & ; d'ou par r4currence,~une suite ~ e formules ~q~i (autonmtique~aent, et pam & ~ c e & un deus ex machina !) ~ont apparaltre 9J~a l'expression co~anue pour le bord d'un simplexe. Je formule la question comme suit: Disons qu'on aura d4fini un boumAd'ordre mPchaque fois qu'a tout syst&me d'indices il,...~im on aura attach4 un 414ment k(~1 = A~,~ ~ de G p ( A i i ~ ...~ Aim) ; ~ ~ m 2 J ~ i x x a ~ i x ~ x ~ on n'aura ~ consid4rer que des boums altern4s (c'est-a-dire que ~ change de signe pour route permutation nnpaire d'indices, et s'an~ule quand deux i~dices sont 4~aux). On intro,auit l'op4rateur (co-bord) par la formule: (9]~) 1), il faut et il suffit qu~on ait % ~ = O. C'est ce que j'ai d4montr4 plus haut pour m = 2; dan~ le cas g4n4ral on proc~de par r4currence sur m. Supposons le th4or~me vrai pour m' ~ m. Supposons les ~ , ~_~ d4j& d4termin@s pour routes les valeurs des indices i~ < r; il s'agit de savoir s i l e s conditions auxquelles ies J*~,...~_,~ doivent satisfaire~ lorsqu'on donne ~ i ~ . . . ~ i ~ . a routes les valeurs < r ~ sont co~apatiOles; or cela r4sulte de l'hypoth~se de r4currence (th4or~me suppos4 vrai pour m-l). Venons-en au groupe de de Rham. C'est~ par d4finition~ le quotient du groupe des bourns ~ fm d'ordre i~ de degr4 r,

;I

--

.

.

.

9

.

.

3.

p, satisfaisant ,h d ~ = c ~ ) = O, par le sous-groupe des bourns de la fonne ~ : d ~ , of~ ~ est d'ordre i, de degr4 p-l, et satisfait ~ ~ = O. ~iais, si d ~ ~ = O r on a ~ = d ~ , o~ A est un boom d' ordr@ i, de deer4 ~-i; il s'ensuit que le groupe de de Rham en question est isomorphe au quotient du groupe des bourns ~ d'ordre i~ de degr4 p-l, satisfaisant a d ~ ~ = 0 r par le eompos4 des sousgroupes respeetivement d~finis par d ~ = O et par :~ ~ = O. ~ontrons~par r4currenee sur ~ ~ que ce groupe es% isomorphe au groupe a~aui~g~u~ d~fini d'une mani~re analogue au moyen des bo~as d'ordre et de degr4 p- ~ ~ quel que soit ~ 4 p ; en effet, d 3 k = 0 entra~ne que O l est de la forms d ~ e avee ~ d'ordre ~ I e% de degr4 p - v - i <si I est d'ordre 9 [ de degr4 p - ~ ); la relation 9~ = 0 entra[ne ~ d ~ = 0; d'ofl l'assertlon2en appliquant les th~or~mes d'isomorphis~ne. Done le groupe de de Rham en question est isomorphe au quotient du groupe des boums d'ordre p~ de degr~ 0 r sa%isfaisant ~ d % A = O~ par le eompos4 des sous-groupes d4finis pa~ d A = 0 et l~ar % A = O~ en posant ~ A = ~ ~ on a d~& = Or de sorts que ~ est u~e co-cha[ne~de distension p~ et ~ ='O r c'es~ ~-dire que ~ est un co-c2~le; en appliquant de m~me que plus haut les th4or~me-s d'isomorphisme~ on volt qu'on aboutit bien au p-i~ms groupe de cohomologie, pour le nerf du recouvrement consld4r4; en vertu des h~,Jpoth~ses faites pour ce recouvrement~ e'est bien la le p - i ~ e groups de cohomologie de la vari4t4 V. A part sa rien Sache~ puisqt~e tu ~ il convient ~ ~ z '

P.s

de neuf~ pour l'instant. La famille va bien. tiens, que le b464 s'appelle ~ieolette~ comme ~ ~ . Meilleures amities ~ tous

[ 1952b] Sur les "formules explicites" de la th6orie des nombres premiers L a th6orie a n a l y t i q u e des n o m b r e s c o n n a i t quelques formules qui relient des s o m m e s 6tendues ~ t o u s ] e s z6ros n o n t r i v i a u x de la fonetion z6ta ~ des s o m m e s 6 t e n d u e s aux puissances de h o m b r e s p r e m i e r s ? I1 n ' e s t pas difficile d ' 6 t e n d r e ces formules, dites >>explicites)>, 5~ des cas b e a u e o u p plus g6n6raux, et ee n ' e s t 1~ 5~ vrai dire q u ' u n simple exereiee. Si je me p e r m e t s de l'offrir en h o m m a g e / ~ Marcel R1Esz, e'est que d ' u n e p a r t il m e t en v a l e u r le r61e que joue dans la question une t r a n s f o r m 6 e de F o u r i e r qui n ' e s t pas une fonetion mais une distribution au sens de L. S c h w a r t z , et que d ' a u t r e p a r t la s t r u c t u r e formelle des formules ainsi o b t e n u e s semble pr6senter quelque int6r6t, de sorte que p a r 1~ elles m6r i t e n t d ' 6 t r e mises "~ la disposition des ehercheurs. R a p p e l o n s d ' a b o r d la d6finition des fonctions L de H e c k e ~mfit Gr6sseneharakteren~>. 2 Soit /c u n corps de n o m b r e s alg6briques, de degr6 d sur le corps des rationnels; s i v est une v a l u a t i o n de ~', on d6signera par ~:,~le corps d6duit de ]c p a r eompl6tion p a r r a p p o r t g v, et p a r ]c~ le g r o u p e multiplieatif des 616ments n o n nuls de k~. S i v est une v a l u a t i o n discrbte, elle c o r r e s p o n d g u n id6al p r e m i e r ~ de/c, et on se d o n n e r a le droit d'dcrire ~ au lieu de v, done l%, kl) au lieu de k , k;; on d6signera en ee eas p a r U~ le g r o u p e (compact) des unit6s du corps ~-adique lc,. let.,. On d6signera p a r vQ ( l _ < O < r l ) les v a l u a t i o n s archim6diennes r6elles de lc, p a r v, ( r l + l ~ t ~ r l + r 2 ) los v a l u a t i o n s a r c h i m 6 d i e n n e s complexes de k, et p a r /ca le eompl6t6 de k p a r r a p p o r t & va ( l < 2 G r l q - r 2 ) ; k o est done p o u r t o u t e le corps des r6els, et k, est p o u r t o u t , le corps des complexes; on posera ~]~=1, ~ , - - 2 . 1 v. par exemple A. E. lngham, T h e distribution o/ p r i m e r~umbe.rs (Cambridge Tracts, n~ Cambridge 1932), Chap. IV, dans ee qui suit, nous nous r6f6rerons /~ ce livre par le sigle DP. 2 E. Hecke, E i n e neue A r t yon Z e l a j u n k t i o n c n . . . , Math. Zeitschr. 1 (1918), p. 357, et 5 (1919), p. 11. Reprinted by permission of the editors of Meddelanden Frgm Lunds Universitets Matematiska Seminarium.

48

[1952b1

49 Sur les d o r m u l e s explicites~ de la th6orie des n o m b r e s p r e m i e r s

253

On sait q l f o n entend par un id61e de ]c un 616ment a (a~) du groupe [1/c~ tel que ap~ Up pour presque tout p (c'est-g-dire pour tout p & un c

nombre fini d'exeeptions pr6s); le groupe des id61es, topologis6 de la mani6re maturelle~), a sera d6sign6 par ik; P e 6t.ant le groupe des id61es principaux, soit C~=I~/P~. Soit Z un earaet6re de Ca., ou, ee qui revient au m~me, un caract6re de I~ prenant la valeur 1 sur P~. S i p est un id6al premier de k, soit m (p) le plus petit des entiers m > 0 tels que Z ( a ) = 1 pour a~ Up (e'est-&-dire ap( U~, et a . = l pour vCp) et a~=~l rood. ~)m; de la continuit6 de ~ sur I~, il %sulte que m (p) est toujours fini, et nul pour presque tout p, done que ~= Hp "~(p) est un id$al de It; ~ s'apelle le conducteur de Z. D ' a u t r e part, sur le sous-groupe I] k~ de I~, Z e s t de la ).

forme Z (al, .*., a , l + r2) = I1 A=]

la~]

~

oh chacun des 10 est 6gal A 0 ou 1, off les/, sont entiers, et oh les q~z sont %els. Soit a=(a~) un id61e; pour ehaque p, ap d6termine un id6M principal (ap)=p n (P) de kp; par d6finition des id61es, n(p) est 0 pour presque tout p, donc IIp n(p) est un id6al entier ou fractionnaire de k, qu'on d6signera par (a). P a r d6finition de ~, on a Z ( a ) = 1 si ( a ) = 1, a a = 1 pour tout 2, et av=l pour t o u t diviseur premier ~ de ~; donc, si a a = l pour tout 2, et ap = 1 pour t o u t diviseur premier p de ~, Z (a) d @ e n d seulement de l'id6al a = ( a ) ; on a ainsi d6fini une fonction Z(a)=z(a) des id6aux a de /c premiers g ~. On pose Mors:

L,s,

~ " Z(a)

~ [

"~(V) ~--1

Cl

off la somme est 6tendue A tous les id6aux entiers a de/c premiers g f, et le produit ~ t o u s l e s id6aux premiers p de k qui ne divisent pas ~. Si Z est le caract6re Z0 p a r t o u t 6gal & 1, L (s) est la fonetion z6ta ~ (s) du corps ]~. On tire de (2):

L'/L(s)----~, ~, log (Np) Z(p) ~ (No) -'~.

(3)

Si on pose s=g~-it, les s6ries et produits ci-dessus convergent absolum e n t e t uniform6ment dans tout demi-p]an g ~ l q - a , pour a)O; done, dans un tel demi-plan, L (s), L (8)-1, et L'/L (s) sont born6es. 3 Cf. A. ~,Veil, S u r la lhdorie du corps de classes, J o u r n . M a t h . Soc. J a p a n , vol. 3 (1951) p. 1.

50

[1952b] 254 Pour

Andr6 V~reil

a ( I k et a = ( a ) , soit Ila]I=lI[a~i,~N(a) ~. Comme Ila[[=l pour

a~Pk, zl(a)=Ha[]~x(a) est encore un earaet6re de C~; la fonction L associ6e s ;/1 est L~ ( s ) = L (s+ir); on dira que Z, X~ sont associ6s. Parmi tous les caract6res associ6s ~ un caraet6re donn6, il y e n a un et un seul pour lequel les exposants ~ qui figurent dans (1) satisfont s ~ a ~ = 0 ; sans restreindre la g6n6ralit6, on pourra donc supposer d6sormais que cette condition est satisfaite. S o i t d le discriminant de/c; soit A=(2n)-d]AIN~. On posera:

s~:sq-iq~.

(l~2~rl+r2)

G(s)=(2~IA)~i2~F( el(8): o6 m

~

L)

(4)

G(1--8):(2rlA) 12s [2]r (~(1--~2)~L-!/~D

d6signe l'imaginaire conjugu6, puis

(5) X(~+~o.)

s? ( s ) = G ( s ) / G ~ ( s ) = 2

~

~

A

~ll ~r

E n v e r t u des propridtds connues de la fonction F, IG(s)l 1 est O(ecltl) uniform6ment dans toute bande ao~a_~al, pour c>~d/4, et, dans la mSme bande, ItS(s)] est O(]t[ -v) p o u r N d(a~--12) si on exclut un voisinage des p61es de ~ (s). Posons m a i n t e n a n t A (s)=G(s) L(s). Un thdor6me fondamental de Hecke (loc. cir. 2) dit que A (s) est une fonction mdromorphe dans t o u t le plan, a y a n t ses p61es en s = 0 , 1 si g=X0, sans p61e si ZCZ0, et qu'elle satisfait h l'dquation fonetionnelle

A (1--~)=s A (s)

(6)

off s est une constante, et Is i= I. La dgmonstration se fair en exprimant zl (s) au moyen d'une int6grale d6finie; cette expression montre en m6me temps que A (8) est born6e dans route bande a0 l , D 6 r a n t u n e c o n s t a n t e ; a p p l i q u a n t a l o r s p a r e x e m p l e le t h 6 o r ~ m e D de D P , p. 49, ~ d e u x cercles de r a y o n s c o n s t a n t s r, R e t d e c e n t r e l-~-a~-it, off a e s t fixe > 0 , on v o i t q u e le h o m b r e de zdros de L 1 (s) d a n s le p l u s p e t i t de ces cercles e s t 0 (log ]t[). C o m m e , e n v e r t u de (2) e t de (6), t o u s les zdros de A (s) s o n t d a n s l a b a n d e 0 < a < l , 4 il s ' e n s u i t que le n o m b r e d e s zdros de A (s) s a t i s f a i s a n t ~ T < ] t ] < T - ~ 1 e s t 0 (log T); il y a d o n c u n e c o n s t a n t e a, et, p o u r t o u t e n t i e r m t e l q u e / m I ~--2, u n T m e o m p r i s d a n s l ' i n t e r v a l l e m < T ~ < m ~ - 1, t e l s q u e A (s) n ' a i t p a s de z6ro d a n s la b a n d e [t--Tm[>formules explicites>>. Nuturellement, on pourrait 61argir sensiblement les hypoth6ses fuites sur F. Nous ~llons appliquer ces r6sultuts g une t r a n s f o r m u t i o n de l ' h y p o th6se de R i e m ~ n n qui n ' e s t peut-~tre p~s sans int6r~t. P o u r cel~, nous nous a p p u y e r o n s sur le l e m m e suivant:

Pour que L (8) satis/asse (z l'hypoth&e de Riemann, il /aut et il su//it que la valeur commune des deux membres de (11) soit ~ 0 pour toute /onction F cle la /orme. +co

F(x)

Fo(x) * Fo(--x)=fFo(X§

s (t) dr,

- oo

olt F o est une /onction satis/aisant (z (A), (B). F est ulors continue et est 1~ primitive d'~ne fonetion F ' continue p u r t o u t sauf en un n o m b r e fini de discontinuit6s de premiere esp~ce (co sont les points %--aj si les ai sont les points de discontinuit& de F0); F satisfuit g (B); et, si q)o est l~ ~)trunsformde de Mellin~ de F o (of.6), celle de F est Cb(s)=C)o(S ) ~0(1--~). Si done t o u s l e s zdros o9 de L(s) duns lu 1

bunde critique sont sur ~ = ~ , le premier m e m b r e de ( l l ) est ~ 0 pour co choix de F et q~. Supposons au contraire que L (s) uit duns cette b a n d e un zdro eOo~fio~i7o uvec fl0# 2. Posons Z

i'

1

-

de sorte q u ' o n u gJ(z)=gao(Z) gao(z) et que Fo(x ) e~r0x, F ( x ) e.i'/~ sont les trunsform6es de Fourier des fonctions induites p a r g~o (z), ~ (z) sur ]'axe rdel. C o m m e lu trunsformde de Fourier de route fonction de la f o r i n t P(x) e -J-~, oh P est un p o l y n o m e et A ~ 0 , est une fonction de m~me

[1952b1

59 Sur les ~formules explicites,) de la thdorie des nombres premiers

263

forme, il s ' e n s u i t que, si on p r e n d p o u r gso(z) une f o n c t i o n P ( z ) e Az,, Fo(x ) satisfera s (A) et (B). P o u r un tel ehoix de T0(z), ~r(z) sera de la m 6 m e forme, done q~ (s) sera 0 (e -A't~) p o u r t o u t A ' < A , u n i f o r m d m e n t dans 0 < a < l , et p a r suite >7 qs(o)) sera a b s o l u m e n t eonvergente; p o u r a e h e v e r de d 6 m o n t r e r le lemme, il suffira de faire voir que eette s o m m e sera < 0 p o u r u n ehoix e o n v e n a b l e de P(z) et de A. P o s o n s en effet,

L(s) dans la b a n d e ~o=i(coo--~--iyo)=i(flo--}).

p o u r t o u t z6ro co de partieulier

9

]

critique, ~ = ~ ( c o - - g Soit

Q(z)

.

~Y0), et en

le p o l y n o m e a y a n t

p o u r zSros simples t o u s l e s ~] distinets, a u t r e s que ~70 et ~0, qui satisfont I~)t~1 < 2 ; e o m m e , d ' a p r b s (6), les ~7 sent rdels ou d e u x k d e u x imaginaires eonjuguds, on p e u t p r e n d r e Q 5~ coefficients r6els; on p r e n d r a P (z)-zQ (z) Q (-z). Alors, s i m est l'ordre de % c o m m e zdro de L (s), on aura, p o u r A > 1 :

2 q~ (co)= - 2 m (r

(rio)[4e2A(rio :)")@ 2

P (~])2 e-2A'12 ~

l~vl>2 1 2

3A

et il est clair que le dernier m e m b r e est < 0 p o u r A assez grand. On v a m a i n t e n a n t , au m o y e n d u lemme, d o n n e r une condition ndeessaire et suffisante p o u r que r o u t e s les fonetions L eonstruites sur le corps k satisfassent 5~ l ' h y p o t h b s e de R i e m a n n . P o u r cela, soit e o m m e p r d c d d e m m e n t C ~ = I k / P k le g r o u p e des classes d'idbles de ]c; p o u r t o u t e v a l u a t i o n v de It, k~ sera identifid avee le g r o u p e des id~les d e n t t o u t e s les e o m p o s a n t e s sent dgales b~ 1 saul au plus celle relative s v; e o m m e l ' h o m o m o r p h i s m e c a n o n i q u e de I k sur C~ i n d u i t sur ee g r o u p e /c~ un i s o m o r p h i s m e de k,~." sur son i m a g e dans Ck, eette i m a g e sera, elle aussi, identifide h ~:~. On a ddfini p r d e d d e m m e n t la f o n e t i o n [Jail sur Ik; c o m m e elle est 6gale ~ 1 sur P~:, elle d d t e r m i n e sur Ck, p a r passage au quotient, une f o n e t i o n q u ' o n n o t e r a I1~11;~§ est un h o m o m o r p h i s m e de C k sur le g r o u p e multiplicatif Y des rdels > 0 , d e n t le n o y a u C O est c o m p a c t (cf.3). L a f o n c t i o n I1~II i n d u i t Ixl sur k~, Ixl 2 sur k~, et Nt) ~(~) sur kz si n(x) est ddfini p o u r x(k~ p a r (x)=l) '~(x). D ' u n e manigre gdndrale, si ~v est u n h o m o m o r p h i s m e s n o y a u c o m p a c t d ' u n g r o u p e G sur le g r o u p e 7, on p e u t n o r m e r la mesure de H a a r sur G p a r la condition que la m e s u r e de la p a r t i e e o m p a e t e de G ddterminge p a r l < ~ 0 ( ~ ) < M soit log M; si ~v est un h o m o m o r p h i s m e ~ n o y a u comp a c t de G sur u n sous-groupe discret 7' de 7, on p e u t n o r m e r la m e s u r e de I-Iaar sur G p a r la condition que la m e s u r e de la partie e o m p a e t e (ouverte et fermde) de G ddterminde p a r l_ 1 a n d ~ a p r i m i t i v e m - t h r o o t of Received by the editors February 25, 1952. (1) For a bibliography on this subject, see my article Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc. vol. 55 (1949) p. 497; the numbers in brackets will refer to the bibliography at the end of that paper, which will be quoted as NF. 487 Reprinted from Transactionsof the American MathematicalSociety, V1.73, pp. 487 495 by permissionof the American Mathematical Society. 9 1952 by the American Mathematical Society.

63

64

[1952d1

488

ANDR]~ WEIL

[November

u n i t y over the field Q of rational n u m b e r s (e.g. ~"=e~'~1m). If t is a n y integer prime to m , ~ - + ~ determines an a u t o m o r p h i s m gt of Q(~') over Q; the Galois g r o u p of Q(~') over Q consists of all ~t a n d therefore is isomorphic with the multiplicative g r o u p of integers prime to m m o d u l o m. Let ~ be a n y prime ideal prime to m in Q(f), and p u t q = N p ; t h e n q ~ l (mod m). T h e m-th roots of u n i t y ~-a, for O < = a < m , are all incong r u e n t to each o t h e r m o d ~ and therefore are all the roots of the c o n g r u e n c e X = = I (mod ~0) in Q(~'). For every integer x prime to ~ in Q(~-), x (~-1~1"~ is a root of t h a t congruence, and so there is one a n d only one m-th root of u n i t y X~(x) satisfying the condition Xp(x) =-- x (q-1)lm (mod ~).

For x--=0 (mod ~0) we p u t X ~ ( x ) = 0 . T h e n X~ is a multiplicative c h a r a c t e r of order m of the field of q elements consisting of the congruence classes in

Q(f) mod ~0. L e t r be a n y integer > 1 ; the really significant case is r = 2 , since the quantities we shall c o n s t r u c t are trivial for r = 1, a n d those c o r r e s p o n d i n g to r > 2 can all be expressed in terms of those belonging to r = 2 a n d r = 1. Let a = (ap)l~.~r be a set of r integers ap m o d u l o m, i.e. an element of the direct p r o d u c t G r of r groups all identical with the a d d i t i v e g r o u p of integers m o d u l o m; the c h a r a c t e r s on the g r o u p G * are the functions on G r of the form ~-~apup, where u = (u.) is also an element of G'. N o w write

(r)

yo(~) = ( _ 1),+1

~2 Xl,

x~(x~) ~

9 9 9 mr rood

x~(x,) ~

11

where the x, run over complete sets of r e p r e s e n t a t i v e s of the c o n g r u e n c e classes m o d u l o ~ in Q(f) subject to the condition ~--~;=1 x o - - 1 (mod I~). F o r a given ~, this is a function of a C G ' . If, for a n y u = ( u , ) , we d e n o t e b y N ( u ) the n u m b e r of distinct sets of c o n g r u e n c e classes (xp) m o d u l o la satisfying ~];=1 xp-------1 (mod ~) and X ~ ( x p ) = f " p for l < p < r , t h e n we h a v e

(1)

yo(~) = ( - 1)~+1~2 N(~)~ ~~ u

which gives the expression of J.(p) as a function on G ~ in terms of the characters on G'. B y induction on r it is easily seen t h a t we h a v e (2)

Jo(~) = q-l[1 -- (1 -- q)@

W h e n some b u t n o t all of the ap are 0, e.g. if a~+l . . . . . a. = 0 and none of t h e al, 9 9 9, a. is 0, t h e n it is easy to see t h a t J.(~) reduces to the s u m J.,(p) similarly built up from a ' = (ax, 9 9 9 , a.); in particular, if all the ap except al are 0 a n d a l ~ 0 , t h e n J = ( p ) = X ~ ( - - 1 ) ~. If we p u t % = a o / r n for l 2 can be expressed in t e r m s of those for r = l and 2; in fact, using (7), once easily gets the relations Ja,....at(a) = Ja,(a)Ya,....,a,(a)Na if al+a2--=0 (mod m), and

Ja,,.,

.,at(Q)

=

Jal+a,(a)Jal,a2(a)Jal+a2,a3,...,ar(a )

if ax+a2 ~0 (rood m). As m 2 is a defining ideal for r = 1 and for r = 2 it follows b y induction on r t h a t it is also a defining ideal for all r. I t seems doubtful whether m s is ever the true conductor of the characters ar,(a). For m = 4 , one finds t h a t the conductor is 4; when m is an odd prime one finds t h a t the conductor is either ( 1 - ~ ' ) or (1-~')2; actually it is the latter in the numerical examples which I h a v e examined. A general investigation of this question might lead to results of some interest. 2. H a s s e ' s conjecture. Consider the plane algebraic curve (III)

Y' = 7 X / - k 6

where e, f are integers such t h a t 2 ) ; hence the element of ro which belongs to the same coset of y as >,. is e((x', r) - - (y', J r } ) e ( ( x ' @ J'y', r ) ) . Now denote t h a t coset by f ( z ' ) ; f is a homomorphism of t?" onto r / 7 . Call o~ the canonical homomorphism of A' onto O" = A ' / • and, for z ' = cc" + iy', p u t g(z') = x' + J'y'; o' and r/~/ being identified as in no. 4, what we have just shown amounts to f = o ) o g. As we have g(iz') = - - y' -+ J'x" = J'(x" @ J'y'), g is a complex-analytic homomorphism ; so is ~; the same, therefore, is true of f. Conversely, if J'l is an automorphism of A'o, defining on A'o a complex structure such t h a t f, hence also g, are complex-analytic mappings, then we must have g(iz') ~ J'~g(z'), hence J', = J'. 6. F r o m the theory of abelian varieties and of theta-functions ( [ 8 ] , [9] ; cf. [ 1 1 ] ) , we borrow the following facts. I n order that there should be q algebraically independent meromorphic functions on the complex torus O ~ A / • it is necessary and sufficient fhat there should exist a r e a l - v a h e d a l t e r n a t i n g bilinear form E ( x , y ) on Ao X Ao, such that E ( r , s ) ~ 0 m o d l whenever r c A, s c A, and t h a t the bilinear form F ( x , y) = E ( J x , y) is symmetric and positive-definite (i. e. satisfies F ( x , x) > 0 for all x =/= 0). Such a form E will be called a R i e m a n n form for O; if we write it as E ( x , y) = ( E x , y ) , where E is a linear m a p p i n g of Ao into A'o, the condition that it should be a l t e r n a t i n g is expressed by tE ~ - - E ; this being assumed, we have F ( x , y) <EJx, y ) ~ - - ( E y , J x ) , and the condition t h a t F should be symmetric is expressed by E J ~ t ( E J ) = J'E, hence it is equivalent to the condition t h a t E should be a complex-linear m a p p i n g of A into A'. The condition E(r, s) ~ 0 rood 1 for r c • s a • is equivalent to E ( • C • Hence E defines a complexanalytic homomorphism of O into its dual O'. Finally, if F is positive-definite, we must have Ex 5/= 0 for x =/~ 0, hence E is invertible. Therefore the existence of a Riemann form implies that O and | are isogenous; if moreover E maps A onto • they will be isomorphic, but they need not be so otherwise. A 12iemann form E being given, one can define (by the explicit con-

[1952e]

77 O ~ PICARD V A R I E T I E S .

869

struction of so-called theta-series) theta-functions belonging to 0 and E, i.e. entire functions ~ of the complex coordinates on A, satisfying periodicity conditions which, suitably normalized, may be assumed to be O(x + r) = # ( x ) e [ - - 89

x) + ~E(r, Jx) - - 88176 r) + -~E(r, Jr)]

for all x e A, r e A, where E ~(x, y) is a symmetric bilinear form on Ao X Ao such that E ~ rood2 for all reA, saA. Then, for any a e A , the function

~(x ~- a)O(x)-le[89

x) - - ~E(a, Jx)]

is a multiplicative meromorphic function of the complex coordinates on A, with the multiplicator-set e[E(a, r ) ] ~ e((Ea, r)). As E maps Ao onto A'o, this, in conjunction with the results of no. 4, shows that there exist meromorphic multiplicative functions with arbitrarily given multiplicator-sets. Now consider a maximal set of linearly independent theta-functions v~j(x) belonging to O and the Riemann form NE, N being an integer ~ 0. These must be in finite number; and, if they are taken as homogeneous coordinates of a point P(x) in a suitable projective space, we have P(x-~-r) ~ P(x) for r s A, so that the mapping x--> P ( x ) induces a complex-analytic mapping 4) of o into that projective space. By a theorem of Lefschetz [8], as soon as N is large enough, 9 is a one-to-one mapping of O onto an algebraic variety ~(O), and its Jacobian matrix has everywhere the rank q, so that (o) is without multiple points; ~ can then be used to identify O with (I)(O). For that reason, we shall say that a complex torus O is an abelian variety whenever there exists a Riemann form for O. I f O is an abelian variety, so is every complex torus isogenous to O, hence in particular the dual torus O'. In fact, let O1 ~ At~A1 be isogenous to O; then there is an invertible complex-linear mapping L of Ai onto A, such that L(A1)CA. Let E(x, y) be a Riemann form for O; then the bilinear form E(LxI, Lyl) is alternating, and is ~ 0 rood I for Xl e Ai, y~ e Az ; also, as LJ~-~-JL, we have

E(LJ~z~, Lye) ~ E(JLxl, Lye) ~ F(Lx~, Lyl), where F is defined as before; this is symmetric, and ~ 0 whenever Lx~ ~ Lyl 0, hence, since L is invertible, whenever xl ~ y~--~ 0. 7. Now let V be a K~ihler manifold, i. e. a connected compact complexanalytic manifold with a Kiihler metric; I shall use the notations of a previous note of mine ( [ 1 0 ] ; cf. [3], Part I I ) .

78

[1952e]

870

ANDRI~ WEIL.

If O is the fundamental group of V, and O' the commutator-group of G, H ~ G/G' is the one-dimensional homology group of V with integral coefficients; it is the direct product of its torsion group T, i. e. of the finite group of its elements of finite order, and of a free abelian group, whose rank, by IIodge's theorems, is an even number 2q. If we consider H as a module over the ring of integers, and extend s its ring of operators to R, we obtain a vector-space Ao of dimension 2q over R, which is the one-dimensional homology group of V with real coefficients; we shall denote by ~ the canonical homomorphism of H into Ao; the kernel of X is T, and the image a = X(H) of H by X is a discrete subgroup of rank 2q of Ao. Let V be the covering manifold of V belonging to the subgroup G' of G; every element ~ of H determines an automorphism of V, transforming each point 37 of l~ into a point ~2~ lying over the same point M of V as 3?. If 0 is any closed differential form of degree i on V, a n d / 5 3? are two points on J~, the integral

y,o

0 (taken along any differentiable path from /5 to ~/ on fz)

defines, when/5 is fixed, a function f (3?) such that df ~ 0; we have simplified notations here by writing 0 also for the inverse image of 0 on !~. For any (re//, p~f(~M)--f(M) is a constant, which we can also write as p,, ~ J , 8, the integral being taken along any closed differentiable path of class ~ on V. Then e ~ p~, is a representation of H into the additive group R ; this must be 0 on T, and it can be extended, in one and only one way, to a linear form on Ao; this form will be denoted by p(0, x), so that we have

p

=fo 8 =

Now let B be the vector-space of real harmonic differential forms of degree I on V; then p(O,x), as a function of (0, x ) e B }< Ao, is a bilinear form, which, by ttodge's fundamental theorem, determines a duality between B and Ao; hence we may use it to identify B with the dual space A% of Ao, i.e. with the one-dimensional cohomology group of V with real coefficients; and we write, from now on, (8, x> instead of p(0, x) when 0 is harmonic, i.e. when 8 s A%. If 0 is harmonic, and if P, 3? are two points on ~', the integral

r;

0 is a

real number depending linearly upon 0, which, when/3 is fixed, can be written as , with /~(M).a Ao. Then /~ is a mapping of V into Ao, satisfying ~ o X is an isomorphism of the group F of multiplicator-sets belonging to O onto the group F' of special multiplicator-sets for V; we shall denote by F'o, 7' the images, by this isomorphism, of the subgroups 1~o, ), of r as defined ia no. 4; elements of ~' will be called the trivial multiplieator-sets for V. I f $ is any holomorphic multiplicative function without zeros, belonging ^ ^ to O, and # is its multiplicator-set, then t~ ~ ~/, # o X r 7% and r ] is a holomorphic multiplicafive function without zeros on ]?, with the multiplicator-set # o X. Conversely, let r be a holomorphic multiplicative function

[1952e]

81 ON PICARD VARIETIES.

873

without zeros on V; then ~ = 2--~d(log 4)) is a holomorphic differential on V, hence of the form ~ ~ ~ - - i C O , with 0 e A'o; then we have

with

r

= e[L(z) + c ] ,

L(z) =--~=--~,

where c is a constant; as L is a complex-linear form on A, this shows that the multiplicator-set of 4) is trivial. I f r is any meromorphic multiplicative function belonging to O, with the multiplicafor-set ~, one can choose a constant a ~ A so that ~ [ F ( M ) -4- a] is not everywhere 0 or ~ , and then this is a meromorphic multiplicative function on V with the multiplicator-set ~ o A; in conjunction with the results of no. 6, this shows that there exist meromorphie multiplicative functions on with an arbitrarily given special multiplicator-set. 10. One can define ~he divisor on V of any meromorphic multiplicative function on V, or more generally of any meromorphic function 4) on V such that, for every ~ ~ H, 4)(~i~/)/4)(3I) is a holomorphic function without zeros on V; the definition is briefly as follows (ef. X. Kodaira, [3], P a r t I I ) . I f zl," 9 ", z. are local complex coordinates in a neighborhood of a point P of V, and i 5 is a point lying over P on V, Zl,' - ' , z ~ can be used as local coordinates on V a t / 5 and 4) can be expressed as a product 4) = E(z)II s~j(z)'~ J of a unit-factor E(z) (i. e. a holomorphic function, :J= 0 at P ) and of powers of irreducible holomorphic functions 4)~(z) (irreducible, that is, in the ring of holomorphic functions at 15). Replacing/5 by another point ~ib of V lying over P will merely affect E(z), but not the 4)i(z) or the mj. The divisor of 4) on V will then be defined locally, in a neighborhood of P, as ~ miWj, where t Wj is the irreducible algebroid variety, of complex dimension n - - 1 , defined by r with the orientation determined by the condition ~ - ~ > 0. Then the divisor (4)) of 4) on V will be defined globally, by means of any suitable finite open covering of V; it will be of the form ~ apZp, where the p a~p are integers, and the Zp are irreducible compact analytic (i. % everywhere algebroid) subvarieties of V, of complex dimension n - 1, with the orientation defined above. More generally, any such expression will be called a divisor (more precisely, an analytic divisor) on V.

82

[1952e] 874

AI~DR]~ WELL.

11. Now, let again r be a meromorphic multiplicativc function on V, and put ~ ~ - ~ d ( l o g r it follows from the multiplieative property of that this is a meromorphic differential form on V (more accurately, it is the inverse image on l~ of such a form). By analytic continuation, one sees that

~(M) = e (

f;

~+c),

where c is a constant; the integral is taken along

any differentiable path, from the fixed point/3 on J~ to !~r, which does not meet ( 4 ) ; hence, for ~ e H , we have 4 ( ~ 3 i ) = 4 ( M ) e ( j i ~ ) .

If r is factored

as q~~ E(z)II 4j(z) mj in terms of local coordinates in the neighborhood of J a point of V, then ~ - - ~. mj~2d(log 4'J) is holomorphic in that neighborhood; 3

from this, one deduces, by first considering the ease of a singular 2-cell in such a neighborhood and of its boundary, that, if S is any 2-dimensional singular chain in V, whose boundary C is a differentiable singular chain and does not meef (q~), the intersection-number of S and (q,) is equal to .I t

~; in

the language of the theory of currents (of. [3]), this shows that the cycle (4) of dimension 2r~--2 on V is the differential of the current ~. This implies, firstly, that (q~) is homologous to 0 with real coefficients, hence also (since it is an integral cycle 4) with rational coefficients. In particular, a holomorphic multiplicative function 4' on l~ cannot have zeros: for otherwise the integrM of a n-~ on (q~) would be > 0, and, as this is a closed form, (40 could not be homologous to 0. Also, if ~-e T, the linking coefficient of r and (r

is equal to . ~ ~ mod 1 (el. Igusa [7]) ; for, if t is any differentiable

singular cycle of class ~ which does not meet (4), that linking coefficient is , by definition the intersection-number of (r and of any 2-dimensional singular chain with real coefficients, with the boundary t, reduced rood 1. As it is known, by Poincar@'s duality theorem, that the linking coefficient defines the torsion groups of V, of dimensions 1 and 2n - - 2, as a dual pair, it follows that

(r

is

homologous to 0 with integral coefficients if and only if J i ~ ~ 0 rood 1

for all r e T, i. e. if and only if the multiplicator-set of r is special. 12. Let now Z be any analytic divisor on V. We can cover V with open sets U~, so small that, in each Uh, Z can be written as ~] msWj, w h e r e J As to this and other " obvious" homological properties of the subvarieties of ]7 and in particular of the divisors on V, they can best be justified by the definitions and results in N. Hamilton's forthcoming thesis [4].

[1952e]

83 ON PICARD V A R I E T I E S .

875

W~ is the variety of zeros of an irreducible holomorphic function q~(z) in Uh, the z's being local complex coordinates in Uj~. Define the differential form ~ in U~ by ~h = Y~ mj -~Td(log q~j) ; then ~h - - ~ is h01omorphic in U~ N U~; i

hence, by the main theorem of [10], if Z is homologous to 0 with real coefficients (and only in that case), there will be a meromorphic differential form on V, such that, in each U~, $ - - ~7~ is a closed holomorphic form. Analytic continuation and the monodromy principle then show that there exists a meromorphic multiplicative function ~ on I?, such that, in each U~, 4, = E ( z ) I I q~j(z)"J, where E(z) is a unitZfactor in Uh, and that ~ ~ 2-~7~d(log q~), J

then we have (q6) ~ Z. By the results of no. 11, Z is homologous to 0 with integral coefficients if and only if the multiplicator-set of q~ is special. Now suppose that r q~ are two meromorphic multiplicative functions such that (q~) = (q~) = Z ; then q~/~ is a holomorphic multiplicative function without zeros, hence its multiplicator-set is trivial. Therefore, if ~ is the group of the analytic divisors which are homologous to 0 with real coefficients, we have attached to each Z e ~ a coset c(Z) of the group 7' of trivial multiplicator-sets in the group of all multiplicator-sets, consisting of the multiplicator-sets of all meromorphic multiplicative functions q~ such that (~) ~ Z. This coset will consist of special mnltiplicator-sets, i.e. it will be in r ' / T ' , if and only if Z is in the subgroup ~ of ~ consisting of the divisors which are homologous to 0 with integral coefficients. We have c ( Z ) ~ - / ' if and only if Z is the divisor of a function q~ with the multiplicator-set 1, i. e. of a meromorphic function on V; then we say that Z is linearly equivalent to 0; the group of such divisors will be denoted by ~ ~. But r ' / 7 ' can be canonically identified with P/y, hence also with ro and O'. Therefore the mapping Z---> c(Z), restricted to ~a, defines a canonical homomorphism of ~ onto the Pieard variety O' of V, with the kernel ~z; we may also say that we have defined a canonical isomorphism between ~ / ~ and the character-group ro of A: this is Igusa's first duality theorem. Similarly, the mapping Z---->c(Z) determines a canonical isomorphism between ~ a / ~ z and a subgroup of finite index of the character-group of H ; this subgroup will be the whole character-group of H whenever every character of H of finite order is the multipticator-set of some meromorphic mnltiplicative function on 1~', or, what amounts to the same in view of the above results, whenever every homology class of dimension 2 n - - 2 and of finite order on V (with integral coefficients) contains an analytic divisor. This is Igusa's second duality theorem. By a theorem of Lefschetz ( [ 8 ] ) , V will have that property whenever it is a non-singular algebraic subvariety of a projective

[1952e]

84 876

AZ~DR~ WEIL.

space; indeed, it seems likely that, for such variety V, every eovering manifold of V with a finite number of sheets is again an algebraic variety.

w II.

Construction of a system of representatives for the group of divisor-classes.

13. F o r the proof of the m a i n result of this w we shall need various ]emmas on theta-funetions. As explained in no. 10, if ~ is a theta-funetion (other t h a n 0) belonging to an abelian variety O, one can define the divisor (v~) of ~ on | this is a positive divisor, i. e. all its components have positive eoeffieients. I f a point u of | lies on a component of ( 4 ) , we shall say t h a t is 0 at u. Periodicity conditions for theta-funetions will always be understood to be in the normalized form given in no. 6. As this still depends upon the choice of a symmetric form E ~ when .O and E are given, we shall further normalize this by choosing a set of generators r , , . 9 ryq for • and t a k i n g E~ rk) = E(rj, rk) for 1 ~ j O, the bilinear form

• ~ A X~X'; real spaces of written, more X A'o. Theft,

E"N(x, x' ; y, Y') = E"(x, x' ; y, y') + N E ( x , y) is a :Riemann form for 0 " ; let As be the vector-space of all theta-funetions belonging to | and E"~. Take any u' e 0 ' ; we shall now prove that the codimension of the subspace of A..... consisting of the functions in it which are 0 on Z X u*, is > = q q - a . I n fact, as E " is the sum of two Riemann forms, the result in no. 13 shows that we can choose a function 0'o in A| which is not 0 at ( z , u ' ) , z being an arbitrarily chosen point in Z. },row, 4) being a homogeneous polynomial of degree r in Uo," 9 ", UM, put

0 % ( . , x') = a'o(., .')4)[~o(X),. - . , o , , ( . ) ] ;

86

[1952e] ANDRE ~VEIL.

878

the mapping q5--->y ' r is a linear mapping of the space of such polynomials into Amr; and vq", is 0 on Z X u' if and only if r 9 ",v~i) is 0 on Z. As the codimension of the inverse image of a linear subspaee by a linear mapping is at most equal to that of the subspace, it follows that the subspaee Of Amr, consisting of the functions in it which are 0 on Z X u', has at least the codimension ~(r). 16. N o w we shall prove that there is a function in Am, which is not 0 on any of the sets Z X u', where u' is any point of | I n fact, let (O"o)o.

[1952e]

87 ON

PICAIRD VARIETIES.

879

The dual of A"o ~ A| X A'o being A'o X Ao, this form can be written as <E"ox", y">, where x " = (x, x'), y " = (y, y'), and where E"o is the linear mapping E"o =

(o _/,)

of A| )< A'o onto A'o )< Ao, I and I' ~ tI denoting

again the identical automorphisms of Ao, A'o. The scalar multiplication by i in

A"~

A • X' indnees on A " o = Ao • A'o the automorphism J " = ( J0 j , ) "

Then we have E"o (J"x", y") ~ ) ,

we have

r

+ r) = r

r

= r

r>),

where L~,,~,(x) is a complex-linear function on A, depending upon a ' , ~ ' ; r is a meromorphic function on A, which, by the properties which we have assumed for # and tY', is not everywhere 0 or 0o on /?~(l~) ; and it is a multiplicative function, with the multiplicator-set e(( u'. There is, in a case such as this, no difficulty in defining the intersection-multiplicities, by function-theoretic or topological means.

[1952e]

89 ON P I C A R D V A R I E T I E S .

881

Therefore we have D(u') ~ W o - W(u') ; and what we have done shows that this is a divisor in the group ~ , which has the image u" in O' by the canonical homomorphism of &~ onto O'. As O' is an algebraic variety, we may express the relation W ( u ' ) X u ' = W. ( V X u'), where the 9 denotes the intersection, by saying that the W(u') are an algebraic family of positive divisors, parametrized by the variety | The canonical homomorphism of ~ onto O', with the kernel ~z, can then be defined by saying that it maps each divisor X e ~ , onto the (uniquely determined) point u' of O' such that W(u') is linearly equivalent to W o - - X . This implies of course that W(0) is linearly equivalent to Wo, so that, in this last statement, Wo may be replaced by W(0).

w III. The main theorem. 20. By an analytic subset of a complex-analytic manifold, we understand a dosed subset of it which, in some neighborhood of everyone of its points, is the set of conlmon zeros of a finite nmnber of holomorphic functions; an analytic subset which is irreducible (i. e. which is not the union of two other such sets) is called a subvariety. By a divisor L(, we understand, as above, any formal sum of subvarieties of maximal dimension, with integral coefficients; if the latter are positive, X is said to be positive, and we write X } - 0 ; we write X > Y f o r X - - Y > 0 . By the carrier / X I o f X , weunderstand the union of the components of X. In all this w we shall denote by V a connected compact complex-analytic manifold of complex dimension n. If S is a complex-analytic manifold, and X a divisor on V )K S, the relation X(s);K s = X . (V)< s) defines a divisor X(s) on V for every s e n such that V X s is not contained in I X l; as I X ] is closed, the set S' of these points is an open subset of S, which may be considered as a complex-analytic manifold; we shall say that the set of all divisors X(s), for se S', is an a,talytic family of divisors on V, parametrized by S', or having S' as its parameter manifold. Notations being as above, one may always assume that X has no component of the form V X T, where T is a subvariety of S of maximal dimension; in fact, adding such a component to X does not modify X(s) when s is not on T, and makes X(s) undefined if s is on T. That being assumed, we shall show that the set S - - - S ' of points s of S for which X(s) is not defined is an analytic subset of S, of (complex) dimension 0. Each point M X So of V X S has a neighborhood where X can be written as

90

[1952e] 882

ASrD~s

WEIL.

the divisor ( ~ ) of some holomorphic function (I); let z l , ' ' ' , z ~ be local coordinates on V in a neighborhood of M ; then, if a point s, sufficiently near to So, is such t h a t V X s C I X I, we m u s t have ~ ( z , s ) ~ 0 identically in z near z ~ 0; therefore the points of S - - S ' near so are common zeros of all coefficients of 9 when r is expanded into a power-series in zl," 9 ' , z~ with coefficients in the r i n g of holomorphic functions of s at s ~ So; conversely, if a point s, sufficiently near to so, is a common zero of all these coefficients, then it is easily seen, by analytic continuation along V X s, t h a t V X s is contained in I X 1 . A p p l y i n g a theorem of H. Cartan, 5 one sees t h a t S - - S ' can be denned, in a sufficiently small neighborhood of so, by equating to 0 a finite number of the coefficients of r therefore it is an analytic set. I f now T is any component of S - - S', V X T is contained in I X I ; therefore, if T were of dimension p - - 1, V X T would be a component of X, which is against our assumption. I n particular, if S is connected, S' also is connected. Then the homological theory of subvarieties of a complex-analytic m a n i f o l d ( [ 4 ] ) shows t h a t any two divisoi's of the form X ( s ) are homologous to each other on V; in particular, if X ( s ) is homologous to 0 for some s, the same is true for all s s S'. The m a i n purpose of this w is to show that, if V is a Hodge manifold,

the canonical mapping c of the group ~ of the divisors on V, homologous to 0 on V, into the Pieard variety | of V (w I, no. 12) induces an analytic mapping into (9' of every analytic family of divisors belonging to ~ on V. I n other words, if S, X, S' and X ( s ) are as above, and if X ( s ) is homologous to 0 for all s ~ S', the m a p p i n g s---> c[X(s)] is a complex-analytic m a p p i n g of S ' into |

Assuming for a moment t h a t this is so, we shall show t h a t

this mapping can be extended to a complex-analytic mapping of S ;~7~ | I n fact, write as before | A'/A', where A" is a complex vector-space and A' a discrete subgroup of A'; if x' is a variable point of A', we may consider dx' as a vector-valued differential form on A ' (with values in A ' ) ; as this form is i n v a r i a n t by translations in A ' and in p a r t i c u l a r by A', it induces on o ' a vector-valued form which, in terms of the complex structure of O', is a holomorphie f o r m ; its inverse image ~o on S" by the m a p p i n g s--> c[X(s)] is therefore a holomorphic form on S'. As S - - S ' is an analytic subset of S and has no component of dimension p - - 1 , it follows, by H a r t o g s ' theorem, t h a t ~ can be extended to a holomorphic form on S. As every 1-dimensional cycle on S can be deformed into one on S', the periods of ~o on S are the 5 ,, Thdor~me a " of [1], p. 191.

~952e]

91 883

ON PIC'ARD V A R I E T I E S .

same as those on S ' ; as the l a t t e r m u s t belong to a ' , the same is therefore true on S ; as in w I, one concludes from this that the m a p p i n g ~ - +

s

~ of

the universal covering of S into A ' (for a fixed ~) induces a holomorphic m a p p i n g of S into O', which, on S', coincides (up to a translation) with the given one s - + c [ X ( s ) ] . This proves our assertion. Even so, our result will r e m a i n incomplete in one i m p o r t a n t respect: we shall not prove it for families of divisors which are parametrized by algebroid varieties. By an algebroid variety, we u n d e r s t a n d one on which each point has a neighborhood isomorphic to a locally irreducible analytic subset of a complex-analytic manifold. I t seems likely t h a t the proof which is to be given here of our m a i n theorem could be applied to this more general problem with no more t h a n slight modifications, or at any rate that the more general result could be shown to follow from ours, if only the foundations of algebroid geometry were not even now as shaky as those of algebraic geometry once used to be2 21. L e t X ( s ) and Y ( t ) be two analytic families of divisors on V, respectively p a r a m e t r i z e d by the manifolds S and T. We shall now prove t h a t the set of points (s, t) of S >( T such that X ( s ) ~ Y ( t ) is an ar~alytiv subset of S > ( T . I f we p u t i ~ = X } Z T , ~ is a divisor on V > ( ( S X T ) , and we have :g(s, t ) = X ( s ) ; operating similarly on Y, we see t h a t it is enough to show that, if X ( s ) , Y ( s ) are two families, parametrized by the

same manifold S, the set of points s e S such that X ( s ) > - Y ( s ) is an analytic subset of S. I f we write X ~ X" - - X", Y ~ Y' - - Y", where X', X", Y'. Y " are positive divisors, X ( s ) ~ Y ( s ) is equivalent to X'(s) + Y " ( s ) ~ X " ( s ) q - Y ' ( s ) ; therefore it will be enough to prove our assertion for positive X and Y. As the question is purely local so far as S is concerned, we m a y assume t h a t S is a neighborhood of 0 in the space of p complex variables. Let M be any point on ]7; take local coordinates (w, zl," 9 ", z~_l) V in a neighborhood of M, such t h a t the subvariety (of complex dimension of t h a t neighborhood defined by z1 . . . . . z~_~ ~ 0 is not contained I X ( 0 ) [ nor in I Y ( 0 ) l " Then, by Weierstrass's lemma, X can be written a neighborhood of M X 0 on V X S as the divisor ( r of a function r the form

on 1) in in of

6 It is to be hoped that someone will soon undertake the taxing but necessary task of consolidating them or rather building them up anew. As some at least of the main difficulties have been removed by the recent work of H. Cartan, W. L. Chow, ]g. Hamilton, K. Oka and others, the time seems ripe for such an undertaking.

92

[1952e] 884

ANDRE WEIL. (I) (W, 2:, s)

= wo + ~ w~-'4,(~, s),

where the (~ are holomorphic and 0 at the origin; similarly, Y can be written as (~) in a neighborhood of M X 0, with ,I, of the form b 9 (W~ Z,

s) = w~ + Z w~-J~i(z, s), .i=1.

where the ~ are holomorphic and 0 at the origin. Considering 4) and 9 as polynomials over the ring of holomorphi~ functions of z, s at the origin, we can divide ~ by ,I,; let R be the remainder, which is of the form b-1

R(w, z, s ) ~ Y~ w1'Xl,(z,s), where the Xh are holomorphic at the origin.

Put

h=0

[ z I ~supl---Y(s). By Cartan's theorem, s this completes our proof. I f X ( s ) , Y(t) are again two analytic families of divisors on V, then

the set of points (s,t) of S X T such that X ( s ) ~ Y(t) is an analytic subset of S X T; in fact, it is the intersection of the two analytic subsets determined by X(s) >-Y(t) and by Y(t) }-X(s). 22. I f X is any divisor on V, we shall denote by L(X) the set of all meromorphic functions s~ on V which are either identically 0 or such that (s~) ~ - - X ; this is a vector-space over the field of constants. We shall now prove that, for every divisor X on V, the vector-space L(X) is of finite

dimension. If X = X ' - - X " ,

with positive X" and X", we have L ( X ) C L ( X ' ) ; so it will be enough to prove our theorem for a positive divisor X. Then every point M on V has a neighborhood U ~ U(M) in which X can be written as the divisor (cDM) Of a function CM, holomorphie in U; and the set OM" L(X) of all functions r 1 6 2 with S~s L(X), consists of functions which are holomorphic in U. Consider the ideal generated by O~L(X) in the ring of holomorphic functions at M; by Cartan's theorem, 5 this ideal is generated by a finite number of elements ~r of r and there is an open neighborhood U ' = U'(M) of M and a constant / c = ] c ( M ) with the following properties: the closure U' of U' is contained in U; and every ~ e OML(X) such that I P l ~ l i n ~ c a n b e expressed as r ~ ~ ~pi (m), where the X, are r

holomorphic and such that I x ~ l ~ / c on 8'. Take a covering of V by a finite number of neighborhoods U ' . ~ U ' ( M . ) ; to each there belongs a function c[X(s) --X(so)] is an analytic mapping of S into O'. Now, in w I I , we have constructed an analytic family W(u') of positive divisors on V, parametrized by 0% such that u ' = c [ W ( 0 ) - W ( u ' ) ] . This implies that there is one and only one point u' on 0% given by u' ~ c[X(s) - - X(so)], for which W ( 0 ) - - W ( u ' ) is linearly equivalent to X(s)--X(so), i.e. W(u') + X(s) to W(0) + X(so). Let ~o," " ", ~ be a basis for the vector-

[1952e]

95 ON

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VARIETIES.

887

space L [ W ( 0 ) ~ - X ( s o ) ] ; then, if s and u' are as we have said, there is one and only one point ~ in the projective m-dimensional space P"~ such that, if (~o," " ',~m) is a set of homogeneous coordinates for ~, the divisor of the m

function ~ ~ F, ~

is

i=o

(r

=

W(u') + X(s)

--

[w(0) + ;;(So)].

For every S~= • ~i~ in L [ W ( 0 ) -4- X(so)], put Z(~) ~ (~) + W(0) d- X(so), where ~ denotes the point with the homogeneous coordinates (~o,"" ", .~m) in pro; one sees at once that Z(~) is an analytic family of divisors parametrized by P ' . By the final result of no. 21, the set of all points (s, u', ~) in S X O ' X P~ such that Z(~) ~ W(u') d - X ( s ) is an analytic subset r of S X o ' X pro. But we have just shown that, for every s e S, there is one and only one point (u',~) in O ' X P "~ such that (s,u',~) is in F; as r is closed and 'O' and P"~ are compact, this implies in the first place that P is the graph of a continuous mapping f of S into O ' X P'~; it is then easily seen that f induces a holomorphic mapping on every subvariety of complex dimension 1 of any open subset of S; from this, using elementary results on functions of several complex variables, one deduces easily that f is holomorphic on S itself. This completes the proof. The same method can be applied to the determination of all positive divisors in a given homology class. Consider a homology class which contains at least one positive divisor Xo; let r " " ", r be a basis for the vector-space L [ W ( 0 ) - 4 - X o ] ; for every ~ ~ ~ ~ in that space, put Z ( ~ ) ~ ( ~ ) d - W ( 0 ) d- Xo, ~ being the point in pm with the homogeneous coordinates (~o, ' " ", ~ ) . Then we see, as above, that for every positive divisor X homologous to Xo there is one and only one point (u', ~) of O ' X P"~ such that Z(~) ~ W(u') d - X ; therefore this formula determines a one-to-one correspondence between such divisors X and the points (u',~) of | X P~ such that Z(~) }-W(uP). As we have shown, the latter relation determines an analytic subset of O ' X pro, which, since ~P and pm are compact, is the union of a finite number of subvarieties of | X pro; moreover, as | is an algebraic variety, these are algebraic subvarieties of O ' } ( P'~ by Chow's theorem ( [ 2 ] ) . This shows that

the positive divisors in a giv ~,n homology class on V ma/se up a finite number of algebraic families, i.e. of families parametrized by algebraic varieties; of course the latter may have singular points and so need not be complexanalytic manifolds, but are algebroid varieties in the sense defined at the end of no. 20. One may also observe that for a given u' in O' the points ~ of P ~

96

[1952e] 888

ANDI~I~ WEIL.

such that Z(~) ~ W(u') are those in a linear subvariety of P "~, corresponding to the subspaee L[W(O) ~- X o - W(u')] of the vector-space L[W(O) -[- Xo] ; in particular, each one of the above families of positive divisors may be considered as a fibre-variety over a subvariety of 0', the fibres being projective spaces.

w IV.

The higher Jacobian varieties.

24. I shall take this opportunity for making explicit some results implicitly contained in ttodge's book ([5] ; cf. also [6]), as this can be done very simply in the language which has been introduced above in w I and throws some additional light on the results of that w As before, let V be a K~hler manifold of complex dimension n; take p such that 0 ~ p =< n - - 1 ; call now Ao and A'o the homology groups of V over the real number-field R for the dimensions 2p ~ I and 2 n - - 2 p - - 1 respectively. These are vector-spaces of finite dimension over R ; call A and • the subgroups of Ao and of A'o consisting of those homology classes which contain cycles with integral coefficients (i. e. the images in Ao and in A',, of the homology groups of V with integral coefficients for the same dimensions) ; A is a discrete subgroup of Ao of maximal rank, i. e. of rank equal to the dimension of Ao, so that Ao/A is a torus of that same dimension; and the same is true of A'o and A' By the Poincar5 duality theorem, Ao and A'o are put into duality with each other by the intersection-number (or "Kronecker index "), which induces a bilinear form (x', x) on A'o X Ao ; moreover, by the same theorem, A aud A' are associated to each other in that duality, i. e. A' consists of all the elements of A'o such that (r', r ) ~ 0 mod I for all r s A. We may therefore identify A'o with the dual space to Ao, i. e. with the cohomology group of V of dimension 2p d - 1 with real coefficients, and similarly Ao may be identified with the cohomology group of V of dimension 2 n - - 2 p - - 1 with real coefficients. In particular, Ao and A'o may be identified with the de Rham groups of degrees 2n--2p--1 and 2p ~ - 1 respectively; by the de Rham group of degree d, we understand the group of closed differential forms of degree d on V modulo the exterior differentials of forms of degree d - - 1 . Then • A' are the classes of forms of degrees 2 n - - 2 p - - 1 and 2p @ 1 with integral periods; and if elements x, x' of Ao, A'o are the cohomology classes of two closed forms % ~' of respective degrees 2 n - - 2 p - - 1 and 2 p - t - 1 , we have

(x',

x} ~ IJ "g (oP(o.

[1952e]

97 ON

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VARIETIES.

889

By Itodge's existence theorems, each cohomology class contains one and only one harmonic form; so Ao may also be identified with the space of harmonic forms of degree 2 n - - 2 p - - 1 . As in no. 7, consider the operator C derived from the complex structure of V; if ~ is a form of degree d, we have C(Co)) = ( - - 1 ) % , so that C induces an operator such that 6 ' 2 = - - I on forms of any odd degree. As C commutes with the operator • ~ d3 if- ~d, it transforms harmonic forms into harmonic forms; call J the operator induced by - - C on the space Ao of harmonic forms of degree 2 n - - 2 p 1; this is an endomorphism of Ao satisfying J ' ~ I ; therefore Ao must be of even dimension, and J can be used to define a complex structure on Ao ; with that structure, Ao becomes a vector-space over the field of complex numbers, which will be denoted by A. In a similar manner, A'o becomes a vectorspace A' over complex numbers by means of the operator J' induced by - - C on harmonic forms of degree 2p q-1. Let o~, ~' be two harmonic forms of respective degrees 2 n - - 2 p - - 1 , 2p q - 1 , as C is an automorphism of the ring of differential forms, we have C ( J o ~ ) ~ C~o'. Coo; as every form ~ of degree 2~ can be written as fdz~dh~.. 9dz,,dh~ in terms of local coordinates, we have C~ ~ ~ for such a form, and in particular C(Jo~) = ~',~. This gives J % ' . Jo)~oAo, whence <J'o/, J~> ~ <J, ~>, or, if we replace ~o by J-l~, <J%',~> = < d , J - ~ > ; this shows that J" is the transpose of J-~ for the duality determined by the bilinear form <x', x>. According to our definitions in w I, no. 1, this makes A" the " a n t i d u a l " space of A; and, if we put 0 ~ A / • 0 ' ~ A'/A', these are two complex toruses, dual to each other. ~5. We shall now give a definition of d, as an endonlorphism of the de Rham group of degree 2n - - 2p - - 1, whieh depends only upon the complex structure of V and not upon the ehoiee of a K~hler mettle; the existence of a K~hler metric on V, however, is essential to our purposes. Take any d such that 0 _< d--< 2~; in eaeh eohomology class of degree d, there are forms such that do ~ d(C,~) = 0; in fact, the harmonic form of that class has that property. We now prove that, for d~ = 0 and d ( C ~ ) ~ O, the homology class of Cm depends only upon that of o. In fact, assume that o~is homologous to 0 ; let ~ be any harmonic form of degree 2e--- d, and put ~' = C-1~; then ~' is harmonic; we have ~ % ~ C ( ~ ' ~ o ) since ~'~ is of degree 2n, and C(~%) ~ C~" C~o~ ~(Co~), and so ~'~ ~ ~(Co). As 0~ is homologous to 0, we

have.[, ~'o~0;

this gives j ;

~ ( C ~ o ) = 0 ; as this is true for all har-

monie forms ~, i.e. for representatives of all eohomology classes of degree

98

[1952 e ] 890 2n--d,

AND~ WEIS. and as C~o is closed, it implies that C(o is homologous to 0. 7 I f now

x is a cohomology class of degree d, and ~o is a representative of that class such that do~ = 0 and d(Cr

~ O, we shall denote by Cx the class of the form

C(o; this is an endomorphism of the de t l h a m group of dimension d which satisfies C ~ = ( - - 1 ) a I and depends only upon the complex structure of V; for d = 2 n - - 2 p - - 1 , the operator J defined above is J = - - C . The complex torus O defined above by means of the cohomology group Ao of degree 2 n - - 2p - - 1, oi the complex structure determined by J, and of the subgroup • of classes of Ao with integral periods, will henceforth be denoted by | with this notation, the torus denoted above by O~ now appears as O,_~_~. Instead of A, • we now write A(P), A(P), so that Op = A~)//x(p). Thus, if V is any connected compact complex-analytic manifold of complex dimension n on which there exists at least one K~hler metric, we have defined a sequence | | ", | of complex toruses invariantly attached to it, in such a way that O~ and O~_~_~ are dual to each other for 0 de Ch. Ehresmann ~quivaut au cos particulier de la th6orie expos~e ci-apr~s qui s'obtiendrait en se restreignant 6 la seule consideration des alg~bres locales isomorphes 6 des alg~bres Rn{rn) ; cette restriction a l e grave inconvenient que les sousalg~bres, les alg~bres quotients, et (ce qui est essentiel dons la th6orie du proIongement) les produits tensoriels d'alg~bres de ce type ne sont plus des algebras du m~me type. 2. Soit V une vari~t~ diff~rentiable (dons tout ce qui suit, on convient de dire > au lieu de > ou autrement dit > et m~me le plus souvent d'omettre ce qualificatif ; d~signons par D (V) I'alg~bre des fonctions num~riques diff~rentiables sur V. Soitx un point de V ; soit I (x) I'idEal maximal de D (V) form~ des fonctions nulles en x ; alors I (x)rn*'est I'id~al de D (V) form~ des fonctions qui s'annulent en x ainsi qua toutes leurs d~riv~es d'ordre ~ m, c'est-6-dire qui ant en x un contact d'ordre m avec la constante 0 ; I'alg~bre quotient D (V) I (x)rn§ ~qu'on d~signera par DCrn)(v ; x), est une alg~bre locale isomorphe 6 Rn(rn)sJ o est la dimension de V ; on obtient un isomorphisme entre ces algebras en choisissant sur V un syst~me de coordonn~es locales (une >) au voisinage de x . On observera que le quotient de D (V) par I'id~al f'orm~ des fonctions qui s'annulent en x ainsi qua TOUTES leurs d~rivEes est isomorphe 6 Rn ; autrement dit, il existe sur V des fonctions diff~rentiables dont le d~veloppement (formal) de Taylor au point x est une s~rie formelle ARBITRAIRE dons les coordonn~es locales au point x ; mais nous n'aurons pas 6 f'aire usage de ce fair. Si x est donn~, on pourraJt, dons tout ce qu'on vient de dire, remplacer D (V)

[19531

105 113

par I'alg~bre des germes de fonctions diff6rentiables en x sans qu'il y e0t rien de chang6. DI~FINITION. -- SOIT A UNE ALGEBRE LOCALE. SOIENT V UNE VARII~T{: DIFF~RENTIABLE, D (V) L'ALG~BRE DES FONCTIONS NUM~RIQUES DIFF~RENTIABLES SUR V, x UN POINT DE V. ON APPELLERA A-POINT DE V PROCHE DE x , OU POINT PROCHE (OU INFINIMENT VOISIN DEx D'ESP~-CE A SUR V TOUT HOMOMORPHJSME DE D (V) DANS A TEL QUE LA PARTIE FINIE DE L'IMAGE DE f E D (V) DANS A SOIT f (x). Si x' est un tel point, 1'616ment de A que x' associe 6 f E D (V) sera d6sign6, soit par f (x'), soit plus explicitement par I'une des notations suivantes : Af(~') dans ceMe derni~re, x doit f~tre r

,

A~'f(x) comme ~ variable li~e

>>.

Si I est

I'id~al maximal de A, on a par d~finition f (x') ~ f (x) rood. I. Lqmage dans A, par rhomomorphisme f ~ f (x'), de I'id6al I (x) des fonctions nulles en x est contenue dans I, donc I'image de I (x)m§ contenue dans Im*~ et est donc {0} si A est de hauteur m . Autrement dit. tout A-point x' proche de x d6termine si A est de hauteurm, un homomorphisme da'ns A de I'alg~bre D(m)(v ; x), et

r~ciproquement.

Si on 6crit f (x') ---- f (x) -L L (f), f--,- L (f) est une application de D (V) dans I, lin6aire sur R ; dire que f ~ f (x') est un homomorphisme revient alors dire que I'on a, quels que soient {', g :

L(fg) = L (f)g (x)+ f(x) L(g)+ L(f) L(O) donc, r6ciproquement, toute application lin6aire de D (V) dans I ayant cette propri6t6 d6finit un A-point de V proche de,,". En particulier, si A est de hauteur 1, on aura L (f) L (g) ---- 0, et la relation ci-dessus se r6duit 6 L (fg) =

L (f) g (x) § f (x) L (g)

Si, en particulier, I est de dimension 1, engendr6 par un 616ment'z satisfaisant ~ "r2= 0 (A est alors I' >), on voit que la notion de A-point de V proche d e x est identique ~ la notion de vecteur tangent V e n x . C'est I~ essentiellement le point de rue de Fermat dans ses travaux sur le calcul diff6rentiel (Fermat empIo),ait la lettre o I~ o0 nous 6crivons~c). On notera clue I'homomorphisme canoniclue de D (V) sur I'alg~bre locale D{m) (V ; x) d6finit un point proche dex d'esp~ce D (m) (V ; x), dit POINT PROCHE DE x C A N O N I Q U E DE R A N G m . Si on prend pour V la droite num6rique R, et clue x' soit un A-point de R proche de x E R, x' est compl~tement d6termin6 par la connaissance de 1'616ment I. (x') de A que x' associe ~ I'application identique ~ de R sur R consid6r6e comme 616ment de D (R). On conviendra d'identifier x' avec 1'616ment ~ (x') de A: Si I' est une application diff6rentiable d'une vari6t6 V dans une vari6t6 W, g - - g o ~ est un homomorphisme de D (W) dans D (V) ; composant celui-ci avec I'homomorphisme de D (V) dans A ClUi d6finit un A-point x' de V proche

106

[1953] 114 de x E V, on obtient un A-point de W proche de ~ (x), qu'on notera ~ (x'), ou plus explicitement A ~ (x'). 3. On peut, d'une mani~re >, d~finir une structure de vari~t~ diff~rentiable, et plus pr~cis~ment de vari~t~ diff~rentiable fibr~e de base V, sur I'ensemble des A-points de V proches de points de V, A ~tant une alg~bre locale donn~e ," muni de cette structure, I'ensemble enquestion s'appell~ra LE PROLONGEMENT DE V D'ESPI~CE A et se notercr AV.'.Plus g~n~ralement, supposons qu'on se donne une vari~t~ fibr~e de base V, dont les fibres soient des alg~bres isomorphes 6 une alg~bre locale fixe A ; si A (x) est la fibre appartenant 6 x E V, consid~rons rensemble des A (x) -points de V proches de x ; la r~union de ces ensembles, quand x d~crit V, pourra encore ~.tre munie d'une structure de vari6t~ fibr6e de base V, et s'appeilera encore un prolongement de V. Si U et V sont des vari~t~s, on v6rifie imm~diatement que le prolongement d'esp~ce A de la vari~t~ produit U • V s'identifie canoniquement au produit des prolongements d'esp~ce A de U et de V : A A A (UxV) = Ux V Soit ~ une application d'une vari6t~ V dans une vari~t~ W ; si x' E ^V, c'est-6-dire s i x ' est un A-point de V proche d'un point x de V, on a d~fini plus haut le point A ~ (x'), qu'on peut ~crire aussi ~o (x'), comme un A-point de W proche de ~0 (x). On v~rifie que x' --,- A ~o(x') est une application diff~rentiable de^V d a n s ^ W, qui s'appelle le A-prolongement de ~o ; elle se noteA~(ou simplement ~ par abus de langage quand aucune confusion n'est ~ craindre). On notera que, quelle que soit I'alg~bre locale A, on a convenu de consid~.rer R comme plong~e dons A, et que, dons ces conditions, s i x E V, I'application f----* f (x) de D (V) dons R C A d~finit un A-point de V proche dea,, qu'on Jdentifiera toujours avec .~" ; de cette mani~re V se trouve identifi~e ovec une sous-vari~t~ deAV. Si W est aussi une varietY, elle sera de m~.me Identifi6e avec une sous-vari~t~ de ^ W ; et, si ~0est une application de V dons W, i[ est imm~diat que/~ T induit sur V, consid~r~e comme sous-vari(:t~ de AV, I'application ~ode V sur W consid~r~e comme sous-vari~t~ de ^ W. 4. Soit, par exemple, (x, y)--,, f (x, y) une Ioi de composition interne d6finie sur une vari~t~ V, c'est-6-dire une application de V X V dons V ; son prolon. . gement A fest une apphcahon de A V •- ' A V dons A V, c , est-6-dwe une Ioi de composition sur A V ; il est imm~diat que, s i f est associative (resp. commutative), il en est de m~me deaf. Si Vest un groupe de Lie, le prolongement deAV de la Ioi de composition sur V fait deAV un groupe de Lie. Si Vest la droite num~rique munie de sa structure de corps, on peut prolonger 6A Vles lois de composition additive et multiplicative de V ; mais on a vu que dans ce casAV s'identlfie avec I'alg~bre A ; on v~rifie imm~diatement que les lois ainsi obtenues sur A V ne sont autres que I'oddition et la multiplication de I'alg~bre A. Supposons qu'on prenne pour V un espace vectoriel (de dimension finie) sur R. Le prolongement 6AV de I'addition des vecteurs dans V d~termine surAV une structure de groupe ab~lien ; d'autre part, on peut prolonger d A(RxV)

=

ARxAV=

AxAV

[1953]

107 115

I'application (k, x)--4- )~ x de R • V dens V. Au moyen de ces deux lois, on d6finJt surAV une structure de A-module ; on v6rifie facilement (par exemple au moyen d'une base de V) que ce module s'identifie canoniquement avec le produit tensoriel A | V consid6r6 comme A-module. Si de plus on s'est donn6 sur V une multiplication qui fasse de V une alg~bre commutative sur R, le proIongement de cette Ioi b ~V d6termine sur A V, avec les lois pr~c6dentes, une structure d'alg~bre sur A pour laquelle ^ V s'identifie encore canoniquement avec A | V muni de la structure correspondent| 5. On est alors en 6tat de d6montrer le th~or~me fondamental de TRANSITIVITI~ DES PROLONGEMENTS sous la form| pr6cise suivante : THI-'OR|:ME. - - SOIENT A ET B DEUX ALG~BRES LOCALES, V UNE VARI~TI~. IL EXlSTE UN ISOMORPHISME CANONIQUE ENTRE LE PROLONGEMENT D'ESPJ~CE A 9 B DE V ET LE PROLONGEMENT D'ESP~CE A DU PROLONGEMENT D'ESP~-CE B DE V. Soit en effet f E D (V) ; f" se prolonge en une ~pplicationef de e V dans eR ---- B. Ruis celLe,-ci en une application ~(ef) de A ( V ) dansAB = A 9 B ; si x" E: A (BV) ' A(f) (x") est donc un 616ment de A 9 B. ~emontant aux d~finitions, on volt imm6diatement que, si x" | donn6, f - - - A ( f ) (x") | un homomorphisme de D (V) dans A 9 B ; deplus, si x " est proche de x' E eV et x' proche de x ~ V, la partie finle d e ^ ( l f ) (x '), consid6r6e comme 616ment de I'alg~bre locale A | B, | f('x) ; I'homomorphisme en question d6finit donc un (A | B) -point z' proche de x sur V ; et x " ~ z' est, dans ces conditions, une application canonique de ^(I~V) dans le prolongement de V d'esp~ce A 9 B. Rest| cl montrer que cette application est un isomorphisme de la premiere varlet6 sur la seconde ; comme c'est 16 une propri6t~ purement locale par. rapport 6 la base V, il suffit de la v6rifier Iorsque Vest une pattie ouverte d'un espace vectoriel E ; reals, en ce cas. ,~lle r6sulte imm6diatement des isomorphismes pr6c6demment ~tablis entre A ( E ) et A 9 (B | E), d'une part, et entre le (A e B) -prolongement de E et (AeBeE),d'autre part, et de I'associativit~ du produit tensoriel. 6. La notion classique de transformation infinit~simale se g6n6ralise ici comme suit. Si A est une alg~bre locale, et V une vari~t6, une transformation infinit6simale d'esp~ce A sur V "sera, au sens de la th6orie des espaces fibr6s, une section (diff6rentiable) de la varlet6 fibr6e A V de base V, ou autrement dit une application dlff6rentiable de V dans son prolongement AV qui, 6 tout point x sur V, fasse correspondre un A-point x' proche d e x sur V. Si A est I'alg~bre des nombres duaux (n ~ 2), la connaissance de x' 6qulvaut cl celle d'un vecteur tangent 6 V en x, et on retrouve la notion classique. Plus g~n6ralement, une A-apRlication d'une vari6t6 V dens une vari6t6 W sera une application F de V dans^ W ; si, pour x E V, f (x) | le point de W dont F (x) est proche, on dire que F | proche de I'application ?de V dans W. Une transformation Jnfinit~simale est donc une A-application proche de I'application identique. Soient U, V, W des vari~t~s ; soient F une A-application de U dens V, et G une A-application de V dens W. Alors ^G o F est une (A 9 A) -applica-

108

[1953] 116 tion de U dons W, qui fait donc correspondre 6 tout x E U un (A | A) -point de W. Mais I'alg~bre A | A admet un homomorphisme canonique sur A, donn~ par a o a ' ~ aa' ; si on le d~signe paro-, le compos~ de cet homomorphisme avec I'homomorphisme d~finissant un (A 9 A) -point w' de W proche de w E W d~flnira un A-point de W proche de w, qu'on notera o'ww', ou, s'il n'y a pas de confusion 6 craindre, o'w' (plus g~n~raiement, si ~ est un homomorphisme d'une algabre locale A dons une alg~bre locale B, le compos~ de )~ et d'un A-point x' d'une varlet6 X est un B-point de X'not~ XxX' ou "h x', dit sur tout espace fibr~ d~fini d'une mani~re intrins~que 6 partir de la varlet6 de base. D'une mani~re precise, ces espaces fibres sont d~finis comme suit. Salt V une vari6t~ de dimension n ; Rn(m) ~tant I'alg~bre locale d~finie au n ~ 1, 0n point d'esp~ce Rn~m)de V, proche de x E V, est (n ~ 2) un homomorphisme de D (m) (V ; x) clans R n(rn) ; si cet homomorphisme est un isomorphisme de D (m) (V ; x) sur R, cm), le point en question s'appellera un REP~:RE D'ORDRE m EN X SUR V. L'espace des reputes d'ordre m sur V appara/t ainsi comme pattie ouverte du prolongement d'esp~ce Rn(mj de V. Mais, sur cet espace, on peut d~finir une structure d'espace fibr~ principal de base V, le groupe ~tant le groupe de Lie des automorl~hismes de I'alg~bre Rn(m) ; muni de cette structure, cet espace sera not~ PfmJ(v). Tout espace fibr~ de base V associ~ 6 un espace principal p(m)(V) pour un m convenable sera dit C A N O N I Q U E M E N T A'I-I"ACH~ A V (ces espaces sont les e prolongements >7 de V au sens d Ehresmann). C'est sur ces espaces qu'op~rent e naturellement 77 les transformations infinit~simales. La d~finition de P(m)(V) comme pattie ouverte du prolongement de V d'esp~ce Rn(m) permet d'appliquer commod~ment les principes introduits ci-dessus 6 r~tude de ces op~rateurs. II est impossible d'entrer ici clans le d~tail des r~sultats d~j6 obtenus, dont certains sont de d~monstration assez difficile. Indiquons simplement que les notions ci-dessus permettent de loire appara;tre, par exemple, le crochet des chomps de vecteurs comme un cas particulier de la formation du commutateur, les propri~t~s formelles du premier se d~duisant donc directement de celles du second. En particulier, I'application 6 la th~orie des groupes de Lie permet de donner une forme naturelle aux relations entre commutateur et crochet, o0 I'on n'a plus ~ faire intervenir les O. Corollaire 1. Si ~ est comme ei-dessus, a (~'~) - 0 entra~ne ~ = O. Corollaire 2. Soit 2 u n homomorphisme de la ]acobienne J d'une eourbe C dane une varidtd abdlienne A ; soit X un diviseur Bur A. Alors 2 ne ~'annule qu'en u n h o m b r e / i n i de points de l'image 2/cA de A par 2~: dane J. I1 r e v i e n t au m 6 m e de dire que, si B e s t la c o m p o s a n t e de 0 dans le n o y a u de l ' h o m o m o r p h i s m e 2'x de A dans J , et B ' la c o m p o s a n t e de 0 d a n s le n o y a u de l ' h o m o m o r p h i s m e )~2~ de A dans A, on a B ' - B. Sinon, en effet, il y a u r a i t une courbe c o n t e n u e dans B ' et non dans B, courbe q u ' o n p o u r r a i t 6crire / ( C ' ) , oh C' est une courbe a b s t r a i t e (compl6te sans p o i n t multiple) et / une a p p l i c a t i o n de C' dane A. Soit # l ' e x t e n s i o n de / s la j a e o b i e n n e J ' de C'; on a u r a alors Zx,U~C0 et ) , Z x # 0. E n p o s a n t ~ = 2 x / , on a u r a ~ ' = ,u~ 2 (t'apr6s le n ~ 1, done ~'~ = 0, ce qui i m p l i q u e ~ = 0 en v e r t u du eoroll. 1, d ' o h c o n t r a d i c t i o n . Corollaire 3. Soient J~, ,1~ les jacobiennes de deux courbes C1, C~; soit une classe de correspondances entre C~ et C~, c'est-h-dire u n homgmgrphisme de J1

[1954d]

129 CritSres d'~quivalence en g6om~trie alg6brique.

97

dans J2. Alors ~ ne s'annule qu'en un nombre /ini de points de l'image ~'J2 de J2 par ~' dans J r C'est 1s en effet le cas particulier du coroll. 2 qu'on obtient en y rempla~ant C, J , A, X, 2 p a r C1, J1, J2, 02, ~ respectivement. 3. Pour la commoditd du lecteur, nous donnerons aussi la d~monstration du r~sultat suivant, qui est connu 1) : Lemme 1. Soit V '~ une sous-varidtd d'un espace projecti/ pN. Soit L un hyperplan de p.v, gdndrique par rapport ~ un corps de dd/inition k de V. Alors le cycle V . L e s t sans composante de multiplicitd > 1 ; s i n >= 2, il n'a qu'uue seule composante W, et les points multiples de W sont les points de L qui sont multiples sur V e t ceux-lh seulement. Soit x un p o i n t de V ~ L, simple sur V; il rdsulte du crit~re de multiplicit~ 1 (F-VI2, th. 6) que, si L ne contient pas la varidt6 lin6aire t a n g e n t e V en x, x est contenu dans une seule eomposante W de V ~ L e t est un p o i n t simple de W, et que W a la multiplicit6 1 dans V 9 L. Disons suivant l'usage q u ' u n h y p e r p l a n est t a n g e n t ~ V en x s'il contient la vari6t6 lin6aire t a n g e n t e b, V en x; nous allons m o n t r e r que L ne p e u t 8tre t a n g e n t s V en aucun p o i n t simple x de V. Supposons p a r exemple que x ne soit pas contenu dans l ' h y p e r plan X 0 = 0 (en d6signant p a r X 0. . . . . X.v les coordonn~es homog~nes dans pN) ; p r e n a n t celui-ci pour ((hyperplan s l'infinb~, passons s l'espace affine S "v. Dans cet espace, soit F , ( X ) = 0 (1 ~ / z =< m) un syst~me d'~quations pour V N

s coefficients dans k; l'~quation de L sera ~

u i X i - v = 0, oh (u, v) est un

i=l

syst~me de N + 1 variables ind~pendantes sur k. L a vari~t~ lin~aire t a n g e n t e s V en x est d~finie p a r les ~quations V ' ~ F , / ~ x i . ( X i - x i ) = O ; si L e s t i

t a n g e n t e s V en x, il y aura des quantit~s z, telles que u i = ~ z, (~ F,/O xi) , et on aura v = ~ u~x~. Soit ~ un p o i n t g~n~rique de V p a r r a p p o r t ~ k; soient 5 m variables ind~pendantes sur k (~); posons ui= ~ 5~(~F,/O xi), = ~ ~i~i; (x, z, u, v) est alors une sp6cialisation de (u ~, ~, ~) sur k; puisque (u, v) a la dimension N § 1 sur k, (~, ~) doit donc avoir au moins cette dimension. Mais la matrice i I ~ F / ~ xi ]1 est de r a n g N-n, et on p e u t donc ~-n

supposer les F , ranges dans un ordre tel que l'on ait ~ FJO xi = ~ w ~ (~ F,/~ ~ ) , 2r

avec ~ , , ~ k (~), quels soient # et i; on en d~duit u i = . . ~ t ~ ( ~ F ~ / ~ x i ) , avec ~ - - ~Y~~ ;

on a alors k(~, ~) = k (~, [), ce qui montre que k (~,~) a au plus

la dimension n § (N-n) = N sur k. ~) Cf. O. Z~R~SKI, Trans. Am. Soc. ~0 (1941), pp. 48--70, et 56 (!944), pp. 130--140, et aussi T. M~TSUS~K~, Kyoto Math. Mem. 26 (1950), pp. 51--62, et Y. NXK~I, ibid., pp. 185--187. 7*

130

[1954d1

98

ANDR~ W E I L :

Soit d ' a u t r e p a r t x u n p o i n t e o n t e n u d a n s u n e seule e o m p o s a n t e I f de V 9 L, et simple sur I f ; et supposons que W ne soit pas u n e sous-varigt6 m u l t i p l e de V; d'aprgs ee qui pr6c~de, I f a alors la m u l t i p l i c i t 6 1 d a n s V ' L ; p a r suite, d'apr~s le erit~re de multiplicit6 1 (F-VI~, th. 6), x est simple sur V. I1 nous reste s e u l e m e n t ~ d ~ m o n t r e r que, si n ~ 2, V 9 L, ou ee qui r e v i e n t a u m g m e V & L, a u n e seule eomposante, n o n m u l t i p l e sur V. S u p p o s a n t par exemple V n o n e o n t e n u e d a n s X 0= 0, passons k l'espaee affine en p r e n a n t X 0 = 0 pour h y p e r p l a n ~ l ' i n f i n i ; le th. 1 de F - V 1 m o n t r e que V ~ L n ' e s t pas vide (dans SN). Soit ~_~'u i X i - v = 0 l ' 6 q u a t i o n de L ; soit K = /c (u, v); soient I f u n e comi

p o s a n t e de V ~ L dans S ~v, et x u n p o i n t g6n6rique de lV par r a p p o r t ~ K. La prop. 2 de F - V 1 m o n t r e que x est g6n6rique sur V par r a p p o r t ~ ]c (u) et n ' e s t done ni m u l t i p l e sur V n i e o n t e n u d a n s u n h y p e r p l a n X i = 0 s moins que V n ' y soit e o n t e n u e ; il s ' e n s u i t de m 6 m e que, d a n s l'espaee projeetif, V ~ L n ' a a u e u n e e o m p o s a n t e d a n s l'i~yperplan X 0 = 0 puisque eelui-ei n e e o n t i e n t p a s V, done q u ' o n o b t i e n t routes les e o m p o s a n t e s de V ; v L d a n s l'espaee p r o j e e t i f en r a i s o n n a n t d a n s l'espaee affine S x. Mais, t o u j o u r s d ' a p r 6 s la prop. 2 de F - V D V ~ L n ' a d ' a u t r e s eomposantes (dans S N) que I f et ses eonjugu~es sur K ; p o u r m o n t r e r que I f est l ' u n i q u e e o m p o s a n t e de V & L, il suffira done de d 6 m o n t r e r que x a u n lieu sur K, ear ee lieu ne p e u t ~tre a u t r e alors que W. D'ailleurs, d'apr6s ee qui pr6e6de, W a la multiplieit6 1 d a n s V 9 L, e t a done, d'apr6s F-V2, prop. 14, l'ordre d ' i n s d p a r a b i l i t 6 1 sur K, e'est-~-dire que K (x) est u n e e x t e n s i o n s6parable (au sens de N. BOVRBAKI, A l g . , Chap. V, ou encore ((s6parablemen~ engendr6e>> a u sens de F-I) de K = k (u, v). D ' a p r 6 s F-Iv, th. 5, t o u t r e v i e n t done ~ m o n t r e r que K = ]c (u, v) est a l g 6 b r i q u e m e n t ferm6 d a n s K (x)l); d'ailleurs, p u i s q u e v = ~ u i x i , on a K (x) = ]~ (u, x).

i

Soit K 1 la f e r m e t u r e algdbrique de K d a n s K (x); c o m m e K (x) est s6p a r a b l e sur K , il l'est a fortiori sur K i e t est done u n e e x t e n s i o n rdguligre de K i (d'apr~s F-IT, th. 5). Soient t, t' d e u x variables i n d @ e n d a n t e s sur K (x) = K l(x) - k (% x); soit L = / c (u, t, t'); alors L (x) = K l(x, t, t') est u n e ext e n s i o n rdguli~re de L 1 - K i ( t , t ' ) , qui est algdbrique sur L ( v ) K ( t , t'); donc L 1 est la f e r m e t u r e alg6brique de L (v) d a n s L (x); de plus, comme K C K i ( L i , et que K est a l g 6 b r i q u e m e n t fermd d a n s L (v) = K (t, t'), on ne p e u t avoir L 1 = L (v) que si K = K 1 ; donc t o u t r e v i e n t ~ m o n t r e r que L 1 = L (v). Supposons les coordonn6es rang6es d a n s u n ordre tel que/c (x) soit alg~brique s @ a r a b l e sur ]c (x I . . . . . xn). Ddsignons p a r ~ l ' a u t o m o r p h i s m e de L (x) = lc (u, x, t, t') qui laisse i n v a r i a n t s t o u s l e s 616ments de/c (u, x) et qui 6ehange t et t'; d6signons p a r a l ' a u t o m o r p h i s m e d u m 6 m e corps qui laisse i n v a r i a n t s t o u s l e s 6ldments de lc (u~ , . . . , u v , x, t, t') e t t r a n s f o r m e u i en u i + t; posons y = v ~ = v + t x 1 et y ' = v ~ = v + t ' x 1. On & i a = L, doric L (v) ~ = i a ( v ~) = L (y); p a r suite, la fermeture alg6brique de L (y) d a n s L (x) est M = L~, et de m ~ m e eelle de L (y') d a n s L (x) est M ~ = L~ ~ . O n a, d a n s ees c o n d i t i o n s : L (xl, v, x~ . . . . .

x~) = L (y, y', x a. . . .

, x~) ~ M

(y', x~ . . . . .

x~) ~ L (x).

i) Ce r~sultat, et la d6monstration qu'on va en donner, sont emprunt6s en substance O. ZA~SKI, Trans. Am. Math. Soc. 50 (1941). lemma 5, p. 68- g9.

[1954d]

131

Crit6res d'6quivalence en g6om6trie alg6brique.

99

Posons K 0 = k (u); on d 6 d u i t a i s 6 m e n t de F-IVG, prop. 24, que K o ( x ) est alg6b r i q u e s6parable sur K o ( x l , v, x a , . . . , x,~). Au m o y e n de la th6orie de GALOIS (N. BOURBAKI, Alg., Chap. V, w 10, n ~ 4, th. 1, cot. 1), on en eonelut que t o u t corps i n t e r m 6 d i a i r e e n t r e L (x~, v, x 3 , . . . , x,,) et L (x) eat de la f o r m e N (t, t'), oh N eat u n corps i n t e r m 6 d i a i r e entre K o ( x l , v, xa, . . . , Xn) et K 0 ( x ) ; un tel corps eat done t r a n s f o r m 6 en l u i - m 6 m e par r. E n partieulier, le corps M (y', x a, . .., x~) a done e e t t e propri6t6, c'est-s qu'on a M (y', x~ . . . . , x~) = M r ( y , x a. . . . .

xn) 9

Si done on pose M ' M T(x 3. . . . . x~,), on a u r a M ( M ' ( y ) . On a d ' a u t r e p a r t L ~ M ' ( L (x), done, c o m m e L (x) eat une e x t e n s i o n r6guli~re de L, il en eat de m~.me de M ' . C o m m e d'ailleurs le corps L ( y , y', x s , . . . , x~) n ' e s t a u t r e que L (x~ . . . . , x,,) et eat done une e x t e n s i o n p u r e m e n t t r a n s e e n d a n t e de L de d i m e n s i o n n, y est t r a n s e e n d a n t sur L (y', xa, . . . , x,~), done aussi sur M ' qui en eat u n e e x t e n s i o n alg6brique. I1 s ' e n s u i t que M ' ( y ) eat une e x t e n s i o n r6guli6re de L (y); e o m m e L (y) C M C M ' (y), et que M est alg6brique sur L (y), on en c o n c l u t que M = L (y), done L 1 = L (v), ce qui aeh6ve la d 6 m o n s t r a t i o n . 4. L a vari6t6 V e t te corps k 6rant c o m m e d a n s le l e m m e 1, soient L~ des h y p e r p l a n s d6finis r e s p e e t i v e m e n t p a r les 6quations ~7 u ~ i X i = O, p o u r i 1 < v~ . . . c~ L~, d o n e en p a r t i e u l i e r M 0 = p.v ; M~ eat u n e vari6t6 lin6aire de d i m e n s i o n N v, g6n6rique sur k, et on a M , = My_ 1- L v. Soit Vo- V, et d6finissons V, p a r r6currence par V ~ - V, 1" L , ; on v6rifie imm6d i a t e m e n t , par r6currence sur v e t en v e r t u de l ' a s s o c i a t i v i t 6 des intersections, que V ~ - V ' M , . D a n a ees conditions, l ' a p p l i c a t i o n du l e m m e 1 donne, par r6currence sur v, Ie r6sultat s u i v a n t : L e m m e 2. S o i t V '~ une soua-varidtd d ' u n espace projecti/ pA'; aoit M u n e varidtd lindaire de d i m e n s i o n N - r >~ N - n dana P~Y, gdndrique aur u n corps de d d / i n i t i o n k de V. A l o r s le cycle V " M eat dd]ini, et, p o u r O N - n § 1, i

le cycle V . L~ est dd/ini et est une varidtd; s i m composante multiple.

> N-

n, il est dd/ini et sans

C'est t r i v i a l s i n = 0 ou 1 ; nous proc6derons par rdcurrence s u r n . ce q u ' o n a v u plus haut, V ' L ~ est en t o u t cas d 6 f i n i s i r e > N - n ;

D'apr~s soient

[1954d]

133 Crit~res d'SquivMence en g~omStrie MgSbrique.

101

alors Z 1. . . . , Z r les e o m p o s a n t e s de V" L w; on a V' L w = ~Y' a~Ze; nous d e v o n s montrer entralne pouvons k (w),

que m > N de plus r = 1. supposer m < donn6 p a r une

n entrMne aQ= 1 p o u r t o u t ~, et que m > N - n + 1 C'est v r a i p o u r m = 5" d'aprgs le l e m m e 1; done nous 5:. Soit H u n h y p e r p l a n de P-Y, gSn6rique p a r r a p p o r t 6 q u a t i o n ~ z i X i= 0, oh z = (% . . . . . z v ) est u n p o i n t i

de p,~v g6n6rique sur lc (w). D ' a p r g s le l e m m e 1, V" H est une vari6t6 V ~ - : ; le cycle ]7 est r a t i o n n e l sur K = ~ (z), done V e s t d6finie sur K. D'apr~s le l e m m e 1, Z o = Zo. H est une vari6,t6 s i n > 2, une s o m m e de points d i s t i n c t s si n = 2. Mais L = L ~ . H est une vari6t6 lin6aire, et on a H . ( V ' L w ) = (H 9 V) 9L w = V ' L~, puis {V. Lw}pN = { V ' L } H d ' ap r "e s F - \ ' II6, th. 18, eor.; done on a, dans H , V" L = ~Y~ aQZo. P r e n a n t (X: . . . . . X~v) c o m m e coordonn6es e N

homog~nes dans H, soit _37 ~ i X i = 0 l ' 6 q u a t i o n de L dans H ; soit W le lieu du p o i n t de coordonn6es homog~nes ( ~ : , . . . , ~ - ) par r a p p o r t ~ /~ dans un espaee p r o j e c t i f de d i m e n s i o n N - 1 ; soit N la d i m e n s i o n de W. L ' h y p o t h g s e de r6currence, appliqu6e s V, W , L , m o n t r e que ~ > ( N - 1 ) - ( n - 1) =N-n e n t r M n e a o= 1 p o u r t o u t Q, et que N > N - n + 1 e n t r M n e de plus r = 1. P o u r a c h e v e r la d 6 m o n s t r a t i o n , il suffira de faire voir que ~ = m. P o u r eela, soit w le p o i n t (0, ~bl, . .., N~,) dans p,2V ; c o m m e l ' 6 q u a t i o n de p e u t s'6crire N

N

Zo ( ~ ' wiX~) - wo ( ~ z i X i ) = 0 i--0

i=0

les p o i n t s z, w e t ~ sont en ligne droite. Soit w' un p o i n t g~n~rique p a r r a p p o r t K (w) de la droite j o i g n a n t z et ~ ; si ~ a v a i t une d i m e n s i o n < m sur K, w' a u r a i t une d i m e n s i o n < m sur K. C o m m e w e s t une sp6eialisation de w' sur K (~), e t a fortiori sur K, e t a la d i m e n s i o n m sur K, w e t w' seraient alors sp~cialisations g6n6riques F u n de l ' a u t r e sur K, done w' serait un p o i n t du lieu W de w p a r r a p p o r t s K. L a d r o i t e j o i g n a n t z et N serait p a r suite cont e n u e dans IV; mais cela ne p e u t 6tre car on a m < N, done z n ' e s t pas dans W. P o u r n = 1, il n ' e s t pas possible d'am61iorer les inggali%s d u l e m m e 3; le thgor~me de KI~O~EOKEa-C~sTEL~I:OVO r e n d v r a i s e m b l a b l e q u ' o n p o u r r a i t les am~liorer p o u r n > 2 s c o n d i t i o n d ' e x e l u r e certaines vari6t6s (vari~t~s r6gl~es, vari6tSs de SEaRs, p e u t - 6 t r e d'autres). 6. Si V est c o n t e n u e d a n s une vari~t~ lin~aire L 0 de d i m e n s i o n N ' < N , on p e u t consid6rer L 0 c o m m e u n espace p r o j e c t i f de d i m e n s i o n N ' , et V e o m m e vari6t6 plong~e dans ee dernier. I1 s ' e n s u i t que, dans les q u e s t i o n s qui n o u s o c c u p e n t ici, on ne d i m i n u e pas la gSn5ralit6 en s u p p o s a n t que V n ' e s t cont e n u e d a n s a u c u n h y p e r p l a n , ni d o n e d a n s a u c u n e sous-vari~t~ lin~aire de p=v a u t r e que p~V; eela e n t r a l n e que V" L~ est dgfini quel que soit l ' h y p e r p l a n L~. N o u s ferons s o u v e n t c e t t e h y p o t h ~ s e d a n s ce qui suit. On salt q u ' a u m o y e n de la m 6 t h o d e des (ou ) on p e u t assoeier s t o u t cycle p o s i t i f Z de d i m e n s i o n r dans l ' e s p a c e p r o j e c t i f p N u n e f o r m e ~ r + 1 s6ries d ' i n d 6 t e r m i n ~ e s U~(~) (0 ~ i ~ N

134

[1954d] 102

A~D~ W~m :

0 ~ ~o~ r), de degr6 d par rapport g ehacune d'elles si d est le degr6 de Z; la somme de deux cycles correspond le produit des formes assocides; et, si on prend les coefficients de la forme associ6e s Z eomme coordonn~es homoggnes d ' u n point dane n n espace projectif p M les points de pM ~insi ussoci6s aux cycles de dimension r et de degr6 d forment u n > 3x, r,~ dane _P"S (e'est-s l ' u n i o n de sous-wri6t6s de p M en hombre fini), ensemble qui est normulement alg6brique sur le corps premier~); il s'ensuit que lee points de P~" correspondgnt aux cycles positifs de dimension r et de degr~ d duns p~v qui ne sont pas des vari~tgs (resp. qui out gu moins une composante multiple) forment uussi u n ensemble Mg6brique ferm6 3'~v, ~,~ (resp. 3~; ~, d) normulement alg6brique sur le corps premier. Soit alors V~ une vgri6tg de degr~ d duns P~Y, non contenue duns u n hyperplan; soit F (U (~ . . . . , U (~)) lu forme associ~e g V. Soit w u n point du dual P ' ~ de p~v ; soit L~ l'hyperplun de pN correspondent s w. I1 r6sulte des d6finitions que l~ forme assoei~e ~u cycle V" L~ est F (w, U(~ . . . , U(n-1)), forme qui n'est pus identiquement nulle puisque F" L~ est d6fini, et dont les coefficients sont lee coordonn6es homogSnes d ' u n point de 3~Y,~-Ld- Les points w de p,~v tels que ce dernier point soit d~ns 3~'~,~-1,d (resp. duns 3-~", n - L d) forment par consequent u n ensemble alg6brique ferm6, normalement ulg6brique sur le plus petit corps de d6finition de V; lc lemme 3 montre alors que cos compos~ntes sont de dimension ~ N - n + 1 (resp. ~ N - n). Autrem e n t dit : Lemme 4. Soit V '~ une sous-vari~t~ de l'espace pro]ecti[ P~Y, non contenue darts un hyperplan. Pour tout point w du dual p,~v de p x , soit L w l'hyperplan v' w i X i = O. Soit 9~ (resp. 9V) l'ensemble des poi~ts w de P"~ tels que le cycle A.u i

V" Lu ne soit pus une varidtd (seep. ait une composante multiple). A lors 9~ st ~ ' sont des ensembles alg~briques /ermds darts p,N, normalement algdbriques sur tout corps de dd]inition de V; et les composantes de 9~ (resp. de ~ ' ) sont de dimension _~ N - n § 1 (resp. ~ _N -- n). 7. Les lemmes snivunts sent destines s rendre plus m~ni~ble ]~ notion de diviseurs ~>, ou, comme nous dirons plut6t, ~lgdbriquement ~quivalents. Je dois ~ MAx ROSENLICHT la ddmonstration du lemmc 5. Lemme 5. Soient a, b (resp. a', b') deux points sur une courbe C (resp. C'). Alors il existe, sur la sur]ace C • C', une courbe passant par les points a • a' et b• Soient C, C' des courbes eomp]Stes sans points multiples, birutionnellement 6 q u i w l e n t e s respeetivement s C et s C'; on pourra ~,~irr C = [ (C), C ' = / ' ( ( ] ' ) ; et C' ~t~nt completes, i] y aura des points 5, b sur C, ~', b' sur C' t els que a = / (~), b = [ (b), a ' = / ' (~'), b ' = / ' (b'); ulors, si on cons~ruit sur C • C' une courbe p~ssunt par a • 5' et b • b', son image p~r ([,/') sat:isfer~ aux conditions ~) R~ppelons qu'un ensemble ~lg6brique ferm~ ("bunch of V~rieties" dane la terminologie de F-VII} est dit ~Ig6brique sur k si routes ses compos~ntes le sont, et normMement ~lg6brique sur k si de plus eelles-ei sont permut~es entre elles p~r tout automorphisme de l~iss~nt inv~ri~nts tous]es 61~mentsde k.

[1954d]

135 Critgres d'6quivalence en g6om6trie alg6brique.

103

du l e m m e . I1 suffit done de faire la d 6 m o n s t r a t i o n dans le cas oh C, C' s e n t compl6tes sans points multiples. E n v e r t u du th6or6me de RIEMAN~-RocH, il existe slots sur C u n e f o n c t i o n q~ n o n constante, sans p61e en dehors de a et ne s ' a n n u l a n t pas en 5; on p o u t en effet choisir une telle f o n e t i o n p a r m i celles qui o n t en a un p61e d ' o r d r e N, d6s que N e s t assez grand. E n m u l t i p l i a n t F p a r une eonstante, on iJeut supposer F ( b ) = 1. Soit de m 6 m e ~v' une f o n c t i o n non c o n s t a n t e sur C', sans p61e en dehors de a' et telle que qv' (b') = 1. Sur C • C', soit yJ la f o n c t i o n d6finie par ~f ( M x M ' ) = ~0 (M) - ~' (M') p o u r M 4=-a, M'=~ a ,t on a y0 (b • 0. Soit D u n e eomposante passant par b • du diviseur (yo)0 des z6ros de *f; on v a m o n t r e r que D passe p a r a x a'. E n effet, e o m m e ~v' n ' e s t pas eonstante, ya n ' i n d u i t pas la c o n s t a n t e 0 sur b • C', done on a D + b x C ' ; par suite D coupe t o u t e courbe P x C', quel que soit P sur C, en un p o i n t au moins. Mais, si a " est un p o i n t de C' a u t r e que a', ~0' y est dgfinie et 4= oo ; la r e l a t i o n

~)--1= (~--1. (l- (p--1f~t)--i m o n t r e alors que ~ est d6finie e t a la v a l e u r oo en a • a", done a • a " ne p e u t 5tre sur D. P a r suite D coupe a • C' au seul p o i n t a • a', ee qui ach6ve la d6monstration. L e m m e 6. Soit P un point d'une varidtd (abstraite) W, dd/inie sur un

corps lc; soit 31 un point g&~drique de W par rapport h k (P). Alors il existe sur W une courbe passant par M e t P, ayant en M un point simple; si de plus P eat simple sur W, il existe une telle courbe ayant en P u n point simple. C o m m e 2/I a u n r e p r 6 s e n t a n t sur t o u t r e p r 6 s e n t a n t de W, on p e u t passer un r e p r 6 s e n t a n t de W sur lequel P air un r e p r 6 s e n t a n t ; a u t r e m e n t dit, il s u f f i t de se placer d a n s le cas oh W est une vari6t6 affine. S o i e n t n la d i m e n sion de IV, S x l ' e s p a c e affine a m b i a n t ; p a r une t r a n s l a t i o n , on p e u t a m e n e r P en 0; la n o u v e l l e vari6t6 W ainsi d6duite de l ' a n e i e n n e sera ddfinie sur . K = / c (P). Soit L la vari6t6 lin6aire de d i m e n s i o n N - n + 1 d6finie p a r

hi5 X~ = 0 (1 ~< i ~< n - 1), les uij 6 t a n t N (n - 1) v a r i a b l e s i n d @ e n d a n t e s j=l

sur K ; soit C u n e e o m p o s a n t e de W & L p a s s a n t p a r 0 ; si 0 est simple sur W, L e s t t r a n s v e r s a l e g W e n 0, done C est unique, de d i m e n s i o n 1, et 0 est simple sur C; en t o u t cas C est de d i m e n s i o n d > 1, et est d6finie sur la cl6ture alg6b r i q u e K ' de K ' = K (u); soit x = (x 1. . . . . x v) un p o i n t g6n6rique de C p a r r a p p o r t g K ' . C o m m e x + 0, on p e u t supposer p a r e x e m p l e x 1 4= 0, d'o{1 N

u i l = - ~ UihXh/X 1

(l --< i =0, with S ( X ' X ) = 0 if and only if X is equivalent to 0. I n this form, it m a y be regarded as the fundam e n t a l theorem on correspondences; for instance, the so-called R i e m a n n hypothesis for function-fields follows from it almost immediately. Castelnuovo's proof was "geometric"; in other words, it was such t h a t its t r a n s l a t i o n into a b s t r a c t terms was essentially a routine m a t t e r once the necessary techniques h a d been created; in fact, all modern proofs are based upon the ideas i n t r o d u c e d b y him and s u p p l e m e n t e d b y the later work of other I t a l i a n geometers, p a r t i c u l a r l y Severi, on the same subject. I t will now be shown how a r a t h e r simple proof can be given in the classical case b y using t r a n s c e n d e n t a l and topological methods. I n the first place, a n y correspondence Z between two non'-singular varieties V and W over complex numbers induces homomorphisms of the homology groups of V into those of W. If V, W and Z have the same dimension, these h o m o m o r p h i s m s m a p the homology group of V for a n y dimension into t h a t of W for the same dimension. If V is the same as W, the n u m b e r of fixed points of Z (its intersection-number with the diagonal) is given b y Lefschetz's formula as t h e a l t e r n a t i n g sum of the traces of the endomorphisms induced b y Z on the h o m o l o g y groups of V. F r o m this it follows i m m e d i a t e l y t h a t in the case of a curve the integer S (Z) defined above is the trace of the endomorphism induced b y Z on the homology group H of C for dimension 1. If g is the genus of C, H is a free abelian group of r a n k 2g; for a given choice of generators, an endomorp h i s m of H is represented b y an integral-valued square m a t r i x of order 2g. L e t X be a correspondence between two curves C, C'. Let H, H' be the h o m o l o g y groups of dimension 1 for C and C'; let 71. . . . . 72g be generators for ! ! H, a n d 71 . . . . . 729' generators for H'. Call E = II e~ l i the intersection-matrix for the ya, which is skew-symmetric of d e t e r m i n a n t 1 ; call E' the similar m a t r i x ! for the 70- B y a well-known t h e o r e m in topology, the cycles P X C', C x P ' ! a n d 7a X 7Q generate the homology group of dimension 2 for C x C', so t h a t X m u s t be homologous on C X C' to a linear combination !

d. (P x c') + a'. (c x P') + Y % . (r~ x r~). 553

184

[1954 h ]

Put A = lraao IJ. It is easily seen that the matrices of the homomorphisms of H into H ' and of H ' into H induced respectively b y X and by X ' are L=

t(EA)

,

L'=

AE'

where f denotes the transpose of a matrix. This gives L' = E -1. tL . E'.

Consider now on C the harmonic differentials, i.e. the real parts of the differentials of the first kind on C; the vector-space of such differentials is of (real) dimension 2g. Take for this a basis consisting of forms w h respectively homologous to the ),~ in the sense of de R h a m , i.e. such that l e o s = %, r~ B y de

Rham's

ff h A or

theorems,

(4, #

= 1, 2 . . . . . 2g).

E is then also the matrix of the integrals

t a k e n on C.

The differential 0k of the first kind with the real part coa has an imaginary part which is also harmonic and can therefore be written as ~] c,~o,. Put /x

] = II q~ II. From the fact that i~'~ is again of the first kind, it follows at once that ]~ = -- 1, where 1 denotes the unit matrix. We have ~h A ~/~ = 0; integrating this over C and expressing ~h, 0n in terms of the ooa, we find E = t j . E . ] or the equivalent relation t ( E J ) = E J, expressing that E ] is a symmetric matrix. If ~ is any differential of the first kind, we have i~ A ~ > 0 everywhere, C being oriented in the usual manner. Integrating this over C, we find that the quadratic form with the matrix E ] is positive-definite. These statements on E ] are substantially identical with Riemann's bilinear relations and inequalities for the periods of the integrals of the first kind; nor does the proof just given differ in substance from Riemann's. t ! Now let again X be as above; define the forms coo, OQ and the matrix J ' for C' just as cox, 0h, J have been defined for C. The differential form ~ A ~s induces 0 on every component of X since such components are algebraic sub-. varieties of C • C'. Therefore f f ~ A ~;, taken on X, must be 0. Expressing X as above in terms of a homology basis on C • C' and expressing 0k, ~0 in terms of the o h, o)s one finds, by taking the real and imaginary parts of the double integral, two equivalent relations, one of which is (EA)E' = V. (EA). E'J'. Take the transpose of this relation, remembering that t ( E A ) = L and that 554

[1954h]

185

E ' J ' is s y m m e t r i c ; m u l t i p l y to the left b y E '-1 and to the right b y we get L J =- J ' L .

f-1

__

_

j;

This expresses the fact t h a t X induces a linear m a p p i n g of tile complex vectorspace of differentials of the first kind on C into the corresponding space for C', or also a complex homomorphism of the lacobian v a r i e t y of C into t h a t of C'. All this is well-known. The i n e q u a l i t y S ( X ' X ) >= 0 is now easy to prove. I n fact, S ( X ' X ) is no other than the trace of the m a t r i x L'L, which is the same as t h a t of M = J - 1 L ' L ] . This m a y be written as M = j - 1 . E - 1 . *L. E ' L J = (E J) -1 . *L. E ' J ' L . As the q u a d r a t i c form with the m a t r i x E J is positive-definite, it can be transformed into a sum of 2g squares b y a suitable substitution U. This gives t U . E J . U = 1 a n d therefore (E J ) - i = U . *U. The trace of M is the same as t h a t of the m a t r i x N = U - 1 M U = t ( L U ) . ( E ' J ' ) . (LU). P u t E ' J ' = [[ %~ ][, L U = [[ xoa lJ; then the trace of N is Tr(N) = ~ ( ~ so~xo~xo~ ). 1.

O,~

Since E ' J ' is positive-definite, it is clear t h a t the r i g h t - h a n d side is -->__0 and t h a t it is > 0 except when L U = 0 i.e. when L = 0. In order to complete the proof, it only remains to show t h a t L cannot be 0 unless X is equivalent to 0; this is an easy consequence of Abel's theorem. If I m a y be allowed a personal note here, this is precisely how I first persuaded myself of the t r u t h of the a b s t r a c t theorem even before I h a d perceived the connection between the trace S ( Z ) and Castelnuovo's equivalence defect. No one with a n y experience in such m a t t e r s will fail to acknowledge the cog e n c y of such an argument, even though no proof can be based on it. Is it possible to e x t e n d these results to higher dimensions? Many facts point to a generalization of the R i e m a n n hypothesis which can be s t a t e d as follows. L e t V be a v a r i e t y over the field k w i t h q elements. Then there is for each integer v a correspondence I , between V a n d itself such t h a t to each point of V with coordinates x 1. . . . . x y there corresponds b y I v the point with the 9

v

qV

coordinates Xlq . . . . . x N . The fixed points for I v are precisely those which have their coordinates in the extension k~ of degree v of k. Let N~ be the n u m b e r of such points; if V is non-singular, this is the intersection-number of I~ with the d i a g o n a l of V • V. Now in the classical case the numbers N , of fixed points for the successive powers of a given correspondence Z between a c o m p a c t non-singular v a r i e t y V 555

186

[1954h] a n d itself is given b y Lefschetz's formula as being equal to ~n

B~

N~ = X (-- 1) ~ E (~,.)~ h=O

(A)

i--1

where n is tile complex dimension of V, B h its Betti number for the (topological) dimension h, a n d the ehl, for 1 =0; if so, then presumab l y this generalization might a d m i t a c o m p a r a t i v e l y easy proof in the classical case b y means of Hodge's t h e o r y of harmonic differentials. Before coming to the n e x t example, let me recall the concept of numerical equivalence. Two cycles of the same dimension on a non-singular complete v a r i e t y are said to be numerically equivalent if their intersection-numbers with e v e r y cycle of the c o m p l e m e n t a r y dimension are equal whenever t h e y are b o t h defined. I n the classical case, two cycles which are homologous to each other are obviously equivalent in this sense; this implies at once t h a t the group of equivalence classes of cycles of a given dimension is finitely generated. I n the abs t r a c t case, N6ron's theorem shows t h a t this is so for divisors (cycles of dimension n - - 1 on a v a r i e t y of dimension n) a n d therefore also for cycles of dimension 1; to prove it for dimensions between 1 a n d n - - 1 seems still b e y o n d our reach at present. Again in the classical case, more precise results are known under special assumptions. F o r instance, there are varieties whose h o m o l o g y groups are all generated b y algebraic cycles; this implies t h a t t h e y vanish for the odd dimensions. If V a n d W are such varieties, all algebraic cycles on V • W m u s t then be numerically equivalent to linear combinations of cycles of the form X • Y, where X is a cycle on V a n d Y is a cycle on W. This m u s t be so, in particular, if V and W are non-singular r a t i o n a l surfaces, since the homology groups of such surfaces are known to have the p r o p e r t y in question. Making use of N6ron's theorem, we thus get the following p u r e l y algebraic statement. L e t S, S ' be two non-singular r a t i o n a l surfaces; let the X~ be generators for the group of divisor-classes on S modulo algebraic equivalence; let the X ; be the generators for the corresponding group on S'; then every cycle of dimension 2 on S • S' is 556

[1954h]

187

n u m e r i c a l l y equivalent to a linear combination of the cycles P • S', S • P ' (where P is a point of S and P' a point of S') a n d Xi • X~. I t does not seem hopeless to t r y to find an a b s t r a c t proof for this statement. Let us for a m o m e n t assume it to be true. A p p l y i n g it to the diagonal of S • S, one deduces i m m e d i a t e l y from it the v a l i d i t y of Lefschetz's fixed point formula for S in the following form: if X is a correspondence of dimension 2 between S and itself, the n u m b e r of its fixed points is d(X) -? d'(X) ~- S(X), whore d(X), d'(X) are the degrees of X (its intersection-numbers with P • S and with S • P ) and S(X) is the trace of the endomorphism induced b y X on the group of divisor-classes on S modulo numerical equivalence. This can then be applied as above to a surface S defined over a finite field k of q elements and to the n u m b e r N , of points of S with coordinates in the field k, with q~ elements. One finds t h a t N , is of the form i where the ei are the characteristic roots for a certain linear substitution of finit~ order and are therefore roots of unity. U n d e r the same assumption, one can then verify in this case the following general conjecture. Let V be a non-singular complete v a r i e t y of dimension n over an algebraic number-field K; for the sake of simplicity we assume t h a t it is e m b e d d e d in a projective space and write a set of equations for it as T't,(X o, X 1. . . . . XN) = 0, where the F , are homogeneous polynomials with coefficients in the ring of integers of K. Let Bo, B1 . . . . . B~n be the B e t t i numbers of V (with B o = B~n = 1, since V is irreducible, a n d B h -= B2~_ ~ b y the d u a l i t y theorem). Let 9 be a prime ideal in K such t h a t the equations Yt,=0, reduced modulo 9 , define a non-singular v a r i e t y V~ of dimension n over the residue field K ~ of K mod. 9 ; it is not h a r d to show t h a t all b u t a finite n u m b e r of prime ideals in K have t h a t p r o p e r t y . Assuming the v a l i d i t y of a formula of t y p e (A) for V~, a n d assuming (as is the case in all examptes which could be t r e a t e d so far) t h a t the integers B~ in it are no other than the Betti n u m b e r s of V, call ~hi(9), for 0 ~< h ~< 2n, 1 ~< i ~< Bh, the numbers occurring in the r i g h t - h a n d side of the formula (A) for the v a r i e t y V~; as mentioned before, these numbers are of absolute value qh/= whenever t h e y can be calculated. P u t now ~)h(S) - - ~I I I ( i -- ~"hi(9)" x~'~9 s) --1. 5

i

Then our examples indicate t h a t q)h(s) coincides (except for a finite n u m b e r of factors) with the Euler p r o d u c t for a Dirichlet series which can be continued in the whole plane and satisfies a functional equation of the familiar t y p e 557

188

[1954h] 7~(s) =

7~(h +

1 - - s)

where ~ is the product of the Dirichlet series, of a g a m m a factor a n d of an exponential factor. I t is t e m p t i n g to surmise t h a t this is always so, b u t I have little hope t h a t a general proof m a y soon be found. F o r non-singular r a t i o n a l surfaces at a n y rate the results s t a t e d above would i m p l y t h a t q~2(s), e x c e p t for a finite n u m b e r of factors, is the same as a suitable L-function (in the sense of Artin) for a certain extension of K. F o r instance, for a non-singular cubic surface in the projective 3-space, one thus gets an L-function belonging to the extension of K determined b y the 27 straight lines on the surface. The Galois group for this is known; it is a group of order 27 . 3 ~ . 5 and has a simple subgroup of index 2. I n general, therefore, the function ~b2(s) which we m a y expect to belong to a given cubic surface is essentially an L-function of a definitely non-abelian type. Here is a r a t h e r unexpected connection between n u m b e r - t h e o r y and algebraic geometry.

558

[ 1954i] Poincar6 et l'arithm&ique CONFERENCE

D E M. A. W E I L

Poincar~ et l'Arithm6tique.

Qu'il me

soit p e r m i s a v a n t tout d ' a d r e s s e r mes r e m e r c i e m e n t s & nos coll~gues

h o l l a n d a l s , o r g a n i s a t e u r s de cette j o u r n d e consacr6e & H e n r i POINCAR~, p o u r l e u r a i m a b l e i n v i t a t i o n . J'ai a c c u e i l l i celle-ci avee p l a i s l r , car elle va me d o n n e r Reprinted with permission of the publisher, Gauthier-Villars. 189

~t

190

[19540

l'occasion d'attirer l'attention sur des aspects peu connus de l'oeuvre de POINCAa~, dont je suis persuadd qu'on trouverait profit ~ reprendre l'dtude. Malheureusement les limitations de temps qui nous sont imposdes aujourd'hui n'ont laissd aucune place pour un exposd, si bref f~t-il, des travaux de POINCnn~ sur la gdomdtrie algdbrique et sur les fonctions abdliennes; je ne puis ndanmoins me dispenser ici d'en signaler l'importance cap.itale et la profonde influence sur le ddveloppement que ces branches des Mathdmatiques ont pris depuls lors. Les dcrits de POtNCAa~ qui touchent ~ l'Arithmdtique occupent un volume entier (tome V des OEuvres). On ne saurait nier qu'ils sont de valeur indgale. Certains n'ont guard d'autre intdrgt que de nous laird voir combien attentivement Po~ncAaa ~ ses ddbuts a dtudid toute l'ceuvre d'HERMITE et comme il s'en est assimil6 les mdthodes et les rdsultats. On a dit parfois que Po~ncAae lisait peu; ce qui frappe dans le volume de ses OEuvres dont il s'agit, c'est surtout qu'il s'y montre fort peu instruit des travaux en langue allemande; sans doute ne lisait-il l'allemand qu'avec beaucoup de peine. Mais il ne donne certes pas l'impression d'un ignorant ni d'un autodidacte. C'est sous l'influence d'HEnMtT~, bien dvidemment, que PotncAa~. a consacr6 plusieurs de ses premiers travaux ~ la thdorie algdbrique et arithmdtique des formes, et particuli~rement des formes cubiques ternaires et quaternaires. Ses rdflexions sur ce sujet Font amend en particulier ~ une ddmonstration et ~ une extension du thdor~me de JORDAn d'apr~s lequel il n'y a qu'un nombre fini de classes de formes algdbriquement dquivalentes ~ une forme donn@ de discriminant non nul (OEuvres, t. V, p. a99-3o5); cette question, longtemps ndgligde, mdriterait certainement d'etre reprise, par exemple afin d'~tendre le thdor~me de JORDAn aux corps de nombres algebriques; sans doute conviendrat-il pour cela d'avoir recours '~ POINCAR~. Mais laissons la parole ~ notre auteur. Voici comme il parle de ses premieres recherches, dans son cdl~bre rdcit de la ddcouverte des fonctions fuchsiennes

(Science et Mdthode, p. 5 a ) : g(x,u) of W into W is a birational correspondence between W and W. This is equivalent to saying that, if x, u are independent generic points of V and of W, respectively, over /c, then 7c(x, g(x, u ) ) ~ ~(x, u). (TG2) If x, y, u are independent generic points of V, lz and W, respectively, over lc, then g ( f ( x , y ) , u ) ~ g ( x , g ( y , u ) ) . I f (TG1) is fulfilled, (TG2) is meaningful, since in that case g ( y , u ) is generic on W over lc(x), while f ( x , y ) is generic on V over lc(u) by (G1). When (G1, 2) and (TG1, 2) are satisfied, we shall say that g is a normal (external) law of compOsition on W with respect to the pre-group V, and that W, with this law, is a pre-transformation space with respect to V; g will then mostly be written as a multiplication, i.e. as g(x, u) ~ xu; then (TG1), (TG2) appear as lc(x, x u ) = l c ( x , u ) and ( x y ) u = x ( y u ) . Just as before, we note that the concept of a normal law is independent of ~he field of definition, which may be enlarged at will, and that it is birationally invariant ; if V' is birationally equivalent to V, and W' to W, the laws f, g can be transferred in an obvious manner to V', W'; the pair V', W', with the laws f', g' obtained from f, g by transfer, is said to be birationally equivcdent ~o the pair V, W with the laws f, g. In particular, W, just as V, may be replaced by an a~ne model. Take x, y, u as in (TG2) ; put z = x y , v = y u . By (TO1), v is generic on W over lc(x) ; so is xv, again by ( T G 1 ) ; therefore x-l(xv) is defined. But (TG~) can be written as z u = x v , and so we have x - l ( z u ) = x - ~ ( x v ) .

202

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AND~

WEIL.

As x-% z and u are independent generic points of V, V and W over h',, we can apply (TG2) to the left-hand side, which is therefore equal to (x-lz)u, i.e. to yu by Prop. l(iii), i.e. to v. This proves x - ~ ( x v ) ~ v ; as x, v are independent generic points of V, W over ~, this must therefore remain true for any pair of such points. The conditions stated above may be strengthened by assuming "generic transitivity," which means the following condition: ( H ) I f x, u are independent generic points of V and of W, respectively, over ~, then g ( x , u ) is a generic point of W over k ( u ) . I n that case, the pre-group V. g on V X W X W my Foundations; W X W, by Prop.

we say that W is a pre-homogeneous space with respect This condition is equivalent to saying that the graph has the projection W } ( W on W X W (in the sense the set-theoretic projection then contains a open subset 10 of the Appendix).

to of of of

S. The following result shows that a normal law of composition may be obtained from a mapping satisfying much weaker conditions than those stated above. P~OPOSlTION 2. Let V, W be two varieties, defined over a field k. Let g be a mapping of V X W into W, defined over k, satisfying (TG1) and the following condition: (TG2') There are two independent generic points x, y of V over lc and a generic point z of V over tc such that g(z,u) ~ g ( x , g ( y , u ) ) for u generic on W over k ( x , y , z ) .

Let ~(2) be the smallest field of definition containing lc for the mapping u-->g(x,u) of W into W, 2 being a generic point over k of a variety ~. Then one can write 2 = $ ( x ) and g ( x , u ) = g ( 2 , u), where ~ is a mapping from V to ~ and g a mapping from F X W to W, both defined over ~; putting 9 ~ ~(Y), 2 = ~(z), we have ~(2) C k(2, 9) and may write ~ ~ ?(2, Y), where ? is a mapping from ff X P to F, defined over ~. Finally, ? and .~ satisfy the conditions (G1, 2), (TG1, 2) and define ff as a pre-group and W as a pre-transformation space with respect to ft. In the first place, the smallest field of definition containing k for u--> g (x, u) is contained in k (x) and is therefore, by Prop. 3 of the Appendix, a finitely generated regular extension of lc; this may always be written as Ic(~), e.g. by taking 2 as a suitable point in an affine space, which has then

[1955a]

203 ON ALGEBRAIC GROUPS OF T R A N S F O R M A T I O N S .

361

a locus I7 over /r and may be written as r As g(x,u) is then rational over l~(~,u), it may be written as ~(-2, u), or more briefly as 2 u ; ( T G 2 ' ) can then be written as g ( z , u ) = ~ ( g u ) , which shows that the function u--->g(z,u) is defined over /r so that /c(2) C / c ( 2 , 9 ) ; we may then write 2 ~ f(~?, 9), or more briefly 5 ~ 29. I t is clear that g, ] satisfy (TG1, 2) ; we have to show that ~ satisfies ( G 1 , 2 ) . By ( T G 1 ) , if v ~ g ( x , u ) , the mapping u---> v is a birational correspondence between W and W, defined over /~(2) ; its inverse must then be defined over the same field, so that we have k(u) C l~(2, v) and may write u ~ h ( 2 , v). Notations being as before, put w ~ g ( y , u ) ; as this, by ( T G 1 ) , is generic over lc(x) on W, the relation in ( T G 2 ' ) , which can be written as 5u=2w, is equivalent to w ~ h ( 2 , Su). This shows that the mapping u---> w is defined over /c(2, 5) ; since its smallest field of definition i s / c ( 9 ) , we get k(9) C k(2, ~). Similarly, we have w ~ 9u and therefore u ~ h ( ~ , w ) ; then the relation in (TG2') can be written as ~w ~ 2h(9, w), from which we conclude in the same manner that k(2) C ~(~, ft). This shows that f satisfies (G1). Now, if 2~, 22 are any two generic points of IV over /% there is an isomorphism a of le(2~) onto /~(2~) over /c which maps 2~ onto 22. Take u generic on W over ]~(2~,2~), and put u ~ 2 ~ u , u~=2~u. Then a maps the graph of u--> u~ onto the graph of u---> u2. I f u~ = u2, these two functions coincide, and therefore, by Prop. 4 of F-IVy, a must induce the identity on the smallest field of definition of the first function. As this field is ~(2~), we have thus shown that u , ~ u = , implies 2 1 ~ 2 . Now let 2, Y, t, u be independent generic points over /c on ]?, F, I?, W; put 2 ~ ( x y ) t and 2 ~ 2 ( ? ~ [ ) , these being defined because ~ satisfies (G1). We have to show that 2~ ~ x 2 ; by (G1), they are both generic over /c on I?, and u is generic over /c(2~, 22) on W, so that we need only show that 2~u=~2u. By ( T G 1 ) , 2 ( 9 ( [ u ) ) is defined; by ( T G 2 ) , this is the same as (xY)([u), which, again by ( T G 2 ) , is the same as 2~u since 29, t, u are independent generic points of I?, !?, W over ]c by (G1). Similarly 2(fl([u)) is the same as o2((9[)u) by ( T G 2 ) , and this is the same as 22u by (TG2) and (G1). This concludes the proof. The external normal law of composition j constructed in Prop. satisfies, in addition to ( T G 1 , 2 ) , the following condition:

2

(TG3) If x is generic over 1r on V, to(x) is the smallest field of definition containing ~ for the mapping u-->g(x,u) of W into W. Whenever (TG3) is satisfied in addition to (G1,2) say that V operates faithfully on W by g.

( T G 1 , 2 ) , we will

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A~m~ WEIL.

I f V, W and a mapping g of V }( W into W are given, and g satisfies (TG1, 2', 3), then Prop. 2 shows that (x, y)---~z (where x, y, z are the points of V which appear in ( T G 2 ' ) ) is a normal internal law of composition ell V and that g is a normal external law with respect to the pre-group defined by f on V. I n particular, if V, W, f and g are given and f, g satisfy (TG1, 2, 3), then f satisfies (G1, 2~.

w II.

Construction of chunks.

4. Let V, W be a pre-group and a we-transformation space and f, g the internal and external normal laws belonging to them, these being all defined over a field ~; we will mostly write f, g multiplicatively, as has already been done in w Instead of saying that f is defined at a point (s,t) of V X V, we shall frequently say that st is defined; similarly, when we say for instance that s-l((st)a) is defined, for s, t on V and a on W, this will mean the following: (i) f is defined at (s,t), with the value st; (ii) g is defined at (st, a), with the value ( s t ) a ; (iii) x - ~ x -1 is defined at s, with the value s-l; (iv) g is defined at (s -~, (st)a), with the value s-~((st)a). We recall that two expressions, built up from functions which are defined over ~, coincide for all values of the variables for which they are both defined provided they are defined and coincide when the variables are given independent generic values over k. This applies for instance to the formulas in (G2) and (TG2) which express the associativity of f and g. We say that V is a group-variety or a group if f is everywhere defined on V X V and z - o x -1 is everywhere defined on V; then the corollary of Prop. I shows that there is a neutral element e on V with the usual properties. I f V is a group, W will be called a transformation-space with respect to V if g is everywhere defined on V }( W; if, moreover, V operates transitively on W in the usual sense, i.e. if to every pair a, b on W there is an s ~ V such that b ~ sa, then W is called a homogeneous space with respect to V. In w it will be shown that, fo every pre-group V, there is a birationally equivalent group V', and that, fo every pre-~ransformation space W with respect to V, there is a birationally equivalent transformation-space W" with respect to V'. The proof of this will include a proof of the fact that W is biregularly equivalent to an open subset of W p if and only if it fulfills the following condition : (C) I f a is any point of W, and x a generic point of V over k ( a ) , then xa and x-~(xa) are defined.

[1955a]

205 ON ALGEBRAIC GROUPS OF T R A N S F O R M A T I O N S .

363

A pre-transformation space W with respect to V which fulfills this condition will be called a chun/c of transformation-space, or more briefly a chunlc. For similar reasons, if W is a pre-homogeneous space, i.e. if g satisfies (I-I), we say that W is a homogeneous ehun/c if it satisfies (C) and the following : (He)

I f a and x are as in (C), xa is generic over/c(a) on W.

Finally, V itself will be called a group-chunIc if it is a homogeneous chunk with respect to left-translations and if x -~ is everywhere defined on it, or in other words if it satisfies the following: (GC1) I f s is any point on V, and x a generic point of V over /c(s), then xs and x-l(xs) are defined and xs is generic over k ( s ) on V. (GC2)

For every s on V, s -~ is defined.

PROPOSITION 3. Call ~ the set of those points a o~ W such that xa and x-l(xa) are defined for x generic over /c(a) on V. Then ~ is a k-open subset of W ; ~2 and all /c-open subsets of ~ are chunks; if a e ~ , we have x-~(xa) ~ a , / c ( x , a ) ~ / c ( x , xa), and a is a point of the locus of xa over /c(a) on W. Call F the set of points on V X W where g is not defined; by Prop. 8 of the Appendix, this is a /c-closed subset of V X W. Let r be the graph of the mapping (x,u)--->x-lu of V X W into W, i.e. the locus of (x,u,x-~u) over /C for x, u generic and independent over/C on V, W. Call F t the/c-closed subset of V X W X W consisting of all points ( x , u , v ) with ( x , v ) ~ F ; let F " be the union of the projections of the components of 1~ N F" on the product of the first two factors of V X W X W (this being understood as in F-IV~ and F-VII3 ; F " is the closure, in the Zariski topology, of the set-theoretic projection of F N F ' on V X W ; cf. Appendix, Prop. 10). I t will now be shown that ~ is the same as the set ~ of the points a on W such that V X a is not contained in F U F". I n fact, for xa to be defined, it is necessary and sufficient that V X a should not be contained in F ; le~ rto be the set of points a with this property; it contains both a and rh. I f a e g t o - - ~ , x-~(xa) is not defined; as x -~ is generic over /c(a) at the same time as x, this is equivalent to saying that x(x-~a) is not defined; as x-~a is defined, the point (x, a, x~a) is then in I ~N F', and therefore (x, a) ~s in U ' , so that V X a C F " and a r Conversely, if a~f~o--ft~, then (x,a) is in the projection of one of the components of I' N F', and so, if (y, n, v) is a generic point of that component over ~, (x,a) is a specialization of ( y , u ) over ~. As ( y , u , v ) is

206

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ANDR]~ WEIL.

on 1~ and x-la is defined, v has then the unique specialization x-la over (y,u)--->(x,a) with respect to )~; therefore (x,a,x-~a) is in F ' , and so x(x-la) is not defined and a is not in e . This proves t h a t ~ = e l ; the latter set being k-open by Prop. 7 of the Appendix, e is k-open. As x-l(xu) = u for x, u generic and independent over k on V, W, we m u s t have x - ~ ( x a ) = a whenever s left-hand side is defined, and so for a ~ e and x generic over k ( a ) ; this implies t h a t k ( a ) C k(x, xa), so t h a t k(x,a) = k ( x , xa). Let y be generic over k ( x , a ) on V; then (x-~,xa) is a specialization of (y, xa) over k ( x , a ) , as the former point is not in F , and F is k-closed, (y, xa) is not in F , and so y(xa) is defined. As yx is generic over k(a) by (G1) and is therefore a generic specialization of x over k(a), (yx)a is defined and is a generic specialization of xa over k(a). By associativity, we have y(xa) ~ (yx)a since both sides are defined; as x-~(xa) is defined, it is a specialization of y(xa), and therefore also of (yx)a and of xa, over k ( a ) . This shows t h a t a is a specialization of xa over k ( a ) , i.e. that it is a point of the locus of xa over k ( a ) on W. I f now ~ ' is any k-open subset of e , then the set C = W - - g t ' is k-closed, and so, if xa is in C, a m u s t be in C ; in other words, if a is in ~ ' , so is x a ; it is then clear t h a t e ' is a chunk. The locus of xa over k ( a ) could be described as the closure of the orbit of a under V on W. COROLLARY. Notations being as in Prop. 3, call ~ the set of the points a of ~ such that W is the locus of xa over k(a). Then ~ is It-open or empty according as W is pre-homogeneous or not. In the former case, ~h and all ~-open subsets of eh are homogeneous chunks; and, if a, b are any two points of ~ , there are two generic points x, y of V over k(a, b) such that x a = y b . Except for the last assertion, this is an immediate consequence of Prop. 11 of the Appendix, applied to the k-open set ~ of Prop. 3. Let now a, b be in e ~ ; take x, y generic on V over k(a,b), and p u t u = x a , v = y b . Then the loci of u and of v over k(a, b) are W, and so there is an isomorphism of k(a,b,u) onto k(a,b,v) over k ( a , b ) which maps u onto v; this can be extended to an isomorphism a of k (a, b, x) onto some extension of k (a, b, v) ; then x 9 is generic on V over k(a,b) and we have u ~ x ~ a , i.e. x ~ a ~ y b , so t h a t x ~ and y satisfy the conditions stated in the corollary. F i n a l l y , in order to construct a group-chunk from a given pre-group V, one need only observe t h a t the g r a p h V~ of the function x - - > x -~ is a subvariety of V X V, birationally equiva]ent to V, and that, if we transfer that function to V1, we get an everywhere biregular birational correspondence between V~ and itself since it is the same as the function induced on V~ by the

[1955a]

207 O ~ ALGEBRAIC GROUPS OF

TRANSFORMATIONS.

365

mapping (x, y) ---->(y, x) of V X V onto itself. Therefore we may assume that we have started from a pre-group V on which x -1 was everywhere defined; had that not been the case, one would merely have had to replace Y by V1 to make it so. Call now ~h the set of the points s~ V with the property stated in (GC1) ; by the corollary of Prop. 3, this is a k-open subset of V ; as x---->x-~ is an everywhere biregular mapping of V onto itself, it transforms ~ into a k-open set ~ - 1 ; then ~h A ~ - ~ is a gro~ip-chunk. Thus we have constructed chunks for the three kinds of objects under consideration, viz., transformation-spaces, homogeneous spaces and groups. I f W is a pre-transformation space, defined over k, the set W' of simple points on W is a k-open set on W; by applying Prop. 3 to W', we obtain a non-singular chunk. Similarly, one would get an everywhere normal chunk by taking for W' the set of points where W is normal, this being k-open by Corollary 3 of Prop. 8 of the Appendix. I t will presently be seen that homogeneous chunks and in particular group-chunks are always non-singular, so that no special procedure is required to make them such. By Corollary 2 of Prop. 8 of the Appendix, if one has constructed a chunk, one can at once derive from it a birationally equivalent chunk which is an affine variety; this also applies to homogeneous chunks. As to groupchunks, starting from a pre-group which we take to be an affine variety, and replacing it by the graph of the function x -1 on it, we get for our pre-group an affine model V on which x -~ is everywhere defined. Let V' be a k-open set on V which is a homogeneous chunk; let x ~ (x~,- 9 -, xm) be a generic point of V over k; take a polynomial P with coefficients in k which is 0 on V - - V ' but not on V; as x -1 and P ( x ) are everywhere defined functions on V, so is P(x-~). Call V" the locus of

(xl," 9 ", x,,, 1 / P (x), 1/P(x-1) ) over k in affine space; this is biregularly equivalent to the k-open subset determined on V by the inequalities P ( x ) :/~ O, P (x -*) ~= 0. This is a groupchunk. We have thus proved the following: PROPOSITION 4. TO every pre-homogeneous space (resp. pre-group) defined over k, there is a birationally equivalent homogeneous chunk (resp. group-chunl~) which is an a~ne variety, defined over k. To every pre-transformation space W rind every point a on W with the property stated in (C), there is a birationally equivalent chunk W" which is an a~fine variety and is such that the birational correspondence between W and W' is biregular at a; if a is simple on W, W' may be taken non-singular; if W is normal at a, W' may be tauten everywhere normal.

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tion if s are and

A NDR]:] WEIL.

5. PROPOSITION 5. Let V be a group-chunl~ and W a pre-transformaspace with respect to V; let lc be a field of definition for V, W. Then, is any point of V, and u a generic point of W over k ( s ) , su and s-l(su) defined; the mapping u---> su is a birational correspondence between W itself ; and lc (s, u) ~ ~ (s, su).

Take x, y generic and independent on V over k (s, u) ; put y' ~ yx -1" and u ' ~ x u ; by (G1) and (TG1), y' and u' are generic and independent over lc on V, W, so that y'u' is defined; as we have shown x-lu t to be defined and equal to u, we get, by associativity, y ' u ' ~ yu. We now show that the expression obtained by substituting s for y in y'u', i.e. in (yx -1) (xu), is defined. I n fact, since V is a group-chunk, the mapping (z, t) --> t-lz -~, where z, t are generic and independent over k(s) on V, is defined at ( s , t ) , and its value t-ls -~ at that point is generic over k(s) on V ; this implies that the mapping (z, t) --> (t-~z-~) -~ is also defined there; as this is only another expression for the mapping (z, t)--->zt, we conclude that the latter is defined at (s, t), i.e. that st is defined, and that st is generic over k ( s ) ; substituting x~1 for t, this shows that x' ~ sx -1 is defined and generic over ~ (s) on V, and a f o r t i o r i that the mapping y--->yx -1 of V into V is defined at s, with the value x'. The mapping u-->u' is defined at u, with the value u' which is generic over ~ ( x , s ) on W by (TG1). So x' and u" are generic and independent over lc on V, W, and x'u' is defined; more precisely, we have shown that the mapping ( y , u ) - - - > y ' u ' ~ ( y x - 1 ) ( x u ) o f V X W into W, which is defined over ~ ( x ) , is defined at (s,u). As this is only another expression for the mapping (y,u)---~yu, this implies that the latter is defined at ( s , u ) , i.e. that su is defined, and that these mappings have the same value there, i.e. that x ' u ' ~ s u . By ( T G 1 ) , x~u' is generic on W over ~ ( x , s ) ; therefore su is generic on W over lc(s). But then our assumptions on s, u are also satisfied by s-% su, so that it follows from what we have already proved that s-l(su) is defined; its value must then be u, since x - ~ ( x u ) ~ u , and so we have tc(u) C ~(s, su), and therefore k ( s , u ) ~ t c ( s , s u ) ; this means that u--~ su is a birational correspondence between W and W. COROLLARY. Assumptions being as in Prop. 5, assume also that V operates faithfully on W ; let s, s" be any two points of V, and let u be generic on W over ]~(s, s'). Then su ~ s'u implies s ~ s ' . Take x generic over lc(s,s',u) on V. Since V is a group-chunk, xs is defined and generic over ]c(u) on V; and su is defined and generic over to(x) on W by Prop. 5; by associativity, this gives ( x s ) u ~ x ( s u ) . Similarly we

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have ( x s ' ) u ~ x ( s ' u ) . Therefore s u ~ # u implies ( x s ) u ~ ( x # ) u . But then we can repeat the argument used in the proof of Prop. 2 ; there is an isomorphism ~ of k(xs) onto k(xs'), mapping xs onto xs'; as this transforms the graph of the function u--~ (xs)u into itself, and as tc(xs) is the smMlest field of definition for this graph because of the assumption of faithfulness, must be the identity, and x s ~ x s ' . As s ~ x - l ( x s ) and s ' ~ x - ~ ( x s ' ) , this gives s ~ s'. P~OPOSlTION 6. Let V be a group-chunk and W a chunk of transformation-space with respect to V. Let s be any point on V and (a, b) any point on the graph of the birational correspondence u.-->su between W and itself; then the latter is biregulc~r at (a, b), sa and s-~b are defined, and we have sa ~ b, s-lb ~ a. We first show that sa is defined. Take x, y, u generic and independent on V, V, W over k(a, b,s); and consider the mapping (x,u)-->y-: ( ( y x ) u ) , defined over k ( y ) , of V X W into W. By (GC1), x-->yx is defined at s, with a value ys which is generic on V over k(~,a). By (C), the mapping (x,u) -->xu is defined at (ys, a), and so the mapping (x,u) --> (yx)u is defined at (s,a), with the value (ys)a. At the same time, (ys)u is defined since u is generic on W over/c(y, s), and y(su) is defined because su is defined and generic over k(y) by Prop. 5 ; by assoeiativity, this gives ( y s ) u ~ y ( s u ) . As (a,b) is on the graph of u---->su, and (u, su) is a generic point of ~hat graph over k (y, s), (a, b) is a specialization of (u, su) over k (y, s) ; but then the relation ( y s ) u ~ y ( s u ) implies ( y s ) a ~ y b . By Prop. 3, the mapping v-->y-lv is defined at yb, i.e. at (ys)a, with the value b. We have thus proved that (x,u)--->y-~((yx)u) is defined at (s,a), with the value b; as this is but an expression for (x, u)-->xu, this shows that sa is defined and equal to b. Interchanging a, s with b, s-% and making use of Prop. 5, we see from this that s-~b is defined and equal to a. This~ afortiori that the mappings u--->su, u--> s-~u are respectively defined at a, b, with values b, a ; this means that the birational correspondence u-->su is biregular at (a, b). COROLLA~Y. Every homogeneous chunk is non-singular. Let W be such a chunk with respect to a pre-group V; replace V by a birationally equivalent group-chunk. For any a on W, take x generic on V over ~ ( a ) ; then u--e, xu is a birational correspondence between W and itself, transforming a into the generic point xa of W over ~(a), and biregular at a. As xa is simple on W, a must therefore be simple on W.

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w III.

Construction of spaces.

From now on, until the end of w V and W will denote respectively a group-chunlc and a chunlc of transformation-space with respect to V, both being at the same time assumed to be a~fine varieties; ]~ will denote a common field of definition for V and W and for the normal laws given on them. 6. Let n, n' be the dimensions of V, W, and take N ~ 4 n and also 3n-}-n'; take N independent generic points tl," 9 ", t~ over }c on V; put K ~ l c ( t l , " " ",tN). Let u be a generic point of W over K ; put S s ~ W and u s ~ t s u for 1(t~x-~)ts is defined

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at s~; this means that 2-1 is defined and has a representative on G~. Similarly, if we write t ~ t ~ t ~ -1, the representative of 29 on G-t is t ~ x y ~ ((tx~)t~-l)y~; let ~, ~ be two points of G with representatives r~, s~ on G~, G~ respectively; by Prop. 4 of the Appendix, we can choose 7 so that t.~ is generic on V over l~(r~,s~,t~,t~); the same will then be true of t, and also of tr~ and of (tr~)t~ -1 since V is a group-chunk; for a similar reason, this implies that ( x , y ) - - > ((tx)t~-l)y is defined at (r~,s~), and this completes the proof that G is a group. Now, going back to the space S constructed before, we transfer to G, S, by means of the birationa] correspondences 4)o, r the normal law given for V, W; in other words, for x, u generic and independent over K on V, W, and for 2 ~ o ( X ) , ~t~(u), we define ~ ( x u ) , and prove that this makes S into a transformation-space with respect to G. I n fact, the representative of 2~ on S~ is ((tx~)t~-~)u~, where t ~ t.~t~-~ as before; the rest of the proof is then quite similar to the proof given above. Naturally, if W is non-singular, S is non-singular; if W is everywhere normal, S is everywhere normal. Finally, if W is a homogeneous chunk, S is a homogeneous space. I n fact, in that case, let ~, ~ be any two points of S, with representatives as, b~ in S~, S~ respectively. Take x generic over K ( ~ , ~ ) on V; put 2'~2Ct, ~ " ~ 2 ~ . For u generic over K(x) on W, we have ~ I , ( 2 ~ ) ~ (xt~-!)u~; as W is a homogeneous chunk, x ' ~ (xt~-Z)a~ is defined and generic over K(~t, ~) on W, and therefore we have x ' ~ ( 2 ' ) ; similarly we have x" ~ ~I,(2") with x" ~ (xt~ -~) b~ generic over K(~,, 5) on W. That being so, there is an isomorphism of K(~t, b,x p) onto K(d, D, x") over K ( g , 5) which maps x' onto x " ; this can be extended to an isomorphism ~ of K((t, b, ~) onto some extension of K(~, b,x"). Then we have ~ ( ~ 2 " ~ b , and so 5 ~ ~-~2~. 7. From now on, it will be assumed that W and consequently S are everywhere normal. With this assumption, we shall construct an abstract variety S t, defined over k, and a birational correspondence F between S' and W, also defined over ~, so that the birational correspondence r o F, defined over K. between S' and S is an everywhere biregular mapping of S' onto S. This construction can then be applied to V itself, giving a variety G' and a birational correspondence Fo between G' and V, both defined over /~, such that r o Fo is biregular between G' and G. Transferring the normal laws for V, W to G', S" by means of F, Fo, we see that we have thus constructed a group G ' and a transformation-space S', birationally equivalent to V, W over k ; if W is pre-homogeneous and we have constructed S as a homogeneous space, S' will be a homogeneous space.

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ANDRE W E I L .

I n constructing S', we may assmne that V operates faithfully on W; in fact, if this were not so, one could replace V by another pre-group ? satisfying this condition, according to Prop. 2 of w I, no. 3. Notations will now be the same as in no. 6, with the additional assumptions that W and consequently S are everywhere normal, and that V acts faithfully on W, so that G acts faithfully on S. Let /C' be any field containing /C. Let ~2 (s~) be a cycle of dimension 0 4=1

on V, rational over /C', and assume that s~ =2~s~ whenever i J : j . Then, if we put /C"=/c'(sl,. ' ' , s~), /C'" is a Galois extension of/C', i.e. separably algebraic and normal over/C'. Call K " the compositum of K and/C"; let u be a generic point of W over K " , and put w~ ~ s#. I f m is the dimension of the ambient affine space to W, we write w ~ = (W~l, 9 ',w~m). P u t now

~=i

/~=I

where T, U1,'" ', U m are indeterminates; let y be the point, in an afflne space of suitable dimension, whose coordinates are all the coefficients of the homogeneous polynomial y(T,U) except that of Tr; this is the so-called " C h o w point" of the cycle ~ (w~), and y(T, U) is its " C h o w form." As V acts faithfully on W, and the s~ have been assumed to be distinct, the eorollary of Prop. 5, w II, no. 5, shows that the w~ are all distinct. W e can therefore apply to them the following~ general result: LEM~A. I f in (1) we ta/ce the w~ to be any set of distinct points, and ko is the prime field, then the w~ are separably algebraic over ~co(y). By F-I.~, Th. 1, we need only show that a derivation D of the field /co(W~,-" ",wr) over /co(y) must be trivial. I n fact, applying D to (1), we get : r

as the w~ are all distinct, this cannot be an identity in T, U~,- 9 ", U,, unless all the Dwi, are 0. PROPOSITION 7. Notations being as defined above, we have/c'(y) ~ / c ' ( u ) provided the s~ are all distinct and satisfy the following condition: (S) The set of points ~ = ~ o ( s ~ ) any right-translation.

on G is not mapped onto itself by

The cycle ~ (wi) is the image of the cycle ~ (s~) by the mapping

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213 ON A L G E B R A I C GROUPS OF T R A N S F O R ) s

371

x-,xu

of V into W ; it is therefore rational over U(u). By the main theorem on symmetric functions (VA, no. 7, Th. 1), this implies that y is rational over U ( u ) , i.e. that U ( y ) C U(u). On the other hand, the lemma shows that the w, are separably algebraic over U'(y); as we have u=s,-lw, by Prop. 5 of w I I , no. 5, u is therefore separably algebraic over U'(y), hence also over U ( y ) . Let (r be any automorphism over U ( y ) of the algebraic closure of U ( y ) ; as it induces an isomorphism of U(u) onto k ' ( u ~) over U, u ~ is generic on W over k', so that s~u~ is defined by Prop. 5. This gives (s,u)r ~, i.e. w~r162 ~. But the decomposition of the homogeneous polynomial y(T, U) into linear factors is uniquely determined; applying to (1), we see thus that the w,~ must be the same as the w, except for a permutation, i.e. that there is a permutation i--->r such that w ( = w o ( o . This can be written as s,r162 sr as the s.f are the same as the s, except for a permutation, we can write them as sinks,(o, where i--->r(i) is a permutation. Then we have r ~) =r which can be written as ,~(~)q)(u~) =2~(,)~, i.e. (~(u ~) ~ ( ~ ) - ~ g ~ ( o ~ . As G acts faithfully on S, the corollary of Prop. 5 shows that all the elements ~(~)-~2~(~) of G, for 1 ~ i ~ _ r, must coincide; if ~ is their common value, we have 2~(o ~ ( ~ ) T , which shows that the right-translation [ maps the set 2~ onto itself. By (S), this implies that [ is the neutral element of G, so that (~ (u ~) ~ ~, and therefore u ~ u . As u is separably algebraic over U(y), this shows that

~'(u) c k'(y). 8. Proposition 7 shows that we may write y ~ f ( u ) , where f is a birational correspondence, defined over ~,, between W and the locus Y of y over /cp in affine space. I f k'[y] is the ring generated over U by the coordinates of y, it is well-known that the integral closure of k'[y] in ~'(y) is a finitely generated ring over U, i.e. that it can be written as U [ y * ] , where y* is a point in a suitable a:ifine space; call Y* the locus of y* over /d in that Mfine space. As we have /c'(y*) ~ U ( y ) ~ U ( u ) , we may write y * ~ f * ( u ) , f* being a birational correspondence between W and Y*, defined over U. I t is usual to say that Y* is derived from Y by " n o r m a l i z a t i o n " over ~'. By Prop. 14 of the Appendix, since U ' is separably algebraic over k', U'[y*] is integrally closed in Jc"(y*). PROPOSITION 8. With the notations explained above, y* and ~t are: corresponding generic points over K" on Y* and S in a birational correspondence between Y* and S which maps Y* biregularly onto the {("-open set i

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ANDR]~ WEIL.

In the first place, we prove that the coordinates wiz of the w, are all in U ' [ y * ] ; as they are in U ' ( y ) because Qf the relations w,=s~u and k'(u) = U ( y ) , it will be enough to show that they are integral over the ring k"[y], or in other words (e.g. by F-App. II, Prop. 6) that they are everywhere finite on W. I n fact, let ~r be any place of k"(y) such that y0r) is finite. Take r independent variables A1," " ",% over ]d'(y), and extend ~r to a place ~r' of U'(y, hl," 9 ",s at which every one of the r points (Xi, M w , , ' ' ",Mw,,~) is finite and =/=(0,.- . , 0 ) . The relation (1), by which y was defined, can be written

~,~ 9 ~ v (T, U) = II (~,T-- X (~,~,~) U~). i:1

#

Taking the values of both sides at ~', we see that the right-hand side does not become identically 0 at that place; as y(~) is finite, this implies that no ~ can become 0 at v ' ; but then wi~(~r) can be written as (~w~)0r')/X~(v') and is finite. This proves the assertion about the w~. We have thus shown that the mappings y*--)w~ of Y* into W are everywhere defined on Y*; as we have ~ t ~ A - ~ ( w ~ ) , this implies that y*--->~ is everywhere defined and maps Y* into the set ~ defined in Prop. 8. Conversely, the definition of y can be written

y(T, U) = H ( T - - X,I,,,(~,a) U~,) {=I

/z

if we call $~(~) the coordinates of ~(~). As 9 is everywhere defined on 9 (W), this shows that the mapping ~-->y is defined at every point of the set ~. As U[y*] is the integral closure of lc'[y] in U(y), it is therefore contained in the integral closure of the specialization-ring of every point of ~ on 8. But we have assumed that W and consequently S are normal, i. e. that the specialization-ring of every point of S (over any field of definition for 8) is integrally closed. This proves that ~--->y* is everywhere defined on the set ~. In view of what we have proved above, ~t is therefore the set of points of S where this mapping is defined, and is K"-open by Prop. 8 of the Appendix; more precisely, it is K'-open if K ' is the compositum of K and It'. This completes the proof. 9.

Denote now by S any cycle ~ (s~) on V, rational over the ground-

field to, consisting of distinct points st and satisfying condition (S). From such a set S, and taking U ~ k, we can derive as above a point y, which we now write as ys, and furthermore a point ys* such that k [ys*] is the integral

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215 ON A L G E B R A I C GROUPS OF T R A N S F O R ~ s

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closure of /c[ys] in /c(ys) ; as above, we call Ys* the locus of ys* over /C; we write gts for the open subset of S denoted by ~ in Prop. 8. I f we allow S to run through any finite set of cycles with the properties stated above, then all the varieties Ys* will be birational]y equivalent to W and to each other, and we can take the points ys* to be corresponding generic points of these varieties over /C. It is then an immediate consequence of Prop. 8 that the affine varieties Ys* (with empty " f r o n t i e r s " ) , and the birational correspondences between them for which the ys* are corresponding generic points of the Ys* over/C, determine an abstract variety S', and that this is biregularly equivalent over a suitable field (as a matter of fact, over K itself) with the union of the open sets ~s on 8. In order to prove that S' will be biregularly equivalent to S itself for a suitable choice of the cycles S, it is therefore enough, in view of the well-known " c o m p a c t o i d " property of open sets in the Zariski topology, to show that the family of all open sets ~s is a covering of 8. I n other words, we have to prove the following: PRoPosITIo~r 9.

Given any point ~t on S, there is a cycle S ~ ~, (s~)

on V, rational over/C, consisting of distinct points s~ and satisfying condition (S), and such that 2 i a e ~ ( W ) for all i. Assume that ~ has a representative as on S~; take x generic over K ( ~ ) ~ K ( a ~ ) on V, and put u ~ (xt~-~)a~, this being defined because W is a chunk. If we put, as usual, ~ o ( x ) and ( ~ a ~ ( u ) , we have then ~2g, so that u ~ , ~ ( 2 ~ ) . As the mapping x--->2a is everywhere defined on 11, this shows that the mapping x--->u of V into W is defined at the points s of V such that 2aE(P(W), and at those points only. Let F be the closed subset of V where the mapping x--->u is not defined; by Prop. 12 of the Appendix, there is a maximal ~-c]osed subset Fo of V contained in F ; then an algebraic point of V over /C is in F if and only if it is in Fo. Call F~ the union of the conjugates over /C of all the components of Fo; this is a /c-closed set on V, and its definition shows that the cycle S on V will satisfy the last one of the conditions stated in Prop. 9 if and only if it lies in V - - F , Now assume first that the field /C is infinite. Applying Prop. 13 of the Appendix to the variety V ~ ~ V - - F , and to the empty subset of V' X V', we obtain a separably algebraic point Sl over /C on V'; call s ~ , . . . , s~ all the distinct conjugates of s~ over /C; if this set satisfies condition (S), which will be the case in particular if d ~ 1, then it solves our problem. Suppose that this is not so, and therefore that d ~ l . For any r ~ d , let sa+~," " ",Sr be any set of r - - d points on V - - F ~ , distinct from one another and from s~," 9 .,sa; put S ' ~ { g ~ , ' 9 ",,~a} and S"~{2a+~," 9 ",~}. I f the set 12

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ANDRI~ WEIL.

S ' U S " is mapped into itself by a right-translation r other than the identity, one of the following circumstances must occur: (i) r maps each one of the sets S', S " onto itself; then r is of the form s':lt ", with s', t" in S', and there must be two elements s", t" of S" such that t " ~ s " r ; (it) r maps S' into S " ; as d ~ 1, we can choose two distinct elements s', t' in S', and then s " ~ s%, t " ~ t ' r are in S", so that we have t " ~ (t's'-~)s"; (iii) r maps some s'~S' onto some # ' ~ S" and some t" e S' ofito some tlP~ S'; then s " ~ s't'-Itl ". Thus, in order to satisfy the requirements of Prop. 9, it is enough to take as sa+l," 9 ", sr the conjugates over k of a point s ~ sa+l of V - - F ~ , separably algebraic over/c, satisfying the following conditions : (a) no 3z, for d -~ 1 ~(&gj-~2) for lF'(x)w for x, w generic and independent over k on G and H', H ' is a homogeneous space with respect to G. But then the unicity assertion in the main theorem of AG can be applied to H and H ' and shows that they are biregularly equivalent; in other words, H itself, with the law f, is a group. This completes the proof. I t is easily seen that the pair (H,a) is uniquely determined, up to an isonlorphism, by the conditions (i), (it), (iii) in Prop. 2 ; in other words, if H ' and a' have similar properties, there is an everywhere biregular birational correspondence between H and H ' which maps a onto a' and transforms the law of composition between G and H into the law between G and H'. The space H may be called the coset-space determined by G and Z, and may be denoted by G/Z; if ]Z 1 is a normal subgroup of G, the space H, with the group-law 1 This is Theorem 1 of Nakano ( [2] ).

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ANDRE WEIL.

determined by Prop. 2, is called the quotient-group (or factor-group) of G by Z, and is denoted by G/Z. 3. Before making another application of Prop. 1, we will introduce a new condition which a law of composition may satisfy. Let V, W be two varieties, g a mapping of V X W into W, and ~ a field of definition for V, W and g; consider the following condition: (TGI')

v~g(x,u),

If x, u are independent generic points of V, W over ~, and then k ( x , u ) ~ k ( x , v ) ~ k ( u , v ) .

The condition k ( x , u ) ~ ( x , v ) is equivalent to ( T G 1 ) of AG, no. 2. The condition k ( x , u ) - - k ( u , v ) implies that the dimension of V, which is the dimension of x over k ( u ) , is the same as that of v over ~(u), and therefore at most that of W; if the dimensions of V and of W are the same, this implies that v is generic over ~(u) on W, which is condition ( I t ) of AG, no. 2. Let k' be any field of definition containing ~ for the birational correspondence u--> v ~ g(x, u) between W and itself; if u is taken generic over lc'(x) on W, we have M(u) ~ / d ( v ) ; since ~(x) C ~'(u,v) by ( T G I ' ) , we have k(x) C ~'(u). Taking u' generic on W over k ' ( x , u ) , we get in the same manner k(x) C ~ ' ( u ' ) . As k ' ( u ) , k'(u') are independent regular extensions of k', their intersection is ~', so that k(x) C k'. This shows that ( T G I ' ) implies ( T G 3 ) . In view of the results of AG, end of no. 3, this shows that, if g satisfies ( T G I ' ) and ( T G 2 ' ) , or if two mappings f, g of V X V into V and of V X W into W are given and satisfy ( T G 1 r,2), then V is a pre-group and W a pre-transformation space, and V operates faithfully o n W.

I f a pre-group V and a pre-transformation space W satisfy ( T G Y ) , we say that W is a pre-principal space with respect to V; if at the same time V and W have the same dimension, so that, as we have shown, W is prehomogeneous with respect to V, we also say that V is simply pre-transitive o n W.

Let IV be a pre-principal space with respect to a pre-group V; by the main theorem of AG, we can construct a group G and a transformation-space S, birationally equivalent to V, W and defined over the same field k; then S is also pre-prineipal with respect to G. Let T be the locus of (u, xu) over ~ on S X S, x and u being independent generic points of G, S over k; put t ~ (u, xu); then ( T G I ' ) implies that /r C ~ ( t ) , i.e. that we may write x ~ ~b(t), where ~b is a mapping of T into G, defined over/~; conversely, if this is so for a transformation-space S with respect to G, S is pre-prineipal.

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499

The space S will be called a principal space with respect to G if, for x, u generic and independent over/r on G, S and for t ~ (u, xu), we have x ~ ( t ) where ~b is an everywhere defined mapping, defined over k, of the locus T of t over k into the group G. I f at the same time S is homogeneous, it will be called a principal homogeneous space with respect to G. PROrOSITION 3. Let S be a pre-principal transformation-space with respect to a group G, both being defined over a field k. Then there is a l~-open subset P of S which is a princ@al transformation-space with respect to G; if G and S have the same dimension, P is uniquely determined and is homogeneous. Let T and ~ be defined as above; call F the It-closed subset of T where q~ is not defined. We first show that, if (a, b) is in F, (sa, s'b) is in F for all s, s' in G. I n fact, take x, u generic and independent over k(s,s') on G, S ; p u t v ~ xu, ul ~ su, Vl ~ s'v, x~ ~ s'xs-1; then we have vl ~ x~ul, and xl, ul are generic and independent over k(s,s') on G, S, so that (u,v) and ( u , Vl) are generic points of T over k(s, s'), and that x ~ ~(u, v), xl ~ ~(u~, v~) by the definition of ~ ; this gives

~(u,v) --s'-l~(su, s'v)s. I f (a,b) is in T, it is a specialization of (u,v) over lc(s,s'), and therefore (sa, s'b) is also in T ; then the above relation shows that r is defined at (a, b) if it is defined at (sa, s'b), i.e. that (a, b ) c F implies (sa, s'b) 9 F. As (e, u) is a specialization of (x, u) over k, e being the neutral element of G, T contains the diagonal A of S X S. As the projection of A on either factor of S X S is everywhere biregular, the projection of the /~-closed subset F A A of A onto S is a /~-closed subset F ' of S, consisting of the points a 9S such that ~ is not defined at (a,a). From what we have proved above, it follows that, if a~ F', sa ~F" for all s ~ G. For the same reason, if a is in S - - F ' , then ~b(a, sa) is defined for all s e G; as (a, sa, s) is then a specialization of (u, xu, x) over k, and x ~ ( u , xu), this shows that ~(a, s a ) - - s for all a e S - - F ' and all s e G, and therefore ~ ( a ~ s ) ~ l~(a, sa); in particular, if x is generic over/c(a) on G, the locus of xa over k ( a ) has a dimension equal to that of G. I f G is complete, every specialization (a, b) of (u, xu) over k can be extended to a specialization (a,b,s) of (u, xu, x) over k, so that b ~ s a ; in other words, every point of T must be of the form (a, sa), with a~S and s c G; then it follows from what we have proved above that such a point cannot be in F unless a and sa are in F p. Without attempting to decide whether this is still so in the general case, we shall merely show that, if u

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is generic over k on S and (u,u') is in F, then u' must be in F'. In fact, suppose that this is not so; take x generic over Ir on G; call X, X' the loci of xu, xu' over ~(u,u') on S; by what we have proved above, they have the same dimension, which is that of G. By F-VI3, Th. 11, we have T N (u X S) ~ u X X ; at the same time, since we have shown that (u, xu') is in T, u X X ' is contained in T N (u X S ) ; as X and X ' have the same dimension, this implies that X ~ X : . But, as we have shown, since (u, u') is in F, (u, xu') must be in F, and therefore, since F is k-closed, u X X' must be contained in F ; as X ~ X ' , this implies that F contains (u, xu), which is generic on T over ~, and contradicts the definition of F. One may observe that, if S is pre-homogeneous, this again Shows that (a, b) cannot be in F unless a, b are in F ' ; for, if a~F" and x is generic on G over tc(a,b), xa has then over k(a) a dimension equal to that of G, and therefore is generic on S over/c(a) since in the present case the dimensions of S and G are equal; then if (a, b) is in F, so is (xa, b), and so b must be in F'. Now replace first S by S - - F ' ; as F ' is mapped onto itself by all operations of G, S - - F ' is again a transformation-space with respect to G, defined over k, and satisfies our other assumptions. Writing again S instead of S - - F ' , we see that it is enough to prove our result under the additional assumption that / 7 ' ~ . If G is complete or S is prchomogeneous, this already implies that S is principal. Otherwise we observe that, since T is also the locus of (z-lu, u) over /r and since we have ~(x-lu, u) --x, T and F are mapped onto themselves by the mapping (u,v)---. (v,u) of S X S onto itself. Call now F " the "projection" of F on either factor of S X S (in the sense of F-IV3 and F-VIIi, i.e. the closure of the set-theoretic projection); this will be the same, whether we project F onto the first or the second factor, and it is not S by what we have proved above, since F ' is empty; it is therefore a ~-closed subset of S. By what we have proved, F " is mapped onto itself by all operations of G. Then S - - F " is the principal space whose existence was to be proved. Finally, assume that the space S from which we first started was prehomogeneous; this means that T ~ S }(S. Let a, b be any two points in S - - F ' ; then, if x is generic on G over to(a, b), xa, xb are generic on S over k(a, b), and so there is an isomorphism r of to(a, b, xa) onto k(a, b, xb) over ]c(a,b), mapping xa onto xb; then we have x~a--xb, i.e. b=x-~x~a. This shows that S - - F " is homogeneous, and also that an open subset of S which is a transformation-space for G cannot contain a point of S - - F " without containing S - - F ' . Therefore S - - F ' is the only open subset of S which is a principal space with respect to G.

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I f N is a principal homogeneous space, the mapping $ of ~ r , S X S into G which has been defined above will be called the canonical mapping of S X S into G. For any a, b on S and s on G, the relations b=sa, s=ck(a, b) are equivalent; in particular, for any a on S and for x generic over /c(a) on G, the mapping x--->xa~v of G into S has the inverse v--->x= $(a,v); as both are everywhere defined, this is therefore an everywhere biregular mapping of G onto S, defined over /c(a). I n particular, if there is at least one rational point a over k on S, S is biregularly equivalent to G over k. 4. Let G be a group, V and W two varieties, and F a mapping of V ; K W into G, all defined over a field k. We may consider W ? < G as a transformation-space with respect to G, the law of composition between them being ( x , ( N , y ) ) ~ ( N , xy) for any N ill W and any x, y in G. We now apply Prop. 1 of no. 1 to the ease when we take for S this transformationspace W X G and for Z the graph of the mapping N-.-->F(M,N) of W into G, where M, N are independent generic points of V, W over /c. We must then consider the smallest field of definition /e' containing ~ for the mapping N-->xF(M,N) of W into G, where x is generic over lc(M,N) on G. As this mapping is defined over /c(x,M), U is a regular extension of /c, contained in/c(x, M). Then Prop. 1 shows that we may write / c ' = It(u), where u is a generic point over /c of a transformation-space U with respect to G; as /~(u) c / c ( x , M ) , we may write u = f ( x , M ) , where f is a mapping of G X V into U, defined over /~; moreover, as the mapping x.--->f(x,M) of G into U is no other than the mapping F defined in Prop. 1, we see that f is defined at every point (s, M) of G ;4 M, and that f(ss', M) = sf(s', 3f) ; taking s ' = e , and writing f(M) instead of f(e,M), this gives f ( s , M ) = s f ( M ) , and in particular u ~ xf(M). As the mapping N--~ xF(3i, N) is defined over lc(u), xF(M,N) is rational over / c ( u , N ) ; similarly, if y is generic over lc(x,M,N) on G, the mapping N..->yxF(M,N) is defined over l~(yu), and so yxF(M,N) is rational over lc(yu, N). As we can write

y = (yxF(M, N) ) (xF(M, N) )-1, this shows that y is rational over k(u, yu, N). If N' is generic on W over lc(x,y,M,N), y must then also be rational over /c(u, yu, N'); thus /c(y) is contained in lc(u, yu, N) and in lc(u, yu, N'); as these are independent regular extensions of tc(u, yu), their intersection is ]c(u, yu), and so we have ]c(y) C lc(u, yu). This means that U is a pre-principal space and may therefore, by Prop. 3, be replaced by a principal space, birationally equivalent to it.

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The most interesting case is that in which there are two mappings F1, F2 of V, W into G, defined over some overfield K of k, such t h a t F ( M , N ) ~ F I ( M ) F 2 ( N ) for M, N generic and independent over K on V, W ; by the corollary of Th. 7, VA-18, this is always so whenever G is an abelian variety. Take x generic over K(M, N) on G, and put z ~ xF~(M) ; then the m a p p i n g N - - > x F ( M , N . ) ~ z F 2 ( N ) is defined over K(z), so that k(u) C K(z). As x, M are generic and independent over K ( N ) on' G, V, u is then generic over K ( N ) on U, and the dimension of U is t h a t of u over K ; the relation K(u) C K(z) shows t h a t this is at most the dimension of G. Therefore U is pre-homogeneous and may be taken to be a principal homogeneous space with respect to G. Moreover, we may write u ~ O ( z ) , where is a m a p p i n g of G into U, defined over K. I f we substitute yx for x, with y generic over K ( M , N , x ) on G, z is replaced by yz, and u by yu; this gives r ~ y ~ ( z ) , which may be written as r and thus shows t h a t ~ is everywhere defined. P u t t i n g now a ~ , ( e ) , we see t h a t a is rational over K and that u ~ z a , i.e. f(M) ~ F I ( M ) a . P u t now g(N) ~ F 2 ( N ) - l a ; g is a m a p p i n g of W into U, defined over K. As we have also g(N) ~ F ( M , N ) - I f ( M ) , g is also defined over the field k(M), and therefore also over k(M') if M ' is another generic point of V over K ; if we take M, M ' generic and independent over K on V, k(M) and k(M') are independent r e g u l a r extensions of k, so t h a t their intersection is k ; hence g is defined over k. Thus we have proved the following r e s u l t : PROPOSlTIO~ 4. Let G be a group, V and W two varieties and F a mapping of V X W into G, all defined over k. Assume that there are two mappings F~, F2 of V, W into G, defined over some overfield K of k, such that F ( M , N ) ~ F ~ ( M ) F 2 ( N ) for M, N generic and independent over K on V, W. Then there is a princ@al homogeneous space U with respect to G, and two mappings f, g of V, W into U, all defined over ~, such that f ( M ) = F ( M , N ) g ( N ) , i.e. F ( M , N ) ~ r where ~ is the canonical mapping of U X U into G. COROLLARY. Notations being as in Prop. 4, U, f and g are uniquely determined by G, V, W and F, up to an isomorphism. I n fact, assume t h a t Up, f', g" have similar properties ; then we have xf'(M)), where r is the canonical m a p p i n g for U'. This shows that the m a p p i n g N---->xF(M,N) is defined over k ( u t) with u ' ~ x f ' ( M ) ; thus, if we put u ~ x f ( M ) as before, we have k ( u ) C k ( u ' ) and m a y write u ~ r where r is a m a p p i n g of U" into U, defined over ~.

xF(M,N) ~ r

[195561

245 ON ALGEBRAIC GROUPS A]qD H O M O G E N E O U S SPACES.

503

Replacing x by yx, with y generic on G over k (M, N, x), we get yu ~ ~ (yu p) ; from this we conclude, in the usual manner, t h a t ~ is everywhere defined. Take any point a' on U', and put a ~ ( a ' ) ; as we have x a ~ r and as the mappings x--->xa, x - o x a ' are everywhere biregular mappings of G onto U and U', defined over k(a'), we see t h a t ~ is an everywhere biregular m a p p i n g of U' onto U. Moreover, we have ~ ( u ' ) ~ xtp(f'(M)), and therefore f ~ ~ o f'; from this one easily concludes t h a t g ~ ~ o g'. This proves our assertion. 5. Let G be a group, defined over a field k. We will now prove t h a t the classes of principal homogeneous spaces with respect to G, for birational equivalence over k, form a set. I n fact, let x, y be independent generic points of G over k; let (~ be the isomorphism of ~ ( x ) onto ~(yx) over ~ which maps x onto yx. Let H be any principal homogeneous space with respect to G, defined over k ; let a be an algebraic point over k on H , and p u t u ~ xa, so that u is generic over 7~ on H. Then ]c(u) is a regular extension of 7c contained in ~ ( x ) and such t h a t ~ ( u ) = ~ ( x ) ; moreover, we have

~(y,u) =k(y,u~)

= ~ ( u , u~)

since u ~ y u . Conversely, let ~ ( u ) be any such extension of k, and call U the locus of u over k; then we may write u r g (y, u), where g is a m a p p i n g of G X U into U, defined over ~; and one verifies at once t h a t this makes U into a pre-principal pre-homogeneous space with respect to G, and thus determines uniquely a class of birationally equivalent principal homogeneous spaces with respect to G. As every such class is determined by at least one such extension, this shows that these classes form a set. I f G is commutative, one can define canonically a commutative groupstructure on the set of classes of principal homogeneous spaces with respect to G. I n order to do this, we first observe that, if H is any transformationspace over a commutative group G, then the law (x,u)-->x-lu, for x~G, u e H , defines on H a structure of transformation-space with respect to G; this will be called the opposite transformation-space to H and will be denoted by H - ; it is a principal homogeneous space if H is such. PROPOSITION 5. Let G be a commutative group, defined over a field k. Let H~, for 1 ~-- i X n, be principal homogeneous spaces with respect to G, defined over k. Then there is a principal homogeneous space H with respect to G, defined over tr and an everywhere defined mapping f of HI }( H2 X " 99X H,~ into H, defined over ~, such that f(s~a,.

9 .,sna~)

=

s~"

. . s,,f(a,.

9

a~)

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for all s~e G and a~eH~. Moreover, H and f are uniquely determined up to an isomorphism of H. Put V ~ W ~ HI X H2 X " " " X 1t,,; call ~ the canonical mapping of H, X H~ into G, so that b~~ sa, is equivalent to s ~ q~,(04, b~) for a~, b~ in H~ and s in G. Let u = ( u ~ , . - .,u~) and v ~ ( v ~ , . 9 .,v,,) be two points of V; put

F(u, v) = fI +,(u,, ~,), i=I

where the right-hand side has a meaning since G is commutative. H~, choose a point 04, and put a ~ ( a , . 9 .,a~). We have

r

vd ~ r

vdr

On each

ud -~

for all i, as one verifies at once, and therefore, again because of the commutativity of G :

F ( u , v ) = F ( a , v ) F ( a , u ) -1. Thus the assumptions of Prop. 4 are satisfied, so that there is a principal homogeneous space U and two mappings f, g of V into U, all defined over /c, such that (1)

f(u) =F(u,v)g(v),

F ( u , v ) = , ~ ( g ( v ) , f ( u ) ),

where ~ is the canonical mapping of U )< U into G. Take any point b on V, and take v generic over k(b) on V; as F is defined at (b,v), the relation (1) shows that f is defined at b. Thus f is everywhere defined. As F ( u , u ) ~ e, the relation (1) gives g ( u ) = f ( u ) , i.e. f ~ g . If s l , ' ' ' , s ~ are any elements of G, and we put s ~ s l " 9 .s.~ and u ' ~ (slu~,. 9 ",s,u,,), we have F ( u ' , v ) ~ s - I F ( u , v ) and therefore, by (1), f ( u ' ) ~ s - l f ( u ) . If we now put H ~ U-, i.e. if we take for H the opposite space to U, H and f will have the properties stated in Prop. 5. Let us now assume that ~ and ~ have similar properties; put F ~ / ~ - . Put 2 = F ( u , v ) - ~ ] ' ( u ) , the multiplication in the right-hand side being that of ~7. I f the s~, s and u" have the same meaning as above, we have ]'(u')=s-~]'(u), so that ~ does not change if one replaces u,v by u',v. Therefore /c(2) is contained both in k ( u , v ) and in k(u',v). If the s~ have been taken generic and independent over lc(u,v) on G, t~(u,v) and lc(u',v) will be independent regular extensions of k ( v ) ; this gives k ( 5 ) C k(v), so that we may write 2 = t T ( v ) , with j defined over k. Then we have ]'(u) = F ( u , v ) g ( v ) ; by the corollary of Prop. 4, ~7, ~ and j must then be

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505

the same as U, f and f, respectively, except for an isomorphism of U onto U. This proves the assertion about unicity in Prop. 5. In Prop. 5, take n ~ 2; call ~1, ~ 2 the classes of H1, H2, and denote by t~l ~ ~ 2 the class of H. This defines on the set of classes of principal homogeneous spaces with respect to G a commutative group-structure. In fact, commutativity is obvious. Call ~ o the class of G, and therefore of all principal homogeneous spaces with respect to G which have a rational point over k. For any principal homogeneous space H with respect to G, the mapping f ( x , u ) ~ x u of G X H into H satisfies the condition of Prop. 5; there-fore we have ~ o ~-t~ ~ 9/ for all classes ~ . If ~ is the canonical mapping of H X H into G, then ~, considered as a mapping of H X H- into G, satisfies the condition of Prop. 5 ; therefore, if ~ - is the class of H-, we have 5~ -~ &t- ~ 9/0. Finally, let H1, H2, H~ be three principal homogeneous spaces with respect to G ; apply Prop. 5 successively to the following spaces: (a) to H~, H2, obtaining a space H12 and a mapping f12; (b) to H12, H3, obtaining H', f'; (c) to H2, H3, obtaining H2~, f2a; (d) to H~, H~3, obtaining H", f"; (e) to H1, H2, H3, obtaining H, f. Then the two mappings

f'(f~,~(u,u~),u,),

f"(u,f~(u~,m))

of H1 }( H2 X Ha into H', H'" satisfy the same condition as the mapping f. By the unicity assertion of Prop. 5, this shows that H', H" are isomorphic to H. This means that the addition &t~ + ~2 is associative. One proves quite similarly, by induction on n, that if :~ and ~ , are the classes of the spaces H, H, in Prop. 5, then ~ ~ ~ . ~ . In fact, let H', f' be the space and the mapping obtained by applying Prop. 5 to H~,. 9 -, H,~, so that ~ ' ~ & q + . . . + 5 ~ _ ~ by the induction assumption; and let H", f" be the space and the mapping obtained by applying Prop. 5 to H', H,, so that ~ " ~ ~ ' - ~ ~ by definition. Then the mapping

(ui,

.,u.) ~f"(f'(~.

,~_~),u,,)

of H~ X" 9 ' } ( H ~ into H " has the properties stated for f in Prop. 5~ so that, by the unicity assertion in Prop. 5, H" is isomorphic to H. From this one deduces that every element ~ of the group we have just described is of finite order. In fact, take on a space H of class ~t any positive cycle ~ a~ of dimension 0, rational over k.

Call H , any space of

class nt~; then there is a mapping f(u~," . ",u,) of the product of n factor~ equal to H into H , with the properties stated in Prop. 5. From the unicity assertion in Prop. 5, it follows that any permutation of the u, will change f

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into sf, with s~ G; as f is everywhere defined, we see that s ~ e by t a k i n g ul . . . . . u~; therefore f is a symmetric function, so t h a t f ( a l , - - -,a~) is rationM by the m a i n theorem on symmetric functions (VA-7, Th. 1). So H~ has a rational point over k, and is therefore isomorphic to G. Now, ~ being as before, put Ho ~ G and take, for each integer n ~ 0, a space H~ of class n ~ so that all the H . are disjoint. On the set | ~ U H~ yt

(which is of course not an algebraic v a r i e t y ) , we will define a commutative group-law f (in the sense of group-theory, not of algebraic geometry) such that Ho ~ G will be a subgroup of @ and that f induces on H ~ X H~, for all m, n, a m a p p i n g fm.~ of H,~ X H~ into H ~ satisfying the conditions in Prop. 5. As there is such a m a p p i n g f~.~ for each m, n, and as it is uniquely determined up to an automorphism of H ~ (i. e. up to left-multiplication by a rational point of G), we merely have to choose the f,~.~ so that the m a p p i n g f of (~ X ~ into @ which coincides with f~,~ on Hm X H , for all m, n satisfies the axioms for groups ; we do this as follows. F o r any n, we take fo,~(x, u) ~ xu for x c G , u~H~. We choose f-l.~ and, for all n ~ 0, f~,~ and f_~_~ so as to satisfy the conditions in Prop. 5. Now, for elements Ul," " ",u~+l of H1 in any number, we define ul" 9 .u~+~ inductively as being equal to ul for n ~ 0 and to f~,~(u~" 9 - u~,u.+~) for n ~ 1, similarly, for elements v~," 9 ',v,+~ of H_~, we define v~. 9 9v~+l as equal to vl for n ~ 0 and to f-~,-l(vl" " "v., v~+~) for n ~ 1. I t is then easily seen that, whenever m, n are both ~ 0, there is one and only one way of choosing fro,, so t h a t it satisfies the condition f ~ , , ( u ~ ' 9 -u~, u~+l" - 9u~+,) ~ ul" - 9u ~ when the u~ are in I-I1, we determine f_~_~ similarly, using H_~ instead of H , Finally, for m => n ~ 0, we choose f,~_~ and f_~,, so as to satisfy the conditions n

f~,~(ul"

9 .u~, vl"

9 .v,)

~

I I f - l . ~ ( v , , ~ , ) 9 u~+~. 9 . u ~

I-~,-(~,"

9 '~m,~"

" .u.)

=fIf-~,l(~,,~)

'v.+l"

9 .~

4=1

respectively, the u~ being any elements of H1 and the v~ any elements of H _ , I t is then a trivial m a t t e r to verify t h a t these choices of the f~., satisfy all the requirements for a commutative group-law on | The points on the H~ which are rational over k form a subgroup g of | As we have shown t h a t there are such points for some n ~ 0 , there is a smallest n ~ 0 for which there is such a point a ~ H ~ , this n is the order of ~ in the group of classes of principal homogeneous spaces with respect to G. Then ~ is the direct product of the group g o n g A G of rational

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points over k on G and of the infinite cyclic group ), generated by a. The quotient-group | may be described as an algebraic group consisting of n components respectively isomorphic to Ho ~ G, H ~ , - - - , H , _ , 6. PROPOSITION 6. Let A be an abelian variety and H a principal homogeneous space with respect to A, both being defined over a field k. Let V1," 9 ", V~ be varieties, and F a mapping of VI X 9 " " X V~ into H, all these being defined over k. Then there is for each i a principal homogeneous space H~ with respect to A and a mapping F~ of V~ into H~, Ht and F~ being defined over k, and there is a mapping f of H1 X " 9 " X H , into H with the properties stated in Prop. 5, such that, for (M1," 9 ",M~) generic over k on V 1 X " 9 " X Vn, we have F ( M I , . 9 .,Mn) ~ f ( F l ( M ~ ) , "

9 ",F,(M~) ).

Moreover, all these are uniquely determined up to isomorph@ms. For n ~ 1, there is nothing to prove. I f the assertion is proved for a product of two factors, then this can be applied to the product V I X ( V 2 X " ' " X V ~ ) of V1 and V2X" 9 " X V ~ , so that the general case follows by induction on n. Thus it is enough to treat the case of two factors V, W and of a mapping F of V X W into H. Call ~ the canonical mapping of H X H into A; let ( M , N ) and ( M ' , N ' ) be two independent generic points of V X W over k; and put x~(F(M,N'),F(M',N')

),

y~(F(M,N),F(M,N')

).

so that we have xy = ~,(F(M, N ) , F ( M ' , N') ). As the mapping ( (M, M'), N p) --> x of (V X V) X W into A has the constant value e on the variety (M, M) X W, Th. 7 of VA-18 shows that x is rational over k ( M , M ' ) ; for a similar reason, y must be rational over k ( N , N ' ) ; in other words, there are mappings (I), $ of V X V, W X W into A, both defined over k, such that x ~ r and y ~ r By the corollary of Th. 7' of VA-18, 9 and q, satisfy the assumptions of Prop. 4, so that there are two. principal homogeneous spaces U~, U.~ with respect to A, mappings F~, G~ of V into U~ and mappings F2, G2 of W into U2, all defined over k, such that F~(M) ~ x G I ( M ' ) , F 2 ( N ) ~ y G 2 ( N P ) ; moreover, as r ~ ( N , N ) are defined and equal to e, we have G1 ~ F 1 , G~ ~ F 2 . Now call H~, H2 the spaces respectively opposite to U1, U2, and apply Prop. 5 to H~, H2: let / t be the principal homogeneous space and ~ the mapping of H~ X H~ into /~ with

250

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ANDRI~ WEIL.

the properties stated in that proposition. As H1, H2 are opposite to U~, U2, w e have

F~(M') ~ x F I ( M ) ,

Put P(M,N)=](FI(M),F2(N)).

F2(N') ~ y F 2 ( N ) ,

multiplication in the right-hand sides being understood in the sense of H1, H2. By the definition of ~, we have then:

P(M',N') = (xy)#(M,N), while, as we have seen above, the same relation holds if F is substituted for iV. But then, as the corollary of Th. 7, VA-18, shows that the mapping

( (M',N'), (M,N) ) ---->xy of (V X W) X (V X W) into A satifies the condition of Prop. 4, the corollary of Prop. 4 shows that H, F must be the same as H, F except for an isomorphism of H onto H. Then, replacing ]~ by a mapping f of:H1 X H2 into H by means of that isomorphism, we have the spaces HI, H2 and the mappings F1, F2, f whose existence was asserted in our proposition. As to unicity, assume that there are spaces Hi*, H~* and mappings F~*, F2*, f* with the same properties. Then, x being defined as above, or equivalently by F(M', N') ~ xF(M, N'), we have

f*(F~*(M'), F2*(N')) ~ xf*(F~*(M), F~*(N')) = f*(xF~*(M), F2*(N')) and therefore F~*(M')= xFI*(M) since the mapping u---->f*(u,v) of H~* into H is, as easily seen. an everywhere biregular mapping of H~* onto H. But then the corollary of Prop. 4, applied to the mapping (M',M)-->x of V X V into A, shows that H~*, FI* are the same as H~, F~ except for an isomorphism. The same argument applied to y instead of x shows that H2, F~ are uniquely determined up to an isomorphism. Then Prop. 5 shows that f is uniquely determined. This completes the proof. 7. The foregoing results will now be applied to the theory of Jacobian varieties. As in VA-35, we consider a complete non-singular curve F of genus g > 0, defined over a field ~. I f a is any divisor on F, Prop. 6 of the Appendix of AG shows that there is a smallest field containing k over which a is rational; this field will be denoted by/r (a). In particular, if M1," 9", M~ are independent generic points of F over k and if we put m ~ ~ M~, then, i

by VA-4, Lemma 1, to(m) is the field k(M1,- 9 .,Mg)~ of symmetric functions of M~,.. ",M~, defined over k, i.e. the subfield of to(M1,. 9 .,Mg) consisting of those elements which are invariant under all permutations of

[1955b1

251 ON

ALGEBRAIC

GROUPS

AND

ts

SPACES.

509

M1," 9 ",Mg; such a divisor m will be called generic over k. As lz(m) is a regular extension of k, we may write it as k(u), where u is a generic point of a variety W over k, and we may write u ~ F ( M 1 , ' 9 .,Mg), with F defined over k ; as F is symmetric in the M~, this may also be written as u ~ F ( m ) . Now let the N~, P~, for 1 ~ i ~ ( p ) determines a representation of r (the Galois group of K over k) as a group of permutations on the gp. F o r a given p, let -/p be the subgroup of r determined by co(p) ~ p ; then, for o e F, o)(p) takes a number of distinct values equal to the index dp of 7p in r . I f Kp is the subfield of K consisting of the elements of K invariant under ~,p, gp is defined over Kp ; therefore, if ( ~ , " 9 ", ~ p ) is a basis of Kp over bo, we may write gp ~ ~ ~hp~, where the hp~, for 1 ~ v x ' ~ f t , ( f t - l ( x ) ) ; and qst induces on Vt, the mapping x'---> x, i. e. the mapping ft o fr -1. Applying L e m m a 2, we see that gt is defined at ( t ' , a ) whenever a is a point of Vt where ft' oft -~ is defined, and that ~bt is defined at every point of Vt, where ft oft, -~ is defined. Therefore gt is biregular at (if, a) whenever a is a point of Vt where fr oft -~ is biregular. Now let Ao be the k(t)-closed subset of T X Vt where gt is not biregular ; and assume first that a is a point of Vt with the property stated in (i). Then (t'~ a) is not in Ao, so that T >( a is not contained in Ao ; let Ax be the (non-dense) k(t, a)-closed subset of T consisting of those points t~ such that ( t , a ) eAo. By [4], App., Prop. 12, there is a ~-closed subset A2 of T containing all ~-closed subsets of T contained in A1; in particular, every point of A~ which is algebraic over k must be in A~. Let As be the union of the components of A2 and of their conjugates over /r put T ' = T - - A 3 ; this is a k-open subset of T such that, if tl is any algebraic point over k in T', gt is biregular at (tx, a). On the other hand, assume, as in (ii), that ft, oft -~ is everywhere biregular. Then gt is biregular at every point of t' X Vt, so that Ao has no point in common with t' X Vt. This implies that the projection of Ao on T is non-dense in T, so that, if we call A~' the closure of that projection, it is a (non-dense) /c(t)-closed subset of T. Let A2' be the maximal J~-closed subset of T contained in A~'; let Aa' be the union of the components of A~' and of their conjugates over k ; put T " ~ T - - AJ. Then T " is/c-open on T ; and, if t~ is any algebraic point over /~ in T", gt is biregular at every point of t~ X Vt. Now let t~ be a separably algebraic point over k in T' (resp. T " ) ; if k is finite, we may take for t~ any algebraic point over 1c in T ' (resp. T " ) ,

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since in that case every algebraic extension of k is separable; if k is infinite, we apply [4], App., Prop. 13. Let ti," 9 ", t~ be the distinct conjugates of tl over k. As they are in T' (resp. T ' ) , gt is biregular at (h,a) (resp. at every point of h X Vt), and a fortiori at (h, x), for 1--~ i -~- ~; therefore it induces on t~ X Vt a birational correspondence g~ between Vt and the locus V~ of the point g ~ ( x ) ~ g t ( h , x ) over l~(t,t~) in the projective (resp. affine) ambient space of X ; and g~ is biregular at a (resp. at every point of Vt). But, as we have already observed, the relation/~ (x) ~ k (t, v) shows that X is birationally equivalent to T X V; we may write x ~ f ( t , v), where f is a birational eorrespondence between T X V and X, defined over ]c; then we have x ' ~ f ( t ' , v ) ; and f is the product of Yt and of the birational correspondence (t', v) --> (t', x) between T X V and T X Vt. As the latter correspondence is biregular at (h, v), and gt is biregular at (t~, x), for I ~ i ~ n, we see that f is biregular at (h, v), and that we have

g~(~) = g, ( t,, ~) = i ( h, ~). As the point f ( G v ) has the same locus over ]~(h) as over k(t, h), this shows that Vt is defined over k ( h ) . As every automorphism of ~ over k can be extended to an automorphism of ~(v) over k ( v ) , this also shows that V~ is the transform of V~ by the isomorphism of ]a(t~) onto ]~(h) over k which maps t~ onto h. Also, if f~ is the mapping of V into V~, defined over k ( h ) , which is such that f ~ ( v ) ~ f ( h , v ) , we have f ~ g ~ o f t ; and f~ is the tran~form of f~ by the isomorphism of ]~(t~) onto ~ ( h ) over ]a which maps t~ onto t~. Now apply Proposition 1 to the variety V, defined over the groundfield k, to the variety V~, defined over k(t~), and to the birational correspondence f~ ; this gives a projective (resp. afflne) variety W and u birational eorrespondence F between V and IF, both defined over k, such that F o f~-~ is biregular wherever all the f~ o f - 1 are defined, i.e. wherever all the gi o gt -~ are defined, Now, in ease (i), all the g, are biregular at a, so that all the g~og~-~ are biregular at the point gJa) ; therefore F o ft-% which is the same as (F o f~-~) o gl, is biregular at a; as this involves merely a local property of W at the image of a by that mapping, we may replace W, in the projective ease, by one of its afflne representatives. Thus we have solved our problem in ease (i). I n ease (ii), g, is biregular at every point of Vt; as we have just shown, this implies that F o f t -1 is biregular at every point of Vt, so that it determines an isomorphism of Vt onto a k(t)-open subset W' of W. The assumption in (ii) implies that 1'11' is invariant under the isomorphism of ]~(t) onto k ( t ' ) over /~ which maps t onto t'. From this and from [4], App., Prop. 9, it follows easily that W' is k-open; thus (IV', F) is a solution of our problem.

[19561

303 T:HE F I E U D OF D E F I N I T I O N

OF A V A R I E T Y .

5~1

COROLLARY. Let t~, T, t and t' be as in (C) ; let V be a variety, defined over lc; let Vt be a variety, defined over k ( t ) ; let ft be a birational correspondence between V and Vt, defined over k ( t ) and such that ft, oft -~ is everywhere biregular . Then, if a is any point of Vt, there is an alfine variety W and a birational correspondence F between V and W, both defined over k, such that F o f t -~ is biregular at a. We may assume that t" has been taken generic on T over k ( t , a ) ; take t" generic on T over k ( t , t ' , a ) . Call a', a" the images of a by ft, oft -~ and by ft" o ft-% respectively. The isomorphism of /~(t, a, t') onto k(t, a, t") over /~,(t,l~) which maps t' onto t" maps a" onto a"; therefore, if Vt,~ is a representative of the (abstract) variety Vt, on which a' has a representative a,', the point a" of Vt,, has a representative a,~" on Vt,,a. Let ft', be the hirational correspondence between V and Vt,~ which is determined by ft'. As ft,, o ft, -~ is everywhere biregular and maps a' onto a", ft"a oft,, -~ is biregular at a~'. Applying Proposition 2(i) to V, Vt,~ and ft'~, we get a solution ( W , F ) of our problem. 6.

~ow we can deal with problems (D) and (C').

THEOREM 5. Problem (D) has a solution if and only if ft, oft -1 is everywhere biregular for t' generic over It(t) on T. The condition being obviously necessary, assume that it is fulfilled. By the corollary of Proposition 2, there is, to every point a of Vt, an affine variety W~ and a birational correspondence Fa between V and We, both defined over /c, such that F~oft-~ is biregular at a; call ~ the /~(t)-open subset of Vt where F~ oft -~ is biregular, and call Wa' its image on Wa by F~ o ft-% which is a /~(t)-open subset of Wa. Then WJ is the subset of W~ where ft o F~-~ is biregular ; as in the proof of Proposition 2, this implies that Wa" is invariant under the isomorphism of /~(t) onto /~(t') over ~ which maps t onto t', and we again conclude from this that W~' is /c-open. As we have a ~ ~2~ for every a e Vt, the open sets r form a covering of Vt; by the well-known " c o m p a c t o i d " property of open sets in the Zariski topology, there must be finitely many points as on V such that the sets ~a~ cover Vt. Then the k-open subsets W~' of the afflne varieties W~o, together with the birational correspondences Fap o F~. -~ between them, define an abstract variety, which, together with the obvious birational correspondence between it and V, solves our problem. THEOREI~[ 6.

Problem (C') has a solution, i.e., problem (C) has a

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solution (V, ft) for which ft is e~,erywhere biregular, if and only if ft',t is everywhere biregular and satisfies eondHiott (i) in Theorem 4. The solution 'is unique up to a k-isomorpl~is.m. This is an immediate consequence of Theorems 4 and 5. 7.

As to the projective or affine embeddability of the solution of prob-

lems (D) and (C'), we have the following result. THEOREM 7.

Let V be a variety, defined over a field k, and projectively embeddable over an overfield K of k. Then V is projectively ( resp. a]finely ) (resp. a~nely) embeddable over lc provided (i) K is separable over k or (it) 1; is everywhere normal with reference to 1,'. The assumption means that there is a birational correspondence f, defined over K and biregular at every point of V, between V and a subvariety of a projective (resp. affine) space; if we regard f as a mapping of V into that space, it has a smallest field of definition ~' containing k; we may replace K by k ' ; after that is done, K is finitely generated over k. I f K is separable over k, it is a regular extension k l ( t ) of the algebraic closure kl of k in K, and kl is a separably algebraic extension of k of finite degree. Proposition 2(it) shows that V is then projectively (resp. affinely) embeddable over k~; by Theorem 2, this implies that the same is true over ~; this completes the proof in case (i). I f K is not separable over k, let ]c* be the union of the fields kP-'~, for n = 1 , 2 , . . . ; then the compositum K* of K and k* is separable over k*, so that, by what we have just proved, V is projectively (resp. affinely) embeddable over k*. In order to deal with case (it), it is therefore enough to prove our theorem in the case in which V is everywhere normal with reference to k, and K is purely inseparable over ]~; I owe the proof for this to T. Matsusaka; it is as follows. We may again assume that K is finitely generated over /c; as it is purely inseparable, it is contained in some field ] d ~ lWq, where q is a power of the characteristic. Then there is a mapping f' of V into a projective (resp. affine) space, defined over Z", such that f' determines a birational correspondence, biregular at every point of V, between V and the closure W' of its image by f'; then W' is a projective (resp. affine) variety, defined over .~', and f' determines a /d-isomorphism between V and a It'-open subset of W'. Call the automorphism ~---)~q of the mfiversal domain; put W ~ W'~; W is then a projective (resp. affine) variety, defined over k. Let x be a generic point of V over /c; then W' is the locus of the point y ' ~ f f ( x ) over /d, and W is

[1956]

305 TItE

FIELD

OF D E F I N I T I O N

OF A V A R I E T Y .

523

the locus of the point y~y"~ over k. As y' is rational over k'(x), y is so over k ( x ' ) ; we may write y ~ g ( x ) , where g is a mapping of V into W, defined over k; as we have k ' ( y ' ) ~ / c ' ( x ) , we have k ( y ) ~ I c ( x ' ) , which implies that k(x) is purely inseparable over k(y). I n the projective case, let U be the projective variety derived from W by normalization in the field k ( x ) ~ ; U is birationally equivalent to V over k; let z be the point of U which corresponds t~ x on V. I n the affine case, we take for z a point in a suitable affinc space such that k[z] is the integral closure of the ring k[y] in the field k(x), and for U the locus of z over /C. I n either case we may write z ~ f ( x ) , where f is a birational correspondence between V and U, defined over k. By definition, U is everywhere normal with reference to k, and the mapping h ~ g o f-~ of U into W is everywhere defined and such that the (settheoretic) inverse image of every point of W for that mapping consists of finitely many points of U. Let a be any point of V; let (a, b) be a specialization of (x,z) over x---)a with reference to k,; then, as h is defined at b, (a,b,h(b)) is a specialization of (x,z,y) over k. As f' is defined at a, g is also defined there, so that we must have h ( b ) ~ g ( a ) ; therefore b is one of the finitely many points of U whose image by h is g(a). As V is normal at a by assumption, with reference to /c, this implies that f is defined at a, and that we have b = f ( a ) . We have g ( a ) ~ f ' ( a ) G hence f ' ( a ) = g ( a ) ~ - ' ; as g(a) = h ( b ) , this shows that f ' ( a ) is the unique specialization of y' over z---) b with reference to k"; as f' is biregular at a, f'-~ is defined at f'(a), and therefore x has no other specialization than a over z---)b with reference to U, hence also with reference to k by F-II~, Prop. 3. As U is normal at b, with reference to /C, this implies that f ~ is defined at b ~ f ( a ) . We have thus shown that f is biregular at every point of V, so that it is a k-isomorphiim between V and a /c-open subset of U. As a special ease (already contained in Proposition 2), we see that, in problem (D), V' is projectively (resp. affinely) embeddable over k if Vt is so over/c(t) ; similarly, in problem (C'), V is projectively (resp. affinely) embeddab]e over /C if Vt is so over /c(t). 8. In [4], the construction carried out in Nos. 7-9 can be advantageously replaced by the application of our Theorem 6 to the situation described in No. 6 of that paper. The application is entirely straightforward, so that no further details need he given; this shows that the recourse to the Lang-Weil 1U is the "derived normal mt~del of W in tile field k(x)" according to Zariski's delinition (!5], pp. 69-70); of. also [3].

306

[1956] 57)~4

ANDRt~ WEIL.

Theorem, J.e.. in substance, to the so-called "Riemann hypothesis" in the case of a finite groundfield (loc. cir., p. 374) was unnecessary; so is the assumption of normality in the final result (loc. cir., p. 375) ; normality had to be assumed there merely because of the use made of the Chow point in the construction on p. 370, whereas in the present paper a different device was adopted (in the proof of Proposition 1). Of course, in the main theorem of [4] (p. 375), parts (i) and (ii) remain unchanged. For the sake of completeness, we give here the improved result by which part (iii) of that theorem may now be replaced: PROPOSITION" 3. Let G be a group and W a chunk of transformation-space with respect to G, both defined over t~. Then there is a transformation-space S with respect to G, and a birational correspondence f between W and S, both defined over lc, with the following properties: (a) f is biregular at every point of W; (b) for every s e G and a e W such that sa is defined, we have f(sa) = s f ( a ) ; (c) every point of S can be written in the form sf(a), with s e G and ae W. Moreover, S is uniquely determined by these properties up to a k-isomorphism compatible with the operations of G. UNIVERSITY OF CHIOAG0.

REFERENCES.

[1] W. L. Chow, "Abelian varieties over function-fields," Transactions of the American M a t h e m a t i c a l Society, vol. 78 (1955), pp. 253-275. [2] S. Lang, "AbeIian varieties over finite fields," Proceedings of the National A c a d e m y of Sciences, vol. 41 (1955), pp. 174-176. [3] T. M a t s u s a k a , "A note on m y paper ' S o m e t h e o r e m s on abelian varieties '," N a t . Se. Rep. Ochanomizu Univ., vol. 5 ( ] 9 5 4 ) , pp. 21-23. [4] A. Weil, " On algebraic g r o u p s of t r a n s f o r m a t i o n s , " A m e r i c a n J o u r n a l of Mathematics., vol. 77 (1955), pp. 355-391. [5] O. Zariski, " Theory and a p p l i c a t i o n s of holomorphic f u n c t i o n s on algebraic varieties over a r b i t r a r y groundfields," M e m o i r s of the A m e r i c a n M a t h e m a t i c a l Society, no. 5 (1951), pp. 1-90.

[ 1957a] Zum Beweis des Torellischen Satzes

Vorgelegt v o n t t e r r n 1~I. D e u r i n g

in d e r S i t z u n g v o m 8. F e b r u a r 1957

Die yon Siegel geschaffene Theorie der hSheren Modulfunktionen kann als Theorie der Moduln ffir abelsche FunktionenkSrper angesehen werden; ihre Anwendung auf das klassische Problem der Moduln algebraischer Kurven beruht auf einem Satz yon 1~. Torelli, der ungefi~hr besagt, dab eine algebraische Kurve durch die Normalperioden der abelschen Integrale 1. Gattung auf der Kurve v611ig bestimmt ist. Der Beweis, der von Torelli selbst ffir diesen Satz gegeben wurde (Rend. Ace. Lincei [V] 22 [1914], p. 98), ist im wesentlichen algebraisch-geometrischer Natur. Eine moderne, vSllig einwandfreie und lfickenlose Darstellung dieses Beweises liegt nieht vor; dasselbe grit auch ffir den durchaus klassisch formulierten und an mehreren Stellen ziemlich skizzenhaften Beweis yon A. Andreotti (Mdm. Acad. Belg. 27 [1952], fase. 7). In der vorliegenden Arbeit soll der Torellische Satz in einer solehen Fassung formuliert und bewiesen werden, dab er auch im ,,abstrakten Fall" (beliebiger Charakteristik) seine Gfiltigkeit behalf. Die ganz einfache Beweisidee, die mit dem Andreottischen Ansatz eng verwandt ist, ist leider durch teehnische Schwierigkeiten etwas entstellt, welche es nStig machen, die F~lle niedrigeren Gesehlechtes besonders zu behandeln. I. Zur abstrakten Formulierung des Torellischen Satzes braucht man den Begriff einer ,,polarisierten" abelschen Mannigfaltigkeit (vgl. meinen Vortrag ,,On complex multiplication", Tokyo Symposium on Numbertheory, 1955, sowie eine demn~chst erscheinende Arbeit yon T. Matsusaka im American Journal o/ Mathematics). Zun~chst sei A eine abelsche Mannigfaltigkeit im klassischen Fall, d.h. fiber dem komplexen Zahlk6rper; der zugehSrige FunktionenkSrper sei der KSrper aller periodischen meromorphen Funktionen in einem n-dimensionalen Vektorraum R fiber dem komplexen Zahlk6rper mit einem vorgegebenen Periodengitter G vom Rang 2 n. Damit G tats~chlieh Periodengitter eines solchen KSrpers der Dimension n sei, ist bekanntlieh notwendig und hinreiehend, dal3 es eine auf G • G ganzzahlige alternierende Reprinted from G6ttingen Nachrichten 1, 1974, by permission of Akademie der Wissenschafien GOttingen.

307

308

[1957a] Andrd Weil

34

Bilinearform B gebe, die Imaginiirteil einer im Raum R positiv definiten Hermiteschen Bilinearform ist. Nun sagt man, dab jede solche Bilinearform B eine Polarisierung der abelsehen Mannigfaltigkeit A ----RIG bestimmt, wobei zwei Formen B, B' dann und nur dann dieselbe Polarisierung von A bestimmen, wenn sie sich nur um einen konstanten Faktor unterseheiden. Man sagt, dab A polarisiert ist, wenn man auf A eine bestimmte Polarisierung gew/~hlt hat. Wenn s/~mtliehe Elementartefler der Form B einander gleich sind, sagt man, dab die dureh B polarisierte Mannigfaltigkeit A zu der Hauptschar der polarisierten abelsehen !~annigfaltigkeiten der Dimension n geh6rt. Wenn eine abelsche Mannigfaltigkeit keine komplexen Multiplikationen (d. h. keine nieht-trivialen Endomorphismen) besitzt, dann ist sie nur auf eine Weise polarisierbar. Man kann sieh leieht iiberzeugen, dab es sieh in der klassischen Theorie der abelsehen Funktionen (z.B. in den Arbeiten von Hermite und Humbert) meistens um polarisierte abelsohe Mannigfaltigkeiten handelt; nur dann, wenn von solehen Mannigfaltigkeiten die Rede ist, durf man von ihren Moduln spreehen. Z.B. ist die Siegelsehe Theorie der hSheren Modulfunktionen niehts underes als eine Theorie der l~oduln ftir die Hauptschar der polarisierten abelschen l~annigfultigkeiten einer gegebenen Dimension, indem ein System soleher !~Ioduln dureh einen Punkt im Fundamentalbereieh der Modulgruppe bestimmt ist. Im ubstrukten Fall sei X ein nicht-ausgearteter positiver Divisor auf einer abelsehen Mannigfaltigkeit A ; darunter versteht man einen positiven Divisor X, der hSehstens durch endlieh viele Trunslutionen in einen dazu linear /~quivalenten Divisor verwandelt wird. Dann sagt man, dub X eine Polarisierung von A bestimmt, wobei zwei solehe Divisoren X, X' dann und nur dunn dieselbe Polurisierung von A bestimmen, wenn es zwei positive ganze Zahlen m, m' gibt, ftir welehe m X und m'X" algebraiseh /~quivalent sind; jeder positive Divisor X" mit dieser Eigensehaft heiBe ein Polardivisor ftir die dureh X polarislerte abelsehe Mannigfaltigkeit A. Aus dem Hauptsatz der Theorie der Thetufunktionen folgt sofort, daft im klassischen Fall dieser Polarisierungsbegriff mit dem oben definierten zusammenf/~llt. Insbesondere sei J die Jacobische Mannigfaltigkeit einer algebraischen Kurve C; die Bezeichnungen seien dieselben wie in meinem Bueh Varidtds abdliennes et courbes algdbriques, Paris 1948 (zitiert als VA). Es sei ~ die (bis auf eine Translation eindeutig bestimmte) kanonisehe Abbfldung von C in J. Wenn a----~aiMi ein Divisor auf O ist, werde ich zur Abkiirzung ~(a)---i

~ a ~ ( M ~ ) sehreiben (statt S[~(a)] wie in VA). Wie in VA wird dureh W~ diejenige Untermannigfaltigkeit yon J bezeiehnet, die aus allen Punkten der Form ~0(a) besteht, wenn a die Gesamtheit der positiven Divisoren vom Grad r auf C durehl/iuft. Start Wg-1 wird meistens 0 gesehrieben; werm f ein Divisor der kanonisehen Klasse auf C ist, besteht O ~ W~_x aueh aus den Punkten (~) - - ~0(m), wenn m die Gesamtheit der positiven Divisoren vom Grad g - - 1

[1957a]

309 Zum Beweis des Torellischen Satzes

35

durchlauft; O wird also durch die Abbfldung u -+ ~ ( ~ ) - u auf sich selbst abgebildet. Start Wg_~ werden wir W schreiben; es sei W* das Bild von W bei der Abbildung u --> ~ (1) - - u yon J auf sich selbst. Wenn X ein Zyklus (und insbesondere eine Mannigfaltigkeit) auf J und a ein Punkt auf J ist, so wird (wie in VA) durch Xa der aus X durch die Translation a entstehende Zyklus bezeiehnet. Wir wollen unter der kanonischen Polarisierung von J diejenige verstehen, welche von O oder (was auf dasselbe hinausl~uft) yon einem beliebigen unter den Divisoren Oa bestimmt wird. Diese Polarisierung ist offenbar invariant bei jeder Abbfldung u --> • u + a v o n J auf sich selbst (dasselbe gilt iibrigens fiir jede Polarisierung einer abelschen Mannigfaltigkeit, wie z.B. sofort aus VA-73, prop. 31 folgt). Im klassisehen Fall gehSrt jede kanoniseh polarisierte Jacobische Mannigfaltigkeit zu der Hauptsehar der abelsehen Mannigfaltigkeiten derselben Dimension. 2. Der eigentliehe Inhalt des Torellischen Satzes besagt nun, daB eine polarisierte abelsche Mannigfa~tigkeit im wesentlichen h6chstens auf eine Weise die kanoniseh polarisierte Jacobisehe Mannigfaltigkeit einer Kurve sein kann. Genauer li~Bt er sich wie folgt formulieren: tIauptsatz. Es seien C, C' zwei Kurven vom gleichen Geschleoht g ~ 1 ; es sei (bzw. q~') die kanonische Abbildung yon C (bzw. C') in ihre Jacobische Mannig]altigkeit J (bzw. J'). Es gebe einen Isomorphismus ~ der kanonisch polarisierten Mannig/altig]ceit J au] die kanonisch polarisierte Mannig/altiglceit J'. Dann gibt es einen Isomorphismus ] yon C au/ C' mit der Eigenscha/t, daft ~ o q~ -~ • qJ o ] + a ist mit konstantem a; dabei sind ], a und das Vorzeichen + dutch ~ eindeutig bestimmt, wenn C und C" nicht hyperelliptisch sind; bei hyperelIiptischen C und C" sind / u n d a dutch ~ und das Vorzeichen -4- eindeutig bestimmt, wobei das letztere beliebig gewahlt werden kann. Der Beweis ergibt sieh dadurch, dab man eine explizite Konstruktion ftir C und ~ angibt, wenn J als gegeben gedacht wird. Zuni~chst wird gezeigt, dab die Sehar {O~} der aus O durch Translation entstehenden Mannigfaltigkeiten durch die Polarisierung yon J eindeutig bestimmt ist; das folgt im klassischen l~all aus bekannten S~tzen aus der Theorie der Thetafunktionen und im abstrakten l~all aus folgendem Satz: Satz 1. E i n positiver Divisor Z au/ J laflt sich dann und n u t dann dutch eine Translation in 0 aber/i~hren, wenn Z ~ 0 ist (wobei X ~ O wie in VA-57 dureh das Bestehen der linearen .~quivalenz X, ~ X fiir alle t definiert ist). Es ist ni~mlich O~ -~ O fiir alle a. Es sei Z ~ O m i t positivem Z; dann gibt es nach VA-62, th. 32, cor. 2 ein a, ftir welches Z ~ O~; indem wir Z durch Z_~ ersetzen, brauehen wir also nur zu zeigen, daB, wenn Z ~ O und Z positivist, Z mit O zusammenfallen muB. Es sei x ein generischer Punkt auf J (in bezug auf einen gemeinsamen Definitionsk5rper ftir alle in Betracht kommenden Mannigfaltigkeiten); dann ist naeh VA-41, th. 20 ~ - 1 ( O ~ ) ~ - ~ P ~ , wo die

310

[1957a] A n d r d Weil

36

P~ (1 ~ i < g) P u n k t e auf C sind mit der Eigenschaft ~ ( P ~ )

~- ~(t) -t- x;

i

die P, mfissen also unabh~ngige generische P u n k t e auf C sein, woraus folgt, dab l ( ~ P,) ~- 1. A u s Z ~ O folgt, dab ~-I(Z~) ein mit ~ Pi linear ~quivalenter positiver Divisor auf C sein muB; wegen l ( ~ P,)--~ 1 fallt er also mit ~ Pi zusammen; demnaeh liegt jeder P u n k t T(Pi) auf einer Komponente yon Z~, und der P u n k t g

auf einer Komponente yon Z. Da dieser Punkt, wie letztere Gleichung zeigt, auf O generisch ist, so mul~ O Komponente yon Z sein; also ist Z - - O positiv und ~ 0, woraus folgt, dab Z - O ~ 0 ist, w.z.b.w. N u n wird jedem Divisor X auf J durch die in VA-45 und VA-48 definierte Abbildung X -~ 8~ ein Endomorphismus ~ yon J zugeordnet; dabei ist 8o der identische Automorphismus 6j yon J ; nach VA-57, th. 30 ist dann und nur dann 8x ~ Or, wenn X ~ Y. Wenn X Polardivisor auf der kanoniseh polarisierten Mannigfaltigkeit J ist, so gibt es nach der Definition zwei positive ganze Zahlen m, m', ffir welche m X mit m ' O algebraisch ~quivalent ist; um so mehr besteht dann die Relation m X ~- m ' O , also m(5'X ~- m'(~j; dann mul3 aber (z.B. nach VA-65, th. 33, cor. 1) n ---- m ' / m ganzzahlig sein, woraus folgt, dab 8x----nOj und X - ~ n O mit ganzzahligem n gilt. Sagen wir, daB ein Polardivisor X auf J minimal ist, wenn es zu jedem Polardivisor Y auf J eine ganze Zahl r gibt, fiir welche Y -~ r X . J e t z t folgt sofort aus Satz 1, dab ein Polardivisor dann und nur dann minimal ist, wenn er der Schar {O~} der aus 0 durch Translation entstehenden Divisoren angeh6rt. I)amit ist diese Schar in invarianter Weise a u f der polarisierten Mannigfaltigkeit J gekennzeichnet. 3. Wir sind jetzt schon imstande, den l~all g--~ 2 zu erledigen. In diesem Fall ist n~mlich O ~ W~ ~ ~ (C). I m Hauptsatz kSnnen sich dann die Kurven a(T(C)), ~'(C') auf J ' hSehstens dutch eine Translation unterscheiden. Da ~, ~' die K u r v e n C, C' isomorph auf ~0(C), ~'(C') abbilden, so muB es also / und a geben, ffir welche ~ o ~ --~ ?' o / + a; dab / und a dabei eindeutig bestimmt sind, ergibt sich daraus, dab (z.B. nach VA-62, th. 32, cor. 2) O durch keine Translation in sich selbst fibergeffihrt wird. Bekanntlich ist jede Kurve vom Geschlecht 2 hyperelliptisch; auf einer solchen Kurve C hat ni~mlich die kanonische Vollschar (bestehend aus allen positiven Divisoren, die mit einem beliebigen kanonischen Divisor ~ linear ~quivalent shad) den Grad 2 und die Dimension 1. Es gibt also einen Automorphismus h yon C derart, dab ftir jeden P u n k t M auf C der Divisor M + h ( M ) der kanonischen Vollschar angehSrt; dann ist ~ ( h ( M ) ) = ~(~) - - T ( M ) . Wenn nun ~, ~', a, ], a dieselbe Bedeutung haben wie oben und /1-----] o h , a l = ~ ( q ~ ( ~ ) ) - - a gesetzt wird, so ist ~ o ~ - - - - - - r o / ~ + a ~ ; dabei sind/~, a~ dureh diese Gleichung eindeutig bestimmt, da sonst / ----/~ o h t

t

[1957a]

311 Zum Beweis des Torellischen Satzes

37

und a durch die Gleichung ~ o ~ : ~ ' o ] + a nicht eindeutig bestimmt w~ren. Damit ist der Hauptsatz im Fall g = 2 vollsti~ndig bewiesen. Die in der Literatur h~ufig vorkommende Behauptung, dab im klassischen l~all eine der Hauptsehar angehSrige abelsehe Mannigfaltigkeit der Dimension 2 ,,im allgemeinen" Jacobisehe Mannigfaltigkeit einer Kurve ist, l~Bt sich jetzt leicht auch im abstrakten Fall in folgender pr~ziseren Fassung formulieren und beweisen: Satz 2. E s sei A eine polarisierte abelsche Mannig/altigkeit der Dimension 2. E s gebe au] A einen Polardivisor X , derart, daft deg (X 9X~) ~- 2 sei ]i~r generisches u. D a n n ist entweder X eine Kurve vom Gesehlecht 2, A die kanonisch polarisierte Jacobische Mannig]altigkeit yon X und die kanonische Abbildung yon X in A die identische; oder A ist das Produlct 1" • 1" zweier elliptischer Kurven 1", 1", und X ist von der F o r m 1" • a' + a • I ~, wo a, a' Punlcte au] 1" bzw. 1" sind. Zur bequemeren Formulierung des Beweises wollen wir (in Erweiterung einer Definition in VA-22) folgendes verabreden: Wenn A eine beliebige abelsche Mannigfaltigkeit und X eine Untermannigfaltigkeit von A ist, so werden wir unter der durch X erzeugten abelsehen Untermannigfaltigkeit yon A die kleinste abelsehe Untermannigfaltigkeit B von A verstehen, die si~mtliche Punkte x - - x" mit x und x' in X enth~lt. Aus VA-21 folgt, dab B die dureh diese Punkte erzeugte Untergruppe von A ist; wenn x ein beliebiger Punkt yon X ist, so ist auch B die im Sinne von VA-22 durch X_~ erzeugte abelsche Untermannigfaltigkeit von A. Z.B. wird die Jacobische Mannigfaltigkeit J einer Kurve C durch die Kurve ~ (C) erzeugt, wenn ~ die kanonische Abbildung yon C in J ist; also wird auch J durch jede der Mannigfaltigkeiten W~ erzeugt. Wegen VA-20, th. 9 folgt daraus, dal3 W~ ftir r ~ g mit keiner abelschen Mannigfaltigkeit birational ~quivalent sein kann. Nach dieser Bemerkung werden wir Satz 2 zun~chst in dem Fall beweisen, wo der Polardivisor X eine Kurve ist, d. h. wo X aus einer einzigen Komponente mit dem Koeffizienten 1 besteht. Es sei i die Injektion, d . h . die identische Abbildung, von X in A. Das Geschlecht g yon X kaim weder 0 noeh 1 sein; im ersteren Fall mfiBte n~mlieh i nach VA-19, th. 8 konstant sein; im letzteren Fall w~re i naeh VA-20, th. 9 bis auf eine Translation ein Homomorphismus, also w~re X bis auf eine Trans]ation eine abelsche Untermannigfaltigkeit von A, und es w~re deg (X. Xu) ~ 0. Es sei J die Jacobische Mannigfaltigkeit yon X; es sei ~ die kanonisehe Abbildung yon X in J ; k sei ein Definitionsk6rper ffir A, X, J, ~. Setzen wir w : ~ (M) + ~ (N) und x ~- i(M) + i(N), wo M, N zwei unabh~ngige generische Punkte yon X fiber k sind. Der Oft (,,locus") von w fiber k ist dann nach unseren iiblichen Bezeiehnungen W~; nach VA-39, prop. 15 ist k(w) der KSrper k(M, N)8 der symmetrischen Funktionen von M, N fiber k; also ist k ( x ) in k(w) enthalten, so dab wir x ~ F ( w ) sehreiben k6nnen, wo F eine Abbildung yon W2 in A ist. Es sei X" der aus X

312

[1957a]

38

Andrd Weil

dureh die Abbfldung u - - > - - u laervorgehende Divisor; wegen VA-61, th. 31, eor. 2 folgt aus der Voraussetzung tiber X, dab deg (X. X~) ~ 2; nach VA-13, th. 4, eor. 2 mul~ also x generiseh tiber lc auf A und k (M, N) algebraisch vom Grad 2 tiber k ( x ) sein; wegen k ( x ) ~ k(M, N)s ist demnaeh k ( x ) -~ k ( M , N ) , ~- k (w). Damit ist gezeigt, dal3 F eine birationale Abbfldung yon W~ auf A ist. Wie oben bemerkt, kann W2 ftir g ~ 2 mit keiner abelschen Mannigfaltigkeit birational ~quivalent sein; es ist also g -~ 2 und W2 : J ; nach VA-20, th. 9 ist dann F bis auf eine Translation ein Isomorphismus von J auf A ; andererseits folgt aus VA-18, th. 7, cor., dab ~ und F o i sieh hSchstens durch eine Translation unterscheiden kSnnen. Damit ist Satz 2 in dem Fall, wo X eine Kurve ist, vollst~ndig bewiesen. Falls X keine Kurve ist, kSnnen wir X ~-- ~ X (~ schreiben, wobei X m . . . . , X (~) Kurven sind. Dann ist ftir generisches u: 2

deg (X. X,)

~ deg (X (~ 9X (~

Da jedes Glied auf der rechten SeRe _~ 0 ist, so folgt unmittelbar aus dieser Gleichung (unter Benutzung von VA-22, prop. 6) erstens, dab jede Kurve X (~ bis auf Translation eine abelsche Untermannigfaltigkeit von A sein muB, und zweitens, dab es nur zwei solche Kurven gibt; dann ist deg (X m .X(,2~) ---- 1. Wegen VA-13, th. 4, cor. 2 ist damit der Beweis beendet. I m klassischen Fall gentigt eine polarisierte abelsche Mannigfaltigkeit dann und nur dann der Voraussetzung yon Satz 2, wenn sie der Hauptschar angehSrt; also ist eine solche Mannigfaltigkeit entweder eine Jaeobische Mannigfaltigkeit oder ein Produkt zweier elliptischer Kurven. Es ist mir nicht gelungen, die entsprechende Frage ftir g -~ 3 zu entscheiden. 4. Zur Behandlung des allgemeinen Falles brauchen wir einige Hflfss~tze. Bekanntlich pflegt man zu sagen, dab ein posit~ver Divisor m a u f der Kurve C in der durch einen Divisor a bestimmten linearen Vollschar ~ (bestehend aus allen m i t a linear ~quivalenten positiven Divisoren auf C) enthalten ist, wenn es einen positiven Divisor m' gibt, ftir welchen m + m' der Schar angeh6rt. Unter Benutzung dieser Sprechweise gilt folgender Hflfssatz: Hflfssatz 1. E8 sei a ein Divisor vom Grad 0 au] der Kurve C; es sei a ----- q~(a) O; es sei ~ die dutch f + a bestimmte lineare Vollschar. D a n n besteht die Punktmenge 0 (-~ O~ aus den P u n k t e n ~(m), wenn m die in der Vollschar enthaltenen positiven Divisoren vom Grad g - 1 durchlau/t. Die Vollschar hat dann und n u t dann einen F i x p u n k t P, wenn a yon der F o r m a -~ q~(P - - Q) * ist, wo Q ein P u n k t yon C ist; in diesem Fall ist 0 ~ Oa = W~(p) ~ W_~(~). Die erste Behauptung folgt unmittelbar aus der Tatsache, dab O, O~ aus den Punkten ~(m) bzw. ~(f + a - m') bestehen, wobei m, m' die positiven Divisoren vom Grad g - 1 durchlaufen. Damit P Fixpunkt der Schar ~ sei, ist notwendig und hinreichend, dab l(I + a - - P) -- l(f + a) sei; nach dem Riemann-Rochsehen Satz l~Bt sich diese Gleichung in der Form / ( P - - a )

[1957a]

313 Zum Beweis des Torellischen Satzes

39

-~ l ( - - a) + 1 sehreiben. D a n n ist 1( P - - a) ~ 1, also P - - a m i t einem positiven Divisor ~quivalent; da P - - a v o m Grad 1 ist, gibt es d a n n einen P u n k t Q derart, d a b P--a ~ Q, also a - ~ ~ ( P - - Q ) ist. U m g e k e h r t , w e n n letztere Gleichung b e s t e h t u n d a ~= 0 ist, ist l ( P - a) = l(Q) -~ 1, l ( - - a ) -~ o; wie oben gezeigt wurde, ist d a n n P F i x p u n k t v o n ~ . I n diesem Fall b e s t e h t ~ aus den Divisoren P + ~, wo ~ die durch ~ - - Q b e s t i m m t e Vollschar durchl~uft. Es sei der positive Divisor m v o m G r a d g - - 1 in dieser Schar ~ e n t h a l t e n ; d a n n gibt es einen posiriven Divisor m ' v o m G r a d g - - 1 derart, d a b m + m ' der Sehar ~ angehSrt; also ist P entweder K o m p o n e n t e v o n m oder K o m p o n e n t e von m ' . I m ersten Fall k a n n m a n m ~ P + n schreiben, wo n ein positiver Divisor v o m G r a d 9 - 2 ist, der in der durch ~ - Q b e s t i m m t e n Vollschar e n t h a l t e n ist. Aus d e m R i e m a n n - R o c h s e h e n Satz folgt aber sofort, d a f alle positiven Divisoren v o m G r a d g - 2 in dieser Vollschar e n t h a l t e n sind; jeder Divisor P + n m i t p o s i t i v e m n v o m G r a d g - - 2 ist also in der Sehar ~ enthalten. Zweitens sei m wie oben in ~ e n t h a l t e n u n d P K o m p o n e n t e v o n m ' . D a n n kSnnen wir m ' -~ P + n ' sehreiben m i t einem positiven n' v o m G r a d g - - 2; w e n a u m gekehrt n' ein beliebiger positiver Divisor v o m G r a d g - - 2 ist, so ist wie oben P + n' in ~ enthalten, so d a f es einen positiven Divisor m v o m G r a d g - 1 g i b t derart, d a f m + P + 11' der Schar ~ angeh6rt; d a n n ist m in ~ enthalten, u n d es ist m ~t+a--P--n" ~ f--rr D a m i t ist bewiesen, daft O r~ Oa aus den P u n k t e n ~0(P + n), ~ ( [ - 1 t ' - - Q ) besteht, wenn u, 11' die Menge aller positiven Divisoren v o m G r a d g - 2 durchlaufen. D a m i t ist der Hflfssatz vollst~ndig bewiesen. 5. Ftir g ~ 5 gilt weiter der folgende Hflfssatz: Hflfssatz 2. Die Bezeichnungen seien dieselben wie im Hil/ssatz 1; die Scha~ sei fixpunkt]rei, und das Geschlecht g yon C sei > 5. Welter ],abe 0 r~ Oa mehr als eine Komponente. D a n n gibt es eine Komponente dieser Menge, die dutch unendlich viele Translationen in sich selbst i~berge/ahrt wird. Es sei k ein algebraiseh abgeschlossener DefinitionskSrper ftir C, J , ~ u n d die K o m p o n e n t e n y o n a; M 1 , . . . , M~_~ seien g - - 2 unabh~ngige generische P u n k t e v o n C fiber/c; wir setzen m ----~ M~. Wegen a ~= 0 ist l(~ + a) ----g - 1, v

also l (~ + a - - m) ~-- 1 ; es ist also [ + a - - m ~ 11 m i t einem durch diese Relat i o n eindeutig b e s t i m m t e n positiven Divisor n v o m G r a d g; 11 ist d a a n rational fiber d e m K S r p e r k ( m ) ---- k(M1 . . . . . Mg_~),, z.B. n a e h VA-41, th. 20. J e d e K o m p o n e n t e y o n O ~ O~ h a t die Dimension g - - 2 u n d ist algebraiseh tiber k, also rational tiber k, weft k algebraisch abgesehlossen ist; folglieh h a t ein in Bezug a u f k generiseher P u n k t einer solchen K o m p o n e n t e die Dimension g - - 2 tiber k. N a e h Hflfssatz 1 li~ft sieh ein soleher P u n k t in der F o r m ~ (~) sehreiben, wo ~ ein in ~ e n t h a l t e n e r positiver Divisor v o m Grad g - 1 ist. E s sei ~ ~-- R1 + - . - -t- Rg-1; d a m R ~ 0:) tiber k die Dimension g - 2 habe, mfissen wenigstens g 2 der P u n k t e B~, z.B. R x , . . . , R~_~, fiber k u n a b h a n g i g

314

[1957a] Andrg Weil

40

sein; es gibt also einen Isomorphismus von k(R1 . . . . . Rg_2) auf k(M1 . . . . . Mg-2) fiber k, der Rt in M~ ffir i ---- 1 . . . . . g - - 2 fiberffihrt. Dieser Isomorphismus kann zu einem Isomorphismus von B ( R 1 , . . . , R~-I) in das Universalgebiet erweitert werden; dadureh wird Rg_l in einen P u n k t N von C fibergeffihrt, also ~ in den Divisor m -[- N ; dann ist ~0(m + N) generiseh fiber k auf derselben K o m p o n e n t e von O ('~ O~ wie ~0(~). D a m i t ist gezeigt, dab es zu jeder K o m p o nente y o n O r-~ O~ einen generischen P u n k t der F o r m ~0(m + N) gibt. Dabei muB der Divisor m + N in ~ enthalten sein; da aber die Relation m + n ,~ ~ + a den positiven Divisor rt eindeutig bestimmt, mug N eine der g K o m p o n e n t e n des Divisors n sein. Da II fiber k (m) rational ist, ist u entweder ein rationaler Primdivisor fiber k(m) oder S u m m e yon mehreren solehen Primdivisoren. I m ersten Fall sind samtliehe K o m p o n e n t e n von n zueinander fiber k(m) konjugiert; dasselbe gilt also ffir die P u n k t e ~(m + N), so dab je zwei dieser P u n k t e generische Spezialisierungen voneinander fiber k (m) und u m so mehr fiber k sind; folglieh haben sie denselben Ort fiber k, so dab O ~ O~ nieht mehr als eine K o m p o nente haben k a n n entgegen der Voraussetzung. Demnaeh ist n yon der F o r m rt-----rtl~...+nh, wo die nt rationale Primdivisoren fiber k(m) sind, mit h ~ 2. Es sei vt der Grad von rtt; dann ist ~ vt = g, also vt ~ g - - 1 ffir alle i wegen i

h ~_ 2. Ffir jedes i i s t ~0(ui) rational fiber k(m), also aueh fiber k ( M 1 . . . . . Ma_2) , u n d laBt sieh also (nach VA-18, th. 7, eor.) in der F o r m yJ(M1) + ~v'(M2) + 999 sehreiben, wo % ~v', . . . g - 2 Abbildungen von C in J sind, die (weil ihre Summe eine symmetrische F u n k t i o n von M1 . . . . . Mg_~ ist) sich voneinander h6ehstens durch Addition einer K o n s t a n t e n unterseheiden. Demnaeh k5nnen wir ~(rt~) in der F o r m g--2

(nt) = ~ ~o~(M~) + c~ = ~t (m) + c~ sehreiben, wo ffir jedes i ~ eine Abbfldung y o n C in J und c~ eine K o n s t a n t e bedeutet; dabei ist jedes ~t bis a u f eine K o n s t a n t e eindeutig bestimmt. Ffir jedes i i s t aber ~t bis auf eine K o n s t a n t e v o n d e r F o r m a~ o % wo at ein Endomorphismus yon J ist; wit dfirfen also annehmen, da$ ~Pt = at o ~v; da k algebraiseh abgesehlossen ist, sind dann die at u n d c~ fiber k definiert. D a / ( n ) --~ 1 ist, so ist u m so m e h r l(rtt) ---- 1 ffir jedes i; folglieh ist ffir j e d e s i n~ algebraiseh (und sogar, wie m a n sieh leieht fiberzeugt, rational) fiber k(~ (ut)). W a r e insbesondere ~ (nt) algebraisch, also rational, fiber k, so ware dasselbe ffir jede K o m p o n e n t e von ut der Fall; dann ware aber ein soleher P u n k t F i x p u n k t der Schar ~ entgegen der Voranssetzung, weil m -4- u die Dimension 9 - - 2 fiber k hat und folglieh ein generiseher Divisor der Vollschar ist. D e m n a e h ist at # 0 ffir jedes i. Wenn At die abelsehe Untermannigfaltigkeit von J ist, auf welehe J dureh ~t abgebildet wird, und dt die Dimension yon A~ bedeutet, so ist also d~ > 0 ffir jedes i.

[1957a]

315 41

Z u m Beweis des Torellischen Satzes

Setzen wir fl---- ~j + ~ ~i. D a n n ist: i

woraus folgt, dab fl(~(m)) fiber k rational ist. D a J dureh den O r t W v o n (m) fiber k erzeugt wird, so muB also fl k o n s t a n t sein. Weft fl ein E n d o m o r phismus y o n J ist, ist d e m n a e h fl = 0, d . h . - - x = ~ a i x ffir jedes x a u f J . i

W e n n x generiseh a u f J fiber k gew~hlt ist, h a t ai x die Dimension d~ fiber k, also ~ ~ x hSehstens die Dimension ~ di. D a m i t ist bewiesen, d a b ~ d~ > g. i

i

i

Andererseits wird J dureh ~v(C), also A t dureh ~ (C) erzeugt; d u r c h I n d u k tion n a c h r ffir 1 < r < 9 - - 2 sieht m a n d a n n leieht, dab die Dimension y o n ~ ( M 1 -t- 999 M,) fiber k gleieh r i s t , solange dieser P u n k t a u f A~ nicht fiber k generisch ist; insbesondere ist die Dimension v o n ~i(m) fiber k gleich v~ -~ inf (g - - 2, d~) ; sie ist d a n n u n d n u r d a n n gleieh di, wenn ~ (m) a u f A ~generiseh ist. N u n ist aber v~ ebenfalls die Dimension v o n q (n~) fiber k; d a dieser P u n k t a u f W~ liegt, so ist v'~ < v~; es ist d a n n und n u r d a n n v~ = v~, wenn ~ (n~) a u f W~ generiseh ist; d a J dureh W~ u n d A~ dureh den O r t von yJ~(m) fiber k erzeugt wird, u n d d a sieh ~i(m) und q(rti) n u r u m die K o n s t a n t e ci u n t e r scheiden, so folgt aus v: - - vi, dab J = Ai, also d~ ---- g u n d v~ -----g - 2 ist. N e h m e n wir an, es sei v'i < g - - 2 ffir jedes i; d a n n k a n n nieht v~ ~--vi sein; also ist v~ < v~; zugleieh m u g aueh v~ ---- di sein, also d~ < v~ ffir jedes i, was der Ungleichung ~ d~ > g = ~ vi widersprieht. Folglieh dfirfen wir z.B. i

i

a n n e h m e n , d a b v~ ---- g - - 2 ist; d a n n ist vl > g - - 2, also % < 2. W e n n v~ = 1 w~re, so w~re, wegen 0 < v~ < v~ ffir jedes i, v~ ~-- 1, also v~ ---- v~ -~ g - - 2 u n d g -~ 3 entgegen der Voraussetzung; also ist v~ = 2. W e n n v~ ~-- 2 w/~re, so w~re v~ = g - - 2 u n d g = 4, entgegen der Voraussetzung; also ist v~ = 1 u n d (wegen g > 5) v~ = d~. So ist gezeigt, dab h = 2, v~ = g - 2, v~ = 2, v~ = d~ ---- 1 ist; der Ort y o n ~%(m) ist A~ u n d h a t die Dimension 1; und der Ort B yon q(n~) e n t s t e h t aus A~ dureh die Translation cz. Setzen wir rt~ ~-- N + N ' ; weder N noeh N ' ist algebraisch fiber k; da aber v~ ~-- 1 ist, so ist N" algebraiseh fiber k (N). U n t e r den Mr muB es d a n n g - - 3 P u n k t e geben, z.B. M~, . . . , Mg_a, die fiber k(N, N') u~abh/~ngig sind. D e r Divisor M~ + 999 A- Mg_a + n~ v o m Grad g - 1 h a t d a n n fiber k die Dimension g - - 2 und ist in der Sehar ~ e n t h a l t e n ; also ist, wenn wit t = ~ ( n ~ ) , w=~(M a+...+M~_a) setzen, t + w generischer P u n k t einer K o m p o n e n t e Z v o n O r~ O~. Dabei shad t, w unabh~ngige generische P u n k t e v o n B bzw. Wg_a fiber k; Z wird also dureh jede Translation, die durch einen P u n k t y o n A~ b e s t i m m t ist, in sich selbst fibergeffihrt. D a m i t ist der Hflfssatz bewiesen. U m festzustellen, ob Hflfssatz 2 aueh ffir g ---- 3 u n d g ---- 4 gtiltig bleibt, w~ren ziemlich langwierige Fallunterseheidungen nStig; ich h a b e sie nieht bis zu E n d e durehgeffihrt.

316 42

[1957a] Andrd Well

Ein positiver Divisor c v o m Grad 2 auf einer K u r v e C vom Gesehlecht g ~_ 2 h e i f e hyperelliptisch, wenn l(r = 2 ist; in diesem Fall heiBe ebenfalls die dureh c bestimmte lineare Vollschar der Dimension 1 hyperelliptisch. Wie bekannt ist, gibt es auf einer K u r v e C vom Gesebleeht g ~ 2 h6ehstens eine hyperelliptische Vollschar, und C heil~t hyperelliptiseh, wenn es auf C eine solche Vollsehar gibt. Hflfssatz 3. Keine der Manniq]altiglceiten W1 = q~(C). . . . . W~_~ = W, Wg_l = O, W* wird in sich selbst durch eine Translation (aufler der identischen ) aberge/i~hrt. Es wird dann und n u t dann W dutch eine Translation in W* tlberge]ahrt, wenn C hyperelliptisch ist; in diesem Fall ist W* ~-- W~(c), wenn r ein hyperelliptischer Divisor au] C ist. DaB die erste B e h a u p t u n g ffir O gilt, ist z.B. in VA-62, th. 32, cor. 2 enthalten. Ffir r < g - - 1 ist O der Ort von v + w fiber k, wenn v, w unabhi~ngige generisehe P u n k t e y o n W~ bzw. W~-~-x fiber k sind; wenn also W~ dureh eine Translation in sieh selbst fibergeffihrt wfirde, w~re das aueh der Fall ffir O. Da W* naeh seiner Definition (in Nr. 1 oben) aus W ---- W~_~ dureh die Abbildung u -~ ~ (~) - - u entsteht, so gilt die erste B e h a u p t u n g aueh ffir W*. N e h m e n wir an, die Translation c ffihre W in W* fiber; es sei c ein Divisor v o m Grad 2 mit ~(r = c. Zu jedem positiven Divisor m v o m Grade g - - 2 m u f es dann einen positiven Divisor m ' desselben Grades geben, derart, dab f - - m ~ m' + r mit anderen Worten, es m u f l ( f - c - m) ~ 1 sein. Das Bestehen dieser Ungleiehung ffir generisches m fiber/c (c) ist aber mit der Ungleiehung 1( ~ - - c) g - - 1 gleiehwertig, und umgekehrt folgt aus der letzten Ungleiehung, dab l(f--c--m) _~ 1 ffir jedes positive m v o m Grad g - - 2 . Wegen des RiemannRoehsehen Satzes bedeutet aber l ( f - - r _~ g - - 1 niehts anderes als 1(r ~ 2. Hflfssatz 4. Es eei V der Oft des Punktes qJ( M - - N ) abet k, wenn M , N zwei unabhangige generische Punkte yon C ~ber k sin& Dann hat V die Dimension 2. Es sei ~ die Menge der P u n ~ e a ~= 0 au] J mit der Eigenscha/t, daft 0 r-~ O~ zwei Komponenten hat, wovon keins durch elne nicht-identische Translation in sich selbet abergeht. Dann ist 9~ = V - - {0}, wenn g ~_ 5 und C nicht hyperelliptisch ist; dagegen gibt es, wenn g ~ 5 und C hyperelliptisch ist, eins Kurve Vx au/ V derart, daft 9~ : V - - V1. Es seien P , Q, P ' , Q' irgend vier P u n k t e auf G; aus ~o(P - - Q ) : ~0(P' - - Q ' ) folgt, daft P - - Q ~ P ' - - Q ' , also P + Q ' ~ P'+Q, und welter, wenn C nieht hyperelliptiseh ist, dab entweder P =- P ' , Q : Q' oder P ---- Q, P ' =- Q'. Die Abbfldnng (P, Q) -> ~ 0 ( P - - Q ) yon G>~ C auf V ist also auBerhalb der Diagonale y o n C • G eineindeutig, falls C nieht hyperelliptisch ist; in diesem Fall hat also V die Dimension 2. Wenn C hyperelliptiseh ist, so gibt es einen Automorphismus h von G derart, dab ffir jeden P u n k t P y o n C der Divisor P + h ( P ) hyperelliptiseh ist; dann ist, wenn c irgendein hyperelliptiseher Divisor ist, ~ (h (P)) -~ r (c) - - ~0(P) ffir jeden P u n k t P ; indem dann N ' : h (N) gesetzt wird, sind M, N ' unabhgngige P u n k t e auf C fiber k, und ~o( M - N)

[1957a]

317 Z u m Beweis des Torellischen Satzes

43

nichts anderes als ~(M + N') --~0(c), also g die aus W2 durch die Translation --~(c) entstehende Fl~ehe. Nun sei a ein Punkt in der Menge 9~; nach den Hflfss~tzen 1 und 2 muB jedenfalls a auf V liegen; es ist also a = ~ ( P - Q) ~= 0, und O ~ Oa ist Vereinigungsmenge yon W~(p) und W*_~(Q). Da die letzteren Mannigfaltigkeiten nach Hilfssatz 3 keine Translation in sich selbst zulassen, so gehSrt umgekehrt dann und nur dann ein solcher Punkt der i~Ienge 9~ an, wenn W~(p) mit W*_~(Q)nicht zusammenf~llt, also (nach Hflfssatz 3) wenn P + Q nieht hyperelliptisch ist. Folglich ist wie behauptet 2 ---- V - - {0}, falls C nicht hypereUiptisch ist; im hyperelliptisehen Fall besteht dagegen 2[ aus allen Punkten yon V - - {0}, die sich nieht in der Form ~ (P - - Q) mit hyperelliptischem P + Q schreiben lassen. Wenn nun P - t - Q hypereUiptisch ist, so ist Q -~ h(P), wo h dieselbe Bedeutung hat wie oben; es sei Vt der Ort des Punktes qJ(M - - h ( M ) ) -~ 2~(M) - - ~(c) fiber/c bei generischem M fiber k auf C. Es gibt bekanntlich Punkte P a u f G derart, dab P ~- h (P); also liegt 0 auf der Kurve Vt. Es ist dann 2 ---- V - - V1, W.z.b.w. 6. I m Falle g > 5 ist nun der Hauptsatz eine unmittelbare Folge der oben bewiesenen Hilfss~tze. Oben ist schon die Sehar {Oa} in invarianter Weise auf der polarisierten Mannigfaltigkeit J gekennzeichnet worden. Es sei T = Ot eine beliebig gew~hlte Mannigfaltigkeit aus dieser Schar. Die Menge der Punkte a :4= 0 auf J mit der Eigensehaft, dab T ~ Ta zwei Komponenten hat, wovon keine dureh eine nieht-identische Translation in sich selbst fibergeht, ist dann nichts anderes als die im Hflfssatz 4 definierte Menge 9/; damit ist aueh die Fl~che V als abgeschlossene HiJlle yon 9~ (in der Zariskisehen Topologie anf J) in invarianter Weise definiert. Die Menge aller Komponenten der Durchsehnitte T ~ Ta, wenn a die Punktmenge V - {0} durehl~uft, besteht dann aus allen Mannigt-r wenn P, Q die Kurve C durchlaufen. Es sei faltigkeiten Wt +v(p), W* nun X eine beliebig gewfihlte Mannigfaltigkeit aus dieser Menge; es sei Y die Menge aller Translationen, die X in eine Mannigfaltigkeit derselben Menge fiberftihren. Aus dem Hflfssatz 3 folgt sofort, dab Y eine Kurve ist, die aus der Kurve ~0(C) dureh eine Abbfldung der Form u -~ + ( u - ~(P)) entsteht, wo P ein Punkt yon Gist. Wir haben also eine explizite Konstruktion angegeben, wodurch auf der polarisierten l~Iannigfaltigkeit J die Schar der aus ~ (C) dutch eine Abbfldung u --> • u + a entstehenden Kurven in invarianter Weise gekennzeiehnet ist. Damit ist der Hauptsatz ffir g > 5 vollst~ndig bewiesen. 7. Um die F~lle g --~ 3 und g ~-- 4 zu behandeln, werden wir zuerst die tangentialen linearen Mannigfaltigkeiten zu den Mannigfaltigkeiten W~ bestimmen. Hilfssatz 5. Es sei F eine Abbildun9 einer Mannig]altiglceit U der Dimension n in eine Mannig/altigkeit V; es sei ]c ein Definitionsk6rper /at U, V, F ; es sei u

318 44

[1957a] Andrd Weil

ein generischer Punier yon U abet k; der Punlct v ~ - F ( u ) sei ein]ach au/ V. Unter diesen Umstdnden ist k (u) dann und nur dann separabel algebraisch ~eber k (v), wenn die zu F i m Punkte u tangentiale lineare Abbildung vom Rang n ist. Bekanntlich ist k(u) dann und n u r dann separabel algebraiseh fiber k(v), wenn es keine niehtverschwindende Derivation von k(u) fiber k(v) gibt. Die Derivationen yon k(u) fiber ]c stehen aber in eineindeutiger Beziehung zu denjenigen Tangentialvektoren zu U im P u n k t u, die fiber k(u) rational sind; wenn ~ ein solcher Vektor ist, wird n~mlich durch die Gleichung D[/(u)] ---d / ( u ; t~), wo ] eine beliebige fiber k definierte numerische F u n k t i o n auf U ist u n d d] das Differential yon ] bedeutet, eine Derivation D yon k (u) definiert; und ]ede Derivation yon k(u) fiber k li~Bt sich so definieren. Die in dieser Weise d e m Vektor ~ zugeordnete Derivation D verschwindet dann und nur dann auf k (v), wenn D durch die tangentiale lineare Abbfldung zu F in u auf 0 abgebildet wird. Die B e h a u p t u n g des Hflfssatzes folgt u n m i t t e l b a r daraus. N u n sei A eine abelsehe Mannigfaltigkeit der Dimension n; es sei T der tangentiale V e k t o r r a u m zu A in O; im Siane der Theorie der algebraisehen Gruppen ist T nichts anderes als die Liesehe Algebra von A, oder genauer der dieser Algebra untergeordnete Vektorraum. Wenn a ein beliebiger P u n k t auf A ist, so wird dureh die Translation u -~ u + a der V e k t o r r a u m T auf den zu A in a tangentialen V e k t o r r a u m Ta isomorph abgebildet; wir werden meistens T~ mit T dureh diesen Isomorphismus identifizieren. Bekanntlich wird dadurch der duale V e k t o r r a u m zu T m i t dem Vektorraum der translationsinvarianten Differentiale (oder, was dasselbe ist, der Differentiale 1. Gattung) a u f A identifiziert. Es sei co ein solehes Differential; es sei F eine Abbildung einer Mannigfaltigkeit U in A. Dem Differential w @ird durch die zu F transponierte Abbfldung $'* ein Differential F ' c o auf U zugeordnet, das durch die Gleichung

F*o~(u; ~) = o~(F(u); F ' , ) definiert werden kann, wo u ein generischer P u n k t auf U, D ein Tangentialv e k t o r zu U in u, und F" die zu F in u tangentiale lineare Abbildung bedeuten. D a n n ist bekanatlich F * co ein Differential 1. G a t t u n g auf U; da wir hier diese Tatsache nur im Falle der kanonischen Abbildung einer K u r v e in ihre Jacobische Mannigfaltigkeit benutzen wollen, so brauehen wir auf ihre Begrfindung nieht n~her einzugehen; in diesem Fall ist sie unmittelbar klar. B e t r a c h t e n wir wieder eine K u r v e C v o m Geschlecht g, ihre Jaeobische Mannigfaltigkeit J und ihre kanonische Abbildung ~ in J ; es sei k ein DefinitionskSrper ffir C, J , ~. Es sei M irgendein P u n k t auf C; da ~ in M biregul~r ist, so wird durch die zu ~ in M tangentiale lineare Abbfldung die Tangente zu C in M auf eine Gerade im tangentialen V e k t o r r a u m zu J in ~0(M) abgebildet. Dieser V e k t o r r a u m ist oben mit dem tangentialen Vektorraum zu J in 0 identifiziert worden; damit ist der Tangente zu C in M eine Gerade durch 0 in T zugeordnet, die wir ira folgenden mit t (M) bezeichnen wollen. Es ist

[1957a]

319 Zum Beweis des Torellischen Satzes

45

leicht einzusehen (z.B. durch die Wahl geeigneter affiner Repr/~sentanten ffir die in B e t r a c h t k o m m e n d e n a b s t r a k t e n Mannigfaltigkeiten), daB, wenn M" eine Spezialisierung yon M fiber k ist, t (M') die eindeutig bestimmte Spezialisierung von t (M) fiber M --->M' in bezug auf k ist; also ist t eine fiber k defiuierte Abbfldung (im Sinne der algebraischen Geometrie) von C in die Malmigfaltigkeit der Geraden dureh 0 in T. Es sei P der (g - - 1)-dimensionale projektive R a u m der unendlichfernen P u n k t e zu T ; die P u n k t e yon P stehen in eineindeutiger Beziehung zu den Geraden dureh 0 in T ; wir wollen mit ~ (M) den der Gerade t (M) zugeordneten P u n k t von P bezeichnen; x ist also eine Abbfldung yon C in P. Satz 3. Der Vektorraum der Di~erentiale 1. Gattung au/ J wird dutch die zu transponierte Abbildung q~* au] den Raum der Di~erentiale 1. Gattung au] C isomorph abgebildet. Da beide R/~ume dieselbe Dimension g haben, so brauchen wir nur zu zeigen, dab ~* o~ ~= 0 ist, wenn das Differential 1. G a t t u n g w auf J nicht 0 ist. Es seien M 1 . . . . . Mg g unabh/~ngige generisehe P u n k t e y o n C fiber k; es sei u = (M1 -t- 999 q- Mg). Da k ( M 1. . . . . Ms) fiber k(u) separabel algebraiseh ist, kSnnen wir Hilfssatz 5 auf die Abbildung (M 1 . . . . , Ms) -~ u von G • 999• G auf J anwenden. Daraus ergibt sieh sofort, dab die Geraden t (M1) . . . . . t (Ms) nicht alle in einer echten linearen Untermannigfaltigkeit von T liegen kSnnen. W e n n n u n q0* eo ~- 0 w/~re, so mfiBte (naeh der Definition von ~* oJ) t (M)ffir j eden P u n k t M auf C in der durch o~ -~ 0 b e s t i m m t e n H y p e r e b e n e in T liegen (wobei wir den R a u m der Differentiale 1. G a t t u n g auf J mit dem zu T dualen R a u m identifiziert haben). Dureh eine /~hnliche B e t r a c h t u n g kann m a n folgenden allgemeineren Satz beweisen, den wir hier nur nebenbei erw/~hnen. Es sei F eine Abbildung einer K u r v e C in eine abelsche Mannigfaltigkeit A der Dimension n; es sei k ein DefinitionskSrper ffir C, A, F ; es seien M1 . . . . . M s n unabh~ngige generische P u n k t e von C fiber/c und u ---- _~ (M 1 ~- 999 -b M~). U n t e r diesen Umst/~nden ist ]c (M1 . . . . . Ms) dann und nur dann separabel algebraiseh fiber/r (u), wenn durch F * kein nichtverschwindendes Differential 1. Gattung auf A auf 0 abgebildet wird. J e d e r H y p e r e b e n e durch 0 in T entspricht im projektiven R a u m P ebenialls eine H y p e r e b e n e ; die H y p e r e b e n e in P, die in dieser Weise der durch o~ ----- 0 definierten H y p e r e b e n e in T entsprieht, werde durch H~ bezeichnet; dabei bedeutet oJ wie oben ein nichtverschwindendes Differential 1. Gattung a u f J (bei Identifizierung solcher Differentiale mit den Linearformen in T). Aus den Definitionen folgt sofort, dab H~ dam1 und nur dann den P u n k t ~(M) enth/~lt, wenn ~*eo im P u n k t M versehwindet. Es sei nun (~ol, . . . , ~og) eine Basis ffir den R a u m der Differentiale 1. G a t t u n g auf J, also ffir den zu T dualen R a u m ; die eo~ kSnnen dann als ein System (gew6hnlieher) K o o r d i n a t e n in T und als ein System homogener K o o r d i n a t e n in P b e t r a c h t e t werden.

320

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[1957a] Angrd Weil

Es ist nach dem obigen klar, dab die Verh~ltnisse der Differentiale ~* w~ im Punkt M yon C, also die Werte ~*co~(M; ~) ffir einen beliebigen nichtverschwindenden Tangentialvektor ~ zu C in M, ein System homogener Koordinaten fiir den Punkt v(M) sind. Die Abbfldung ~ von C in P ist also diejenige, die zur kanonischen linearen Vollschar gehSrt. Bekanntlich folgt unmittelbar aus dem Riemann-Rochschen Satz, dab v ein Isomorphismus von C auf ~(C) ist, wenn C nicht hyperelliptisch ist. I m hyperelliptischen Fall ist, wenn M generisch auf C fiber k gew~hlt wird, k (M) separabel vom Grad 2 fiber k ( ~ ( M ) ) ; der nicht-identische Automorphismus von k(M) fiber k (3 (M)) ist dann derjenige, der oben in Nr. 5 (beim Beweise von Hflfssatz 4) mit h bezeichnet wurde, so dab M + h(M) ein hyperelliptischer Divisor ist. 8. Satz 4. E s 8el a ~ A1 + 999+ Ar ein positiver Divisor vom Grad r ~ g au] C. D a n n und nur dann ist a -~ ~ (a) ein ein/acher P u n k t yon Wr, wenn l (a) 1 ist; in diesem Fall ist die lineare Untermannig/altigkeit yon P, die in P der tangentialen linearen Mannig/altiglceit zu Wr in a entspricht, der Durchschnitt der Hyperebenen H ~ , wenn w die Gesamtheit der Di~erentiale 1. Gattung au] J durchl~u/t, ]t~r welche (q~*o~) ~ a. Zunachst sei a einfaeh auf W~; es sei Tr die tangentiale lineare Mannigfaltigkeit zu W~ in a und L die ( r - 1)-dimensionale lineare Untermannigfaltigkeit yon P, die T~ entsprieht. Wenn M ein generiseher Punkt von C tiber k(A1 . . . . , A~) ist, so enth~lt W~ den Ort yon ~(M + A S + 999+ A~) fiber diesem K6rper; die Tangente zu dieser Kurve im Punkte a ist aber nach dem oben Bewiesenen die Gerade t(A1); also enth~lt T~ diese Gerade, und L enth~lt den Punkt v(A1). Nehmen wir an, es sei l(a) ~_ 2; dann gibt es zu jedem Punkt BI auf C einen Divisor b in der durch a bestimmten linearen Vollsehar derart, da~ b ~ B1; b l~Bt sieh also in der Form B1 + 999+ B~ sehreiben, und es ist ~(l}) -----~(a) -----a. Aus dem oben Bewiesenen folgt nun, dab v(B~) in L liegen muB. Da B~ beliebig ist, ist also die ganze Kurve v(C) in L enthalten, was unmSglieh ist. Demnaeh ist 1(a) ---- 1, wenn a einfaeh ist auf Wr; das Umgekehrte wurde in VA-40, prop. 18 bewiesen. Nun sei l(a) ~-- 1, also a einfach auf W~; wir haben oben bewiesen, da~ L die Punkte v ( A 1 ) , . . . , v(A~) enth~lt. Andererseits ist l ( [ - a) ---- g - r wegen des Riemann-Roehsehen Satzes; m. a. W. bilden die Differentiale co, die der Bedingung (~*r genfigen, einen Vektorraum der Dimension g - - r ; der Durchschnitt L" der Hyperebenen H~, wenn eo diesen Vektorraum dnrehl~uft, hat also dieselbe Dimension r - 1 wie L u n d enth~lt die Punkte ~(Ax), . . . . v (A~). l~alls diese Punkte in keiner linearen Mannigfaltigkeit niedrigerer Dimension liegen, folgt sehon daraus, dab L" mit L zusammenf~Ut. Das ist aber jedenfalls dann der Fall, wenn A~ . . . . . Ar fiber k unabh~ngig sind, da wir beim Beweis von Satz 3 sogar gezeigt haben, dai3 die Geraden t(A~) . . . . . t(Au) in keiner eehten liaearen Untermannigfaltigkeit von T liegen kSnnen, falls A~ . . . . . Ag fiber k unabh~ngig sind.

[1957a]

321 Zum Beweis des ToreUischen Satzes

47

Der allgemeine Fall folgt daraus duroh Spezialisierung. Wenn u ein genefischer Punkt yon W~ fiber k ist, so ist ni~mlieh die tangentiale lineare Mannigfaltigkeit zu W~ in a die eindeutig bestimmte Spezialisierung derjenigen in u fiber u ~ a in bezug auf k. Man braueht also nur zu beweisen, daB, wenn die Behauptung yon Satz 4 ftir die tangentiale lineare Mannigfaltigkeit zu Wr in a = ~ (a) riehtig ist, sie auch ffir jede Spezialisierung a' yon a riehtig bleibt. Es seien nun bl . . . . . bg_~ g - - r fiber k (a, a') unabhi~ngige generisehe positive Divisoren vom Grad g - r - 1; es sei K tier dureh die Komponenten dieser Divisoren fiber k erzeugte KSrper; a' ist noeh Spezialisierung von a fiber K. ' Die Bedingungen (~* to~) ~ a + bQ, (~* we) ~ a ' + bQ, bestimmen dann niehtverschwindende Differentiale %, w~ eindeutig bis auf konstante Faktoren; es ist dana unmittelbar einzusehen, dab bei der Spezialisierung a ~ a' fiber K die Hyperebene H ~ sieh ftir jedes ~ auf H ~ spezialisiert. Wegen der Wahl der b~ sind sowohl die co~ wie die co~ linear unabhangig; der Durehsehnitt der Hyperebenen H ~ wird also auf den Durchsehnitt der H ~ spezialisiert. Damit ist alles bewiesen. 9. Naeh diesen Vorbereitungen kehren wir zum Beweis des Torellisehen Satzes in den F~llen g = 3, g ---- 4 zurfiek. Erstens sei g ---- 3. Naeh Satz 4 ist dann O = W2 im meht-hyperelliptischen Fall singulariti~tenfrei. I m hyperelliptisehen Fall hat dagegen O genau einen singuli~ren Punkt c = ~(c), w o r irgendeinen hyperelliptisehen Divisor auf C bedeutet. Im letzteren Fall ist dana O_r eindeutig dadureh bestimmt, dalt es die einzige Mannigfaltigkeit in tier Sehar {O~} ist, die in 0 einen singuli~ren Punkt hat. Da diese Mannigfaltigkeit aus allen Punkten ~0(M A- N)

--

c

= ~o(M)

--

~

(h (N))

besteht, so ist sie niehts anderes als die im Hilfssatz 4 mit V bezeiehnete F1/~ehe. Der Beweis des Itauptsatzes im hyperelliptisehen Fall g ---- 3 1/Liftsieh darm genauso wie in Nr. 6 dureh Betraehtung der Komponenten der Durehschnitte T c-~ T~ ffir a in V - - {0} zu Ende ffihren, wobei T irgendeine Marmigfaltigkeit aus der Sehar {O,} (z.B. V selbst) bedeutet. Im nicht-hyperelliptisehen Fall sei wieder T = Ot eine beliebig gewKhlte M'annigfaltigkeit aus der Sehar {O~}. Es sei k o ein algebraiseh abgesehlossener gemeinsamer DefinitionskSrper ffir J und T; es sei k ein k o enthaltender DefinitionskSrper ffir C, ~0 und t. Da O dureh die Abbildung u --> ~o(~)- - u anf sich selbst abgebildet wird, so wird T durch die Abbildung u -->F(u) : q~(~) + 2 t - - u auf sich selbst abgebildet. Da O, also aueh T keine nieht-identische Translation in sich selbst zulassen, so ist E die einzige unter den Abbfldungen u -+ c - - u mit konstantem c, die T auf sich selbst abbildet; dadureh ist F nach Wahl yon T in invarianter Weise gekennzeichnet. Es ist klar, dal3 die Tangentialebenen zu T in den Punkten u und F (u) zueinander parallel sind, d. h. dab sie

322

[1957a]

48

Andrg Weil

bei der vorgenommenen Identifizierung der tangentialen Vektorr~ume zu J in u u n d in F (u) zusammenfallen. I m vorliegenden Fall ist 9 ein Isomorphismus yon C auf eine singularit~tenfreie K u r v e 4. Grades in der projektiven Ebene. Es sei u ein generischer P u n k t yon T fiber k; m a n k a n n u in der F o r m u = ~(M1 q- M~) q- t sehreiben, wo M1, M2 zwei unabh~ngige generisehe P u n k t e von C fiber k sind. Die Gerade durch v(M1) und v(M~) ist dann generiseh fiber k, woraus folgt, dab sie zwei weitere S e h n i t t p u n k t e ~ (N1), ~ (N~) mit T (C) hat, und dal~ je zwei ihrer S e h n i t t p u n k t e mit v(C) fiber k unabh~ngig shad; also sind je zwei der P u n k t e M1, M2, N1, N~ 1/nabh~ngige generisehe P u n k t e yon G fiber k; und der Divisor M1 q - M 2 Jr N1 q - N z gehSrt zu der kanonisehen Vollsehar. Es folgt nun unmittelbar aus Satz 4, dab es auf T auBer u genau 5 P u n k t e gibt, in denen die Tangentialebene zu T zur Tangentialebene zu T in u parallel ist; das shad n~mlieh die P u n k t e u" -= q~(N~ -~- N2) + t,

v~j -~ ~ ( M i ~- Nj) -~- t

( i , j ---- 1,2).

Dabei ist u' nichts anderes als E (u). Wenn wir also ffir v ehaen von u und F (u) versehiedenen P u n k t y o n T wahlen, in dem die Tangentialebene zu T zur Tangentialebene zu T in u parallel ist, so kann das nur einer der vier P u n k t e vi~ sein; u n d es ist dann u - v y o n der F o r m ~ ( M - N), wo M, N zwei unabhi~ngige generische P l m k t e yon C fiber k sind. Es ist aber klar, dab ein solcher P u n k t v fiber/co(u ) algebraisch sein muB, weft es sonst unendlieh viele solche P u n k t e g~be; u - v hat also hSchstens die Dimension 2 fiber k0. Da aus Hilfssatz 5 folgt, da~ der Ort von u - v fiber/C die dort mit V bezeichnete Fl~ehe ist, so muB u - - v denselben Ort V fiber Ico haben wie fiber k. Wir haben also die Fl~ehe V als Ort des Punktes u - - v fiber/c o konstruiert; dabei wurde u als generischer P u n k t yon T fiber k gewii~hlt, und v wurde in der oben angegebenen Weise definiert. Es sei nun u 0 ein beliebiger generischer P u n k t y o n T fiber k0; der Isomorphismus yon k o (u) auf/c o (%) fiber/co, der u a u f u 0 abbildet, li~Bt sieh zu einem Isomorphismus der abgesehlossenen Hfille yon /co(u) auf diejenige yon/c0(u0) erweitern. Dadureh ist ersichtlieh, daB es wieder genau vier P u n k t e auBer u0 und F (u0) gibt, in denen die Tangentialebene zu T zur Tangentialebene zu T in u o parallel ist, und daB, wenn v 0 einer dieser vier P u n k t e ist, der Oft von u0 - - Vo fiber/co die Flaehe V ist. Da nun wieder V in vSllig invarianter Weise konstruiert ist, verl~uft der Beweis genauso wie frfiher weiter. 10. Endlieh sei g -~ 4. W e n a C hyperelliptisch ist, so besteht die kanonisehe lineare Vollschar auf C aus den Divisoren der F o r m c + c' -t- c", wo c, c', c" hyperelliptisehe Divisoren shad; folglieh gibt es zu ehaem positiven Divisor a ---- A -{- A' + A " vom Grad 3 nur ehaen zweiten solehen Divisor a' derart, dab

[1957a]

323 Zum Beweis des ToreUischen Satzes

49

a ~ a' kanonisch ist, falls keiner der Divisoren A ~- A ' , A ~ A " , A" ~- A " hyperelliptisch ist. U n t e r Benutzung des Riemann-Roehsehen Satzes folgt dann ans Satz 4, dab die singularen P u n k t e von O genau die P u n k t e ~(r ~ M) sind, wo c einen hyperelliptisehen Divisor und M einen P u n k t auf C bedeuten. Die singularen P u n k t e irgendeiner Mannigfaltigkeit aus der Schar {Oa} bflden also eine Kurve, die sich yon ~ (C) nur dureh eine Translation unterscheidet. D a m i t wird offenbar der hyperelliptische Fall g -= 4 erledigt sein, sobald wir zeigen, wie m a n diesen Fall v o m nicht-hyperelliptischen Fall in invarianter Weise unterscheiden kann. Das wird daraus folgen, da~ im letzteren Fall O n u r einen oder zwei singulare P u n k t e hat. Von nun an setzen wir voraus, dal] C nicht hyperelliptiseh sei. Da v ein Isomorphismus von C auf 9 (C) ist, wollen wir die Bezeiehnungen dadurch vereinfaehen, dal~ wit C mit ~ (C) durch v identifizieren. Es ist dann C eine singularitatenfreie K u r v e v o m Grad 6 im 3-dimensionalen projektiven R a u m P . Aus dem Riemann-Rochschen Satz ist sofort ersiehtlieh, dal~ es wenigstens eine Flache F , vom Grad 2 in P gibt, die C enthalt, sowieeine F l a c h e F a v o m Grad 3, die C aber nicht F , enthalt. Da C v o m Grad 6 ist, so muB dann $'~ die einzige Flaehe v o m Grad 2 sein, die C enthalt; und es ist C ~ F , . F3. W e n n eine Gerade D mit C drei P u n k t e gemeinsam hat, so mug sie in $'3 liegen. Die in F2 enthaltenen Geraden bflden aber eine oder zwei Scharen der Dimension 1, je n a c h d e m F2 ein Kegel ist oder nicht. Folglich kSnnen zwei S e h n i t t p u n k t e einer solehen Gerade mit C n i c h t fiber k unabhangig sein; dabei ist wie frfiher mit k ein Definitionsk6rper ffir C, J , ~ gemeint. Also hat die Verbindungsgerade zweier unabhangiger generischer P u n k t e yon C fiber k keinen dritten Punk~ mit C gemeinsam 1). Die B e s t i m m u n g der singularen P u n k t e von O wird dureh folgenden Hflfssatz geleistet: Hilfssatz 6. E i n positiver Divisor a yore Grad 3 au/ C genCegt dann und nur dann der Bedingung [(a) ~ 2, wenn er sich in der F o r m a : C 9D schreiben lgtflt, wo D eine au/ F2 liegende Gerade ist, und der Schnitt C 9D au/ F~ berechnet wird. Es sei zuerst bemerkt, da~, wenn F2 ein Kegel ist, C den Scheitel von F.~ nicht enthalten k a n n ; wegen C ---- F 2 9~'3 mfil3te namlich sonst dieser Scheitel a u f C singular sein. N u n ist nach dem Riemann-Rochschen Satz die Bedingung l(a) _~ 2 m i t der folgenden gleichwertig: es muB eine Gerade D im R a u m P geben, derart dab fiir jede D enthaltende E b e n e H die Relation H 9C ~ a besteht. D a n n muB offenbar jede K o m p o n e n t e yon a sowohl auf C wie auf D liegen. W e n n D genau zwei P u n k t e A, B m i t F , gemeinsam hatte, so ware D 1 Im Falle der Charakteristik 0 kann bekanntlich nur dann die Verbindungsgerade zweier unabh~ngiger Punkte auf einer Kurve C einen dritten Punkt mit C gemeinsam haben, wenn C in einer Ebene liegt. Im Falle der Charakteristik p ~ 0 gilt dieser Satz nicht mehr; ein Gegenbeispiel wird durch den Ort des Punktes mit den homogenen Koordinaten (1, t~, t~2, t~3) geliefert.

324

50

[1957a] Andrd Weil

zu/~2 in A und in B transversal; eine generische Ebene durch D hatte dann in A und in B hSchstens die Sehnittmultiplizitat 1 mit C; also hatte a hSehstens die zwei Komponenten A, B, jede hSchstens mit dem Koeffizienten 1, was der Voraussetzung widersprieht. Es habe D genau einen Punkt A mit F~ gemeinsam; dann muB A auf C liegen, weft sonst a =- 0 ware; naeh der oben gemachten Bemerkung mull also A ein einfaeher Punkt yon 2' 2 sein. Wenn D nieht die Tangente zu C in A ware, so h~tte wiederum eine generisehe Ebene dureh D die Schnittmultiplizit~t 1 mit C in A, was wie oben zu einem Widersprueh ffihrt. Werm sehlielllieh D die Tangente zu C in A ist, aber nieht in F~ liegt, so nehmen wir fiir H die Tangentialebene zu F~ in A ; dann ist H 9F 2 ~- D' + D " , wo D', D'" zwei Geraden sind; falls F2 ein Kegel ist, ist D' -----D". Nun ist die Sehnittmultiplizit/~t von H mit C in A dieselbe wie die Schnittmultiplizitat yon C mit dem Zyklus D" + D'" in A auf der F1/~ehe F~; sic ist also gleich 2, weft C die Tangente D hat und folglieh sowohl zu D' wie zu D'" in A transversal ist. Demnach kann A hSehstens den Koeffizienten 2 in a haben, was wiederum der Voraussetzung widersprieht. Damit ist bewiesen, dall die Gerade D auf F2 liegen mull. Wenn die Ebene H dureh D geht, so ist t l . F2 = D +4- D', wo D' eine Gerade ist; und H 9C, im Raum P berechnet, ist nichts anderes als der Zyklus (D + D') 9 C, berechnet auf F~. Wenn H generiseh gew~hlt ist, so geht D ' dureh keine Komponente yon a; es mull also D . C ~ a sein; und, wenn es so ist, ist H . C ~ a ftir jede Ebene H dureh D. Wegen C ---- F2 9$'3 ist aber D 9C, auf F~ genommen, nichts anderes als D 9_Fs, bereehnet im Raum P ; D 9C hat also den Grad 3, so dall aus D . C ~ a die Gleiehung D 9C ~-- a folgt. Bei Berfieksiehtigung von Satz 4 zeigt Hilfssatz 6, dall die singularen Punkte yon O die Punkte ~ (D 9C) sind, wenn D die Gesamtheit aller auf F~ liegenden Geraden durehli~uft, und D 9C auf $'2 bereehnet wird. Nun folgt aus dem oben bewiesenen, dall, wenn zwei verschiedene Geraden D, D" auf F2 in einer Ebene H liegen, der Zyklus (D + D ' ) . G, aufF~ bereehnet, derselbe ist wie der Zyklus H 9 C, bereehnet in P ; da die Zyklen der Form H 9C kanonische Divisoren auf C sind, so sieht man, dall unter diesen Umst~nden die Relation D 9C + D" 9C ~ f besteht. Daraus folgt sofort, dall, wenn F2 ein Kegel ist, samtliche Divisoren D 9C miteinander/~quivalent sind; in diesem Fall hat also O genau einen singularen Punkt. Wenn F~ kein Kegel ist, so gibt es auf F 2 zwei Seharen yon Geraden; wenn D und D" nieht derselben Schar angehSren, so liegen sie in einer Ebene. Daraus folgt, dall die Divisoren D . C, wo D eine dieser Scharen durehl~uft, miteinander ~quivalent sind. Also hat dann O entweder einen oder zwei singul/ire Punkte; eine genauere Betraehtung wiirde zeigen, dall es in diesem Fall tatsachlich zwei verschiedene singul/~re Punkte auf 0 gibt. Nun sei H e i n e generisehe Ebene in P in bezug auf k; es sei k (H) ihr kleinster (k enthaltender) DefinitionskSrper; k ( H ) hat dann die Dimension 3 fiber k. Da H generiseh ist, so hat sie bekanntlieh genau sechs versehiedene Schnittpunkte mit C. Es seien M, M ' zwei dieser Schnittpunkte; ihre Verbindungs-

[1957a1

325 Z u m Beweis des Torellischen Satzes

51

gerade ist definiert fiber k(M, M'). Da H dureh diese Gerade geht, so hat k(H) hSehstens die Dimension I fiber k (M, M'); folglieh hat k (M, M') wenigstens die Dimension 2 fiber k; M, M' mfissen also fiber k auf C unabhi~ngig sein. Je zwei der 6 Schnittpunkte yon H mit C sind also fiber k unabh~ngig, woraus folgt, dab keine drei dieser Punkte auf einer Geraden liegen kSnnen. Wenn M, M', M " drei dieser Punkte sind, so bestimmen sie also eine Ebene, die keine andere als H sein kann; demnach mull k(H) in k(M, M', M") enthalten sein, was nur dann mSglich ist, wenn M, M', M " fiber k auf C unabh/ingig sind. Damit ist gezeigt, dall je drei der Schnittpunkte yon H mit C fiber k auf C unabh~ngig sind. Setzen wir H . C ---- ~ M i , und ui~h : 9(Mi + Mj + Mh) ftir jedes i

Tripel (i, j, h). Nach Satz 4 sind die Tangentialhyperebenen zu O in den 20 Punkten ui~-h alle zueinander parallel. Wenn umgekehrt die Tangentialhyperebene zu O in einem Punkt v ---- ~(N + N' + N") zu den Tangentialhyperebenen zu 0 in den Punkten u~j-a parallel ist, so mull naeh Satz 4 H 9C ~ N + N' W N " sein, woraus folgt, d a l l v einer der Punkte ui~h ist. Die Punkte u ~ i h - ucj,h, shad nun die Punkte

O, ~o(M1--M~), q~(MI + M~--Ms--M4), ~o(MI + M2+ M 3 - - M 4 - - M s - - M s ) , sowie diejenigen, die daraus durch eine beliebige Permutation der M~ entstehen. Die Punkte ~ (M, - - M j ) haben die Dimension 2 fiber k; wir werden beweiseh, dall sie dadureh unter den Punkten u i ~ h - ue~,h, charakterisiert sind. Zungchst shad die Punkte uiCh diejenigen, wo die Tangentialhyperebene zn 0 zur Tangentialhyperebene zu k in u~2a parallel ist; da diese Bedingung nur endlieh viele Punkte bestimmt, so sind sie alle algebraiseh fiber k(Ulzs). Da u m gellerisch auf O ist, so haben alle Punkte uijh --ue~,~, h6chstens die Dimension 3 fiber k. Welter ist, weft ~ Mi ein kanonischer Divisor ist: i

~0(M1 ~- M2-~- Ms - - M 4 - M5 - - M s ) -= 2 ~P(M1 -~-M~ + M 3 ) - - q (f) = 2 ux2s--(f); also ist Ulna algebraiseh fiber k ( u l = - u45e); demnach hat der letztere KSrper die Dimension 3 tiber k; dasselbe gilt dann ffir alle daraus dureh Permutation der M, entstehenden Punkte. Der Beweis daftir, dall der Punkt q(M 1 + M ~ - M s - Ma) ebenfalls die Dimension 3 hat, ist etwas umstgndlieher. Er beruht auf folgendem Hilfssatz: Hilfssatz 7. Es gibt eine Spezialisierung (M~) yon (M~) abet k , / a t welch, M~, M'3 abet k unabhangig sind und M~ = M'~ ist. Es seien M~, M~ zwei unabhgngige generische Punkte yon C fiber k. Da Ma, Mi, Ma fiber k unabh/~ngig sind, so kann man sie jedenfalls auf M~, M~, M', spezialisieren und diese Spezialisierung zu einer Spezialisierung (H', M~) yon (H, M~) erweitern0 wo also M~ = M~ ist. Dann ist H' 9C = ~ M'~, also insbesondere H" 9C ~ M~ + M'~, woraus folgt, dall H" dureh die Verbindungs-

326 52

[1957a] Andrd Weil

gerade D yon M~ und M'8 geht, falls diese P u n k t e versehieden sind, und durch die T~ngente D zu C in M~, falls M~ ~ M~ ist; die dadurch definierte Gerade D ist in beiden F~llen rational fiber dem KSrper K = k (M~, M~). N e h m e n wir an, dab die P u n k t e M~, M '3 fiber/r nicht unabh/~ngig seien; dann hat K die Dimension 1 fiber k. Die P u n k t e M'~, und insbesondere M~ und M's, sind alle algebraisch fiber dem kleinsten DefinitionskSrper k (H') ffir H", da M~, M'2 fiber k unabh/~ngig sind, hat demnach k(H') mindestens die Dimension 2 fiber k; H" ist also nieht algebraisch fiber K und ist generisch fiber K in der Schar der durch D gehenden Ebenen. Es seien F ~ 0, F ' = 0 lineare Gleichungen ffir D mit Koeffizienten in K ; die Gleichung ffir H ' 1/~Bt sich in der F o r m F -k tF' = 0 sehreiben, wo t die Dimension 1 fiber K hat. Der P u n k t M~ kann nicht auf D liegen, weil sonst D die Verbindungsgerade der unabh/~ngigen P u n k t e M~, M~ w/~re und keinen dritten P u n k t M~ mit C gemeinsam h/~tte (bzw. nicht die Tangente zu C in M~ sein kSnnte); H" ist dann die durch D und M~ bestimmte Ebene, so dab K(t) in K(M~) enthalten und M' generischer P u n k t von C fiber K ist. Wegen der Gleichung H'.C--M1--M'3=2M~+M~JrM~ folgt daraus sofort, dab M~ fiber K(t) inseparabel algebraisch sein muB, und zwar hSehstens v o m Grad 4. Wenn K (M~) fiber K (t) rein inseparabel (vom Grad 2, 3 oder 4) w/~re, so h/~tten der K 6 r p e r K (M~) und die K u r v e C das Gesehleeht 0. Es muB also K (M~) rein inseparabel vom Grad 2 sein fiber einer separablen Erweiterung K ' yon K (t) vom Grad 2; dann ist aber K ' hyperelliptiseh, und K(M~) ist mit K" fiber K isomorph; also ist C hyperelliptisch, was unserer Voraussetzung widerspricht. Naeh dem Hflfssatz hat also ~ ( M I + M s - M s - M4) die Spezialisierung ~(M~ - - M~) fiber k, mit unabh/~ngigen M~, M 'a fiber/~; letzterer P u n k t hat die Dimension 2 fiber k; wenn der erstere keine gr6Bere Dimension fiber k h/~tte, so w~re die Spezialisierung eine generische; der erstere P u n k t lieBe sieh dann selbst in der F o r m ~ ( N - N') sehreiben mit unabh/~ngigen N, N'; und es w/~re M 1 + M s -k N'~-~ M8 + M4 q- N. Dabei kann nicht l(M 1 -k M2 -k N') > 1 sein; dann mfiBte n~mlich nach Hflfssatz 6 die Verbindungsgerade von M1 und M~ auf F~ liegen, was bei der Verbindungsgerade zweier unabh/~ngiger P u n k t e auf C nicht der Fall sein kann. Es muB also M~ q - M s + N ' - ~ Ms + M4 Jr N sein, was offenbar unm6glich ist. Daraus ergibt sieh folgende invariante Konstruktion ffir die Flgehe V. Man wahle eine beliebige Marmigfaltigkeit T---- Ot aus der Sehar {0~}; es sei ]co ein algebraiseh abgesehlossener Definitionsk6rper ffir J und T ; es sei u o ein generiseher P u n k t von T fiber k 0. U n t e r den 20 P u n k t e n von T, in denen die Tangentialhyperebene zu T zur Tangentialhyperebene zu T in u 0 parallel ist, gibt es 9 P u n k t e mit der Eigenschaft, daB, wenn Ul einer derselben ist, der P u n k t v ~- u~ - - u 0 die Dimension 2 fiber k 0 hat. D a n n ist g der Ort yon v fiber k 0. Das ist n mlich in dem oben Bewiesenen enthalten, falls k o zu

[1957a]

327

Z u m Beweis des Torellischen Satzes

53

gleicher Zeit ein DefinitionskSrper ffir C, q u n d t i s t . W e n n das nicht der Fall ist, w~hlen wir einen k 0 enthaltenden DefinitionskSrper ffir C, q und t. Es i~ndert sich nichts, wenn wir u o durch einen anderen generischen P u n k t yon T fiber k o ersetzen; wir dfirfen also annehmen, d a b uo sogar fiber k generisch ist; m i t denselben Bezeichnungen wie oben kSnnen wir dann u0 = ulna ~- t setzen. Die P u n k t e , wo die T a n g e n t i a l h y p e r b e n e zu T m i t derjenigen in u o parallel ist, sind d a n n die P u n k t e ui~a + t; wenn u~ irgendeiner dieser P u n k t e ist, so ist er algebraisch fiber ko(Uo) ; der P u n k t (uo, u~) h a t also dieselbe Dimension fiber k wie fiber ko, n~mlich 3; folglich h a t er einen Ort fiber k, u n d zwar denselben wie fiber k o; u n d die Dimension yon uo - - u~, sowohl fiber k wie fiber ko, ist nichts anderes als die Dimension des Brides dieses Ortes bei der Abbridung (x, y) -> x - - y y o n J • J a u f J . Oben ist bewiesen worden, dab es 9 P u n k t e u~ gibt, ffir welche dieses Bild die Dimension 2 h a t und dab es dann die Fl~che V ist. D a m i t ist unsere K o n s t r u k t i o n v611ig gerechtfertigt. Den Beweis des H a u p t s a t z e s im vorliegenden Fall k6nnen wir nun genau wie oben fortsetzen. Es b r a u e h t k a u m gesagt zu werden, dab ein einheitlicherer Beweis sehr zu wfinsehen w~re.

[1957b] Hermann Weyl (1885-1955) (avec C. Chevalley)

(~Quand Hermann Weyl et Hella annonc~rent leurs fian~ailles, l'6tonnement rut g~n~ral que ee jeune homme timide et peu loquace, ~tranger aux cliques qui faisaient la loi dans le monde math~matique de GSttingen, efit remport~ le prix convoit~ par tant d'autres. Ce n'est que peu h peu que l'on comprit ~ quel point Hella avait eu raison dans son choix... 1,~ Peut-Stre les v~rit~s math~matiques, comme les femmes, font-elles leur choix entre ceux qu'elles attirent. Est-ce le mieux dou~ qu'elles choisissent, ou le plus s~duisant ? celui qui les d~sire le plus ardemment, ou eelui qui les a l e mieux m~rit~es ? Elles semblent se tromper parfois; souvent il faut du temps pour s'apercevoir qu'elles ont eu raison. Timide, peu loquaee, ~tranger aux cliques, tel apparaissait donc Hermann W e y l / t ses d~buts; tel il devait rester au fond de lui-m~me, en d~pit des succ6s d'une brillante earri~re. Comme beaucoup de timides une fois rompues les barri~res de leur timiditY, il ~tait capable d'enthousiasme et d'~loquence: (~Ce soir-lh, dit-il en raeontant sa premiere rencontre avec celle qu'il devait 6pouser 2, je d~crivis l'incendie d'une grange auquel je venais d'assister; elle me dit plus tard qu'h m'~couter elle s'~tait ~prise de moi aussit6t. ~) Ses propres confidences nous le montrent profond~ment influen~able aussi, jusque dans sa pens~e la plus intime : (~Mon tranquille positivisme 1 E x t r a i t des paroles p r o n o n e e e s p a r C o u r a n t a u x obs~ques de Hella W e y l le 9 s e p t e m b r e i9t~8. Cette c i t a t i o n , e o m m e plusieurs a u t r e s p a r la suite, est tir6e d ' u n e notice inedite eonsaeree p a r H e r m a n n Weu ~t la m 6 m o i r e de Hella Weyl. Nos a u t r e s citations p r o v i e n n e n t des p u b l i c a t i o n s de W e y l . Reprinted from Enseign. Math. IIl, 1957, pp. 157-187, by permission o f the editor.

329

330

[1957b] 158

C. C H E V A L L E Y

ET

A.

WEIL

fut 6bran]6 quand je m'6pris d'une jeune musicienne d'esprit trbs religieux, membre d'un groupe qui s'6tait form6 autour d'un h6g61ien connu... Peu apr6s, j'6pousai une 616re de Husserl; ainsi, ce fur Husserl qui, me d6gageant du positivisme, m'ouvrit l'esprit ~ une conception plus lil~re du monde. )) I1 avail alors vingt-sept ans. C'est ainsi qu'on volt se dessiner, vers l'6poque de son mariage, quelques-nns des prineipaux traits d'une des personnalit6s math6matiques les plus marqnantes et attachantes de la premi6re moiti6 de ce si6cle, mais aussi de l'une des plus difficiles h serrer de pr6s. (~A country lad of eighteen ~, un gars de eampagne de dix-huit ans, ainsi se d6crit-il lui-m~me h son arriv6e h G6ttingen. (~ J'avais choisi cette universit6, dit-il, principalement parce que le directeur de mon lye6e 6tail un cousin de Hilbert et re'avail donn6 pour eelui-ei une lettre de reeommandation. Mais il ne me fallut pas longtemps pour prendre la r6solution de lire et 6tudier tout ce que cet homme avail 6crit. D6s la fin de ma premi6re ann6e, j'emportai son Zahlbericht sous mort bras et passai les vaeances ~ le lire d'un bout ~ l'autre, sans aucune notion pr6alable de th6orie des nombres ni de th6orie de Galois. Ce fnrent les mois les plus heureux de ma vie... 3 )) Un peu plus lard, ce sont les joies de la d6converte: (~Un nouvel 6v6nement fur d6cisif pour moi: je fis une d6couverte math6matique importante. Elle concernait la loi de r6partition des fr6quences propres d'un syst6me continu, membrane, corps 61astique ou 6ther 61ectromagn6tique. Le r6sultat, conjectur6 depuis longtemps par les physiciens, semblait encore bien loin alors d'une d6monstration math6matique. Tandis que j'6tais fi6vreusement occup6 ~ mettre mon id6e au point, ma lampe h p6trole avait commene6 h fumer. Quand je terininai, une 6paisse pluie de flocons noirs s'6tait abattue sur mon papier, sur rues mains, sur mon visage.~) A c e moment, il est d6j& privatdozent /~ GOttingen. Bientbt c'est le mariage, la ehaire ~ Zurich, la 3 ,, De r o u t e rues e x p S r i e n c e s spiritnelles, ~crit-il ailleurs, celles qui m ' o n t c o m b l e de la p l u s g r a n d e joie f u r e n t , e n i 9 0 5 , q u a n d j ' ~ t a i s 5 t u d i a n t , l ' S t u d e d u Zahlberichl ct, en i 9 2 2 , la l e c t u r e de m a i t r e E c k h a r t q u i m e r e t i n t fasein6 p e n d a n t u n s p l e n d i d e h i v e r en E n g a d i n e .

[1957b]

331 HERMANN

WEYL

(1885-1955)

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guerre. Au bout d'un an de garnison h Sarrebruck (comme simple soldat, pr6cise-t-il), le gouvernement suisse obtient qu'il soit rendu ~ son enseignement /~ l'Ecole polytechnique f6d6rale. Mais en 1929, quand GOttingen lui ogre la succession de Hilbert, il se laisse tenter. C'est le nazisme, d'abord imperceptible nuage ~ l'horizon, qui grandit r u e d'ceil, s'abat en trombe sur l'Allemagne en d6sarroi, y recouvre t o u t de bone sanglante. Par bonheur pour H e r m a n n Weyl, l'Institute for Advanced Study de Princeton, nouvellement cr66, ogre de le sauver du d6sastre. I1 h6site. I1 aceepte, il refuse. I1 aceepte de nouveau l'ann6e suivante; c'est de Zurich qu'i] envoie sa d6mission ~ G~ttingen en 1933 et qu'il part pour l'Am6rique. I1 n'eut done pas 5 subir ce stage souvent long, parfois humiliant et p6nible, que les circonstances ont impos6 h beaucoup de savants r6fugi6s aux Etats-Unis. La chaire de l'Institute

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lui assura d'embl6e le confort mat6riel et la situation de premier plan darts le monde scientifique am6ricain auxquels tout, certes, lui donnait droit. Ce furent, dit-il, des ann6es heureuses que cel]es qu'il passa ~ Princeton. Sans doute ne s'aecoutuma-t-il jamais ~ porter ais6ment ce qu'il appelle qu'ils n ' o n t pas lus, qu'ils ne connaissent que par our'-dire! Hermann W e y l se faisait une bien autre et bien plus haute id6e de son m6tier de professeur. I1 vit que Princeton seul, ~ notre 6poque, peut 6tre ce qu'ont 6t6 autrefois Paris, puis GSttingen: un centre d'6ehanges, un ~ clearing-house ~>des id6es math6matiques qui circulent de par le monde. Rappelant l'intense vie math6matique qui s'6tait d6velopp6e autrefois ~ GSttingen sous l'influence dominante de Hilbert, il a 6crit: ;et il ajoute: (~Nous avons assist6 h quelque chose de semblable ici ~ Princeton pendant les premi6res ann6es d'existence de l'Institute for Advanced Study. ~> S'il en a ~t6 ainsi, c'est en grande partie a lui qu'en revient le m6rite. I1 se donna pour tache prineipale de se maintenir au courant de l'actualit6, de renseigner et 6clairer les chercheurs, de leur servir d'interpr6te, de comprendre mieux qu'eux ce qu'ils faisaient ou essayaient de faire; il s'y consacra en route

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modestie, conscient de faire oeuvre utile, conscient d'y ~tre irrempla~able. Dans sa production, qui, pendant toute cette p~riode, reste abondante et d'une extraordinaire varietY, on retrouve la trace de ses lectures, des s6minaires et discussions auxquels il prenait part, des probl~mes sur lesquels de tous c5t6s on sollicitait ses avis. Parmi ces travaux, il n'en est gu~re qui n'61ucide un point difficile ou ne comble une lacune f~eheuse. Cette activit6 s'est poursuivie jusque dans ses derni~res ann6es. Par une supr6me coquetterie peut-~tre, sa dernibre publication aura 6t6 une ~dition rajeunie, compl~tement refondue, de son premier livre, ]ivre toujours utile, encore actuel, auquel par eette r6vision il a donn6 une vitalit6 nouvelle. Qui de nous ne serait satisfait de voir sa carriSre scientifique se terminer de m6me ?

Un Prot6e, qui se transforme sans eesse pour se d~rober aux prises de l'adversaire, et ne redevient lui-m~me qu'apr~s le triomphe final: telle est l'impression que nous laisse souvent H e r m a n n Weyl. N'est-il pas all6, pouss6 par le milieu sans doute, par l'occasion, mais aussi par (~l'inqui6tude de son g~nie ~), jusqu'~ se muer en logiclen, en physieien, en philosophe ? L'axiome Ne sutor ultra crepidam nous interdit de le suivre si loin en ses mStamorphoses. Mais, dans son oeuvre math6matique m~me, il n'est que trop fr6quent qu'il vous glisse entre les mains lorsqu'on eroit le mieux le saisir; et il faut avouer que la t~che de ses leeteurs n'en est pas faeilit~e. I1 est vrai qu'il appartient une p6riode de transition dans l'histoire des math~matiques et qu'il s'en est trouv6 profond~ment marqu6. Souvent il a pu prendre un plaisir grisant ~ se laisser entrainer ou ballotter par les eourants opposes qui ont agit6 eette 6poque, stir d'ailleurs au fond de lui-m~me (comme lorsqu'il s'abandonna un m o m e n t l'intuitionnisme brouw6rien) que son bon sens foncier le garantirait du naufrage. Son oeuvre a grandement eontribu6 ce changement de vision qui a fait passer de la math~matique elassique, fond6e sur le hombre r6el, ~ la math6matique moderne, fond6e sur la notion de structure. L'emploi syst6matique et tout abstrait du rev~tement universel, la notion de vari6t6 analytique

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eomplexe, l'emploi courant et la popularisation, jusque parmi les physiciens, de l'alg6bre vectorielle et du concept d'espaee de repr6sentation d'un groupe, t o u t cela vient a v a n t tout de lui. Mais, s'il 6tait trop 616ve de Hilbert pour ne pas inelure parmi ses outils la m6thode axiomatique, s'il 6tait trop math6matieien aussi pour en d6daigner les sucd6s (son chaleureux 61oge de l'ceuvre d ' E m m y Noether serait lh, si besoin 6tait, pour en faire foi), ee n'6tait pas /~ elle qu'allaient ses sympathies. I1 y v o y a i t . Le probl6me du m o u v e m e n t m o y e n portait sur les sommes d'exponentielles, finies il est vrai, et Weyl en

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avait trait6 des cas particuliers, sans d'ailleurs approfondir la question, qu'il ne devait r6soudre compl~tement, toujours par la m6me m6thode, que lorsqu'il s'y t r o u v a ramen6 vingt-cinq ans plus tard. Mais H. Bohr, alors 61~ve de Landau, s'occupait de s6ries d'exponentielles en vue de l'6tude de ~ (s) dans le plan complexe, probl~me auquel Weyl va bient6t s'int6resser en passant, d6terminant m6me par sa m6thode le c o m p o r t e m e n t asymptotique de ~ (i -k it). D'autre part, les 6]~ves de Hilbert 6talent accoutum6s ~ consid6rer les termes de la s~rie de Fourier comme fonctions propres, et les coefficients de cette s6rie comme valeurs propres, d'op6rateurs convenablement d6finis. I1 semble donc que les voles fussent toutes pr6par6es dans l'esprit de Weyl, lorsque a p p a r u r e n t les premiere t r a v a u x de H. Bohr sur les fonctions presque p6riodiques, pour reprendre la question du point de vue des ~quations int~grales. Mais il est rare qu'un math6maticien, qu'il s'agisse du plus grand ou du plus humble, parcoure le plus court chemin d'un point ~ un autre de sa trajectoire. A v a n t de revenir aux fonctions presque p6riodiqnes ~ l'occasion d'une conf6rence de H. Bohr h Zurich, Weyl avait men6 ~ bien ses m~morables recherches sur les groupes de Lie et leurs representations, et avait congu l'id~e, d'une audace extraordinaire pour l'6poque, de ~ construire ~ les repr6sentations des groupes de Lie compacts par la complete d6composition d'une representation de degr6 infini. Blas6s que nous sommes par l'exp6rience des trente derni6res ann6es, cette id6e ne nous 6tonne plus; mais son succ~s semble avoir fait l'effet d'un vrai miracle ~t son auteur; ~ c'est lh, r6p~te-t-il ~ maintes reprises, l'une des plus surprenantes applications de la m6thode des 6quations int6grales ~>. D~jh I. Schur avait 6tendu au groupe orthogonal, au moyen de l'Ol~ment de volume invariant dans l'espace de groupe, les relations d'orthogonalit~ entre coefficients des repr6sentations que Frobenins avait d6couvertes pour les groupes finis ; mais il y avait loin de lh/~ un th6or6me d'existence. Weyl n'h6site pas h introduire, sur un groupe de Lie compact, l'alg~bre de groupe, toujours congue chez lui comme alg6bre des fonctions continues par rapport au produit de convolution h = f * g,

h(s) = f [ ( s t -1) g(t)dt,

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et dont il fait un espaee pr6hilbertien au m o y e n de la norme (f I] [~ ds)~; les int6grales, naturellement, sont prises au m o y e n de l'616ment de volume i n v a r i a n t que fournit la th6orie de Lie, et que Hurwitz avait sans doute ~t6 le premier h utiliser s y s t 6 m a t i q u e m e n t . Dans cet espace, l'op6rateur / - ~ ~ * ~ * /, off ~ d~signe la fonetion ?~ (s) = ~ (s-l), est hermitien et compl~t e m e n t continu; d'apr~s la th6orie de E. Schmidt, ses valeurs propres forment done un spectre discret, et h chacune correspond un espace de fonetions propres de dimension finie, dont on constate i m m 6 d i a t e m e n t qu'il est i n v a r i a n t par le groupe; c'est done un espaee de repr6sentation de celui-ci. Les th6or~mes de Schmidt fournissent a]ors le d6veloppement de 9 suivant les coefficients des repr6sentations ainsi obtenues, d6veloppement qui converge au sens de la norme. C'est lh une g6n~ralisation directe de la m6thode de Frobenius basSe sur la r~duction de la repr6sentation r6guli~re d ' u n groupe fini; la seule diff6renee, c o m m e l'observe Weyl, e'est l'absence d ' u n 61~ment unit6 dans l'alg~bre d ' u n groupe c o m p a c t ; W e y l y suppl6e par un artifice tir6 de la th6orie des s6ries de Fourier, h savoir l ' a p p r o x i m a t i o n de la masse unit6 plac6e h l'origine p a r une distribution de masses h densit6 continue, concentr6e dans un voisinage de l'origine; la convolution avec celle-ei eonstitue un (~op6rateur r6gularisant)), d'emploi courant aujourd'hui, mais dont c'6tait sans doute la premi6re apparition dans le cadre de la th6orie des groupes de Lie ; Weyl s'en sert pour d6montrer que toute fonction continue p e u t 6tre approch6e, non seulement au sens de la norme, mais m~me uniform6ment, par des combinaisons lin6aires de coefficients de repr6sentations. Bien que le m6moire de W e y l se limit~t n6cessairement aux groupes de Lie, il a v a i t atteint en r6alit~, du premier coup, des r6sultats d~finitifs sur les representations des groupes c o m p a c t s ; apr6s la d~couverte de la mesure de H a a r , il n ' y eut pas un m o t ~ changer ~ son expos6, et, chose rare en m a t h 6 m a tique, il ne vint m6me ~ personne l'id6e de le r6crire. Si, comme nous le raisons aujourd'hui, on consid~re une fonetion presque p6riodique c o m m e d~terminant une representation du groupe additif des r6els darts un groupe compact, et q u ' o n suppose acquise pour celui-ci ]a notion de mesure de H a a r , on d~duit

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imm6diatement des r6sultats de Weyl expos6s ci-dessus le d6veloppement de la fonetion en s6rie d'exponentielles. Les outils manquaient h Weyl, en i926, pour adopter ce point de r u e ; il y suppl6e en rempla~ant l'int6grale par une moyenne sur la droite, d6finie comme limite pour T-+ + zr de la valeur moyenne sur l'intervalle It, t ~ 3?] lorsque cette limite est atteinte uniform6ment par rapport au param6tre t. En 1926, il n'allait pas de soi que la th6orie des 6quations int6grales s'appliqu~t h cette moyenne; Weyl est oblig6 de consacrer une bonne partie de son travail A justifier cette application. I1 convient d'observer d'autre part que, sur un groupe compact, la mani6re la plus simple de construire la mesure de Haar consiste j u s t e m e n t h attacher ~ chaque fonction continue une valeur moyenne, par un proc6d6 directement inspir6 de la th6orie des fonctions presque p6riodiques. Que Weyl, en revanche, air cru voir dans la th6orie de Bohr (~le premier exemple d'une th6orie des repr6sentations d'un groupe vraiment non compact )> (par opposition a p p a r e m m e n t avec les groupes de Lie semi-simples dent les repr6sentations, dans son esprit, se ramenaient, par la (( restriction unitaire )), h celles de groupes compacts), cela montre qu'il se faisait encore quelque illusion sur le degr6 de difficult6 des probl~mes qui restaient ~ r6soudre. Ce n'en est pas moins lui qui a ouvert la voie h t o u s l e s progrbs ult6rieurs duns cette direction.

Sur le reste de son oeuvre d'analyste, nous serons beaucoup plus brefs, d ' a u t a n t plus qu'il a lui-m~me excellemment rendu compte d'une honne partie de cette oeuvre dans sa Gibbs Lecture de t948. D6butant, il partieipa activement au courant de recherches qui se proposait d'approfondir et d'appliquer A des probl6mes vari6s d'analyse la th6orie spectrale des op@ateurs sym6triques. Citons particu]igrement, dans cet ordre d'id6es, sa Habilitationsschri/t de 19i0, off il 6tudie un op@ateur diff6rentiel autoadjoint L sur la demi-droite [0, + ~ ]: d (p(t)d~) L (u) = ~t - - q (t) ~,

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Od p, q sont ~ valeurs r6elles et p (t) > O. Sur tout intervalle fini [0, 1], cet op6rateur, soumis aux conditions aux limites du t y p e habituel, (du/dt)o = hu (0), (du/dt)z = h' u (1), rel6ve de la th6orie de Sturm-Liouville ou, en termes modernes, de la th6orie des op6rateurs compl~tement continus; le spectre est r~el et discret et se compose des X pour lesquels l'~quation L u = Xu a u n e solution satisfaisant aux conditions aux limites impos6es. Le passage ~ la limite l ~ -1- w fair apparaitre, non seulement un spectre continu qui peut couvrir tout l'axe r6el, mais encore des ph6nom~nes impr~vus dont la d6couverte est due ~ Weyl. Les plus int6ressants concernent le comportement des solutions pour l-~-t- ~ lorsqu'on donne ~t X une valeur imaginaire fixe; chose remarquable, ils sont ind6pendants du choix de la valeur donn6e ~ X. C'est ainsi que Weyl est amen~ en particulier ~ la distinction fondamentale entre le cas du (~point limite ~ et le cas du : l'une des propri6t6s caract~ristiques du premier, c'est que l'~quation Lu = Xu y poss+de, quel que soit X imaginaire, une solution et une seule de carr~ sommable sur [0, + w ], tandis que toutes ses solutions le sont, pour X imaginaire, dans le eas du cerele limite. Weyl ~tudie aussi le passage ~ la limite 1-~ + c~ pour les d6veloppements de Sturm-Liouville sur [0, l]; il en tire des formules int6grales o~ apparaissent en g~n~ral des int6grales de Stieltjes, comme on pouvait s'y attendre. Le probl~me des moments de Stieltjes n'est d'ailleurs pas autre chose qne le probl+me aux diff6rences finies, analogue ~ l'6quation Lu ---- Xu sur la demidroite, et Hellinger fit voir par la suite que la m6thode de Weyl s'y transporte prescIue telle qnelle. Mais Weyl put aussi la transposer plus tard ~t un probl~me diff+rentiel o~ le param~tre spectral intervient non lin6airement, ainsi qu'au probl6me aux diff6renees finies correspondant (auqnel il a donn6 le nora de probl6me de Piek-Nevanlinna) ; il apporta m~me ~ cette occasion quelques am61iorations notables ~ son premier expos6. Si celui-ei a donn~ lieu depuis lors ~t des g~n6ralisations assez vari6es, il ne semble pas que la signification v~ritable des r~sultats de Weyl sur les probl6mes ~ param~tre non lin6aire ait jamais 6t6 tir6e au clair. Une autre s6rie de t r a v a u x traite de la r6partition des valeurs propres, dans divers probl~mes de t y p e elliptique. Ils

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reposent principalement sur un principe qui plus tard fut popularis6 par Courant sous la forme suivante: si A est un 0p6rateur sym6trique compl6tement eontinu dans un espaee de Hilbert H, sa n-i6me valeur propre est la plus petite des valeurs ((~minimum m a x i m o r u m ~) que peut prendre la norme de A, c'est-/~-dire le hombre max (Az, z ) / ( z , z), sur un sousespaee de H de codimension n - - 1. Une lois acquise la th6orie des op6rateurs eomplbtement continus, la v6rifieation de ce principe est d'ailleurs imm6diate. Mais Weyl l'adapte en virtuose routes sortes de situations de physique math6matique. Quant au e o m p o r t e m e n t asymptotique des fonetions propres, il avait, nous dit-il en 1948, eertaines conjectures: ~ mais, n ' a y a n t fair pendant plus de trente-einq ans aucune t e n t a t i v e s6rieuse pour les d6montrer, je pr6f6re, ajoute-t-il, les garder pour moi ~; il aura done laiss6 ee probl6me plus diffieile en h6ritage /~ ses suecesseurs.

C'est en 61~ve de Hilbert encore, et en analyste, que Weyl dut aborder le sujet d'un des premiers eours qu'il professa/~ GSttingen comme jeune privatdozent, la th6orie des fonctions selon Riemann. Le eours terrain6 et r6dig6, il se retrouva g6om6tre, et auteur d'un livre qui devait, exereer une profonde influence sur la pens6e math6matique de son si~cle. Peut-~tre s'6tait-il propos6 seulement de remettre au gofit du jour, en faisant usage des id6es de Hilbert sur le prineipe de Dirichlet, les expos6s traditionnels dont l'ouvrage classique de C. Neumann fournissait le modble. Mais il dut lui apparaitre bient6t que, pour substituer aux constants appels ~ l'intuition de ses pr6d6eesseurs des raisonnements corrects et, eomme on disait alors, ~ rigoureux ~ (et dans l'entourage de Hilbert on n'.admettait pas qu'on trich~t l~-dessus), c'6taient avant t o u t les fondements topologiques qu'il fallait renouveler. Weyl n'y semblait gubre pr6par6 par ses t r a v a u x ant6rieurs. I1 pouvait, dans cette tfiche, s'appuyer sur l'ceuvre de Poincar6, mais il en parle h peine. I1 mentionne, comme l ' a y a n t profond6ment influene6, les recherches de Brouwer, alors darts leur premi6re nouveaut6; en r6alit6, il n'en fait aucun usage. De fr6quents contacts avec Koebe, qui d6s lors

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s'6tait consacr6 tout entier h l'uniformisation des fonctions d'une variable eomplexe, durent lui ~tre d'une grande utilit6, particuli~rement dans la mise au point de ses propres id6es. La premibre 6dition du livre est d6di6e ~ F61ix Klein, qui bien entendu, comme Weyl le dit dans sa pr6face, ne pouvait manquer de s'int6resser h u n travail si voisin des pr6occupations de sa jeunesse ni de donner ~ l'auteur des conseils inspir6s de son temp6rament intuitif et de sa profonde connaissance de l'ceuvre de Riemann. Bien qu'il n'efit jamais connu celui-ci, c'6tait Klein qui, ~ GSttingen, incarnait la tradition riemannienne. Enfin, dans Fun de ses m6moires sur les fondements de la g6om6trie, Hilbert avait formul6 un syst6me d'axiomes fond6 sur la notion de voisinage, en soulignant qu'on trouverait lh le meilleur point de d6part pour >/~ 4g c6t6s (g d6signant le genre), et /t la d6termination explicite du groupe fondamental, et on peut regretter qu'il ne l'ait pas fait. Mais la construction mgme des rgtrosections, n~cessairement bas~e sur la triangulation, disparait dans la troisigme ~dition, au profit d'un traitemerit plus purement homol0gique o/1 n'interviennent que des recouvrements. En tout cas, pour tout l'essentiel, ee chapitre constitue une raise au point /~ peu prbs d6finitive des questions qu'il traite. Les th4orgmes d'existence font l'objet du deuxi~me chapitre. Weyl y donne du principe de Dirichlet une d~monstration sireplifi6e, bas6e naturellement sur l'id~e de Hilbert qui consiste, comme on sait, ~ op@er dans l'espace pr~hilbertien des fonetions diff@entiables a v e c l a norme de Diriehlet; mgme dans la troisigme 6dition, il n'a pas cru devoir suivre la variante qu'il avait pourtant contribu~/~ cr6er lui-mgme, et qui consiste R op6rer par projection orthogonale dans le compl6t6 de l'espaee en question, puis ~ montrer apr~s coup que la solution obtenue est diff@entiable. Une fois acquis le principe de Diriehlel,, l'auteur en tire les principales propri6t6s des int6grales ab61iennes et des fonctions multiplicatives, le thbor~me de I/iemann-I/och, puis le th6or~me de l'uniformisation, c'est-Mdire la representation conforme du rev~tement universel de la surface de t/iemann sur une sph@e, un plan ou un disque. Si on laisse de c6t6 les cas de genre 0 ou 1, le rgsultat peut s'exprimer en disant que route surface de t/iemann compacte, de genre > 1, peut se d~finir comme quotient du plan non-euclidien par un groupe discret de d~plaeements sans point fixe. ). Nous dirions aujonrd'hui qu'on a construit pour la surface de ttiemann un module qui est cano-

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nique h u n d6placement pr6s dans le plan non euclidien; autrement dit, on a associ6 canoniquement une structure h une autre. Mais qui saurait mauvais gr6 h Weyl, apr6s avoir achev6 un livre de cette valeur, d'avoir exprim6 d'une mani6re peut-Stre un pen trop romantique son enthousiasme juv6nile ?

C'est en 19t6, pendant la guerre, que Weyl fit paraitre en Suisse son premier m6moire de g6om6trie, sur le c616bre probl6me de la rigidit6 des surfaces convexes. Ici encore, GSttingen lui avait fourni son point de d6part. Sous la direction de Hilbert, Weyl avait collabor6 h la publication des oeuvres compl6tes de Minkowski, off la th6orie des corps convexes tient rant de place. D'autre part, Hilbert avait montr6 comment on peut faire d6pendre les in6galit6s de Brunn-Minkowski de la th6orie des op6rateurs diff6rentiels elliptiques. L'espace t/3 6tant consid6r6 comme espace euclidien, et (x, y ) d6signant le produit scalaire dans R a, soit V un corps eonvexe dans cet espace, d6fini au moyen de la fonction d'appui H; eela veut dire que H satisfait aux conditions II(x + x') ~< H(x) + tt(x'),

H(Xx) ~ ?,H(x)

pour ), >~ 0,

et que V e s t l'ensemble des points y satisfaisant h (x, y> ~< H (x) quel qne soit x. Si on suppose H diff6rentiable en dehors de 0, ]e volume de V est alors donn6 par une formule vol (V) = f H.Q (H) d ~ ,

o6 l'int6grale est 6tendue h la sph6re unit6 S o d6finie par (x, z} ~- i, oh do~ d6signe l'616ment d'aire sur So, et oh Q (H) est une forme quadratique par rapport aux d6riv6es partielles seeondes de H. Soient F, F' deux fonctions, diff6rentiables en dehors de 0, satisfaisant toutes deux h la condition d'homog6n6it6 F (;~x) = kF (x) pour X ) 0; soit B (F, F') la forme bilin6aire sym6trique par rapport aux d6riv6es partielles secondes de F et h celles de F' qui se d6duit de la forme quadratique Q (H) par lin6arisation, c'est-~-dire qni est telle que Q (H) = B (H, H);

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posons aussi LF ( F ' ) = B (F, F'). On v6rifie facilement, au moyen de la formule de Stokes, que l'int6grale I (F, F', F") = j" F".B (F, F')d co, So

o5 F " d6signe une troisi~me fonction satisfaisant aux m~mes conditions que F et F', d6pend sym~triquement de F, F' et F " Cela revient ~ dire que LF, consider6 comme op6rateur diff~rentiel sur les fonctions sur S OprolongSes/~ R 3 par homog6n6itS, est un op~rateur autoadjoint. Si V', V" sont deux corps convexes d~finis par des fonctions d'appui H', H", les formules ci-dessus m o n t r e n t que les d'op~rations unitaires, il est nSeessaire que .~ admette ce qu'on appelle aujourd'hui une forme eompaete, ou pour mieux dire une base telle que ]es eombinaisons lin~aires r6elles des ~lSments de cette base forment l'alg~bre de Lie d'un groupe compact. En examinant ehaque groupe simple s@ar~ment, Cartan avait vSrifi~ dans chaque eas l'existence d'une forme compaete; Weyl en donne une d~monstration a priori basSe sur les propri~t~s des eonstantes de structure de g. Cela fait, il introduit ]e groupe G~ des operations unitaires de G, et son alg~bre de Lie g~. Le groupe G~ est compact, et la th~orie des representations de % est 5quivalente ~ celle des reprSsentations de g. Mais iei se pr~sente une difficult~ nouvelle; du fait que G~ peut n'~tre pas simplement connexe, la th~orie des reprSsentations de gu n'est plus entiSrement ~quivalente h celle des reprSsentations de G~. Si on eherehe h r~tablir l'~quivalence en remplagant G u par son rev~tement universel G~, qui, lui, est simplement connexe, il devient n~eessaire de s'assurer que eelui-ei est compact, et aussi d'en faire un groupe, loealement isomorphe G~,. Ce dernier point, qui devait peu apr~s 5tre ~lueid5 par Sehreier, est compl~tement laiss5 de c6t5 dans le mSmoire de Weyl. Mais c'est dans le premier que r~sidait la vSritable diffieult~. La question revient naturellement h faire voir que G~ a un groupe fondamental fini. Pour cela, Weyl introduit un sousgroupe A~ de G~, qui joue le m~me r61e que le groupe des matrices diagonales dans la th~orie du groupe unitaire special. T o u t ~l~m e n t s de G~ est eonjugu5 ~ un ~l~ment de A~; excluant certains 5lSments s, dits singuliers, qui forment un ensemble a y a n t trois dimensions de moins que Gu, s n'est eonjugu5 qu'/~ un hombre fini d'515ments de A u ; de plus, les 516ments de A u qui ne sont pas singuliers forment dans A~ un domaine simplement connexe A. Supposons que s d~erive dans G~ une eourbe fermSe F qui ne reneontre pas l'ensemble des ~lSments singuliers. Si s (t) est le point de param~tre t sur F, on peut dSterminer par eontinuit5 une courbe a (t) dans A~ telle que, pour t o u t t, a (t) soit eonjugu5 ~ s (t).

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Quand le point s (t) revient /~ sa position initiale s (1) = s (0), le point a (t) vient en un point a (1) qui est un 616ment de A~ conjugu6 de a (0), ce qui ne laisse pour ee point qu'un nombre fini de possibilit6s. Si on a a (1) = a (0), la courbe d6crite par a (t) est ferm6e, et par suite r6duetible a un point dans A; Weyl montre que P e s t alors elle-m~me r6ductible /~ un point. I1 en r6sulte facilement que le groupe fondamental de l'ensemble des 616ments non singuliers de (3~ est fini. De cela, et du fait que les 616ments singuliers se r6partissent sur des sous-vari6t6s a y a n t an moins trois dimensions de moins que On, Weyl conelut (/~ vrai dire sans d6monstration) que G~ lui-m~me a un groupe fondamental tlni. Ce point ~tabli, la voie est ouverte /~ la g~n~ralisation complete au cas semi-simple des r~sultats obtenus pour le groupe lin6aire sp6eial. Weyl d6montre la complete rdduetibilit6 des representations de g, et d~termine explicitement le caract~re et le degr~ d'nne repr6sentation simple de poids dominant donn~. Ici encore, cette d6termination r6sulte des relations d'orthogonalit~ entre caractbres et des propri6t~s formelles de la restriction Z d'un caract~re au groupe A~ qui reeouvre A~ dans le rev~tement simplement connexe G~ de G~. Ce groupe est un tore; Z e s t une combinaison lin~aire finie de caraetbres de ce tore, invariante par les op6rations d'uu certain groupe fini S d'automorphismes du tore qui g6n6ralise le groupe des permutations de z~, ..., z~ dont il a 6t6 question plus haut/~ propos du groupe unitaire special. Le groupe S, dont les dgveloppements ult6rieurs de la th~orie out montr6 qu'il y joue un r61e fondamental, s'appelle m a i n t e n a n t le groupe de Weyl. Enfin la th6orie s'ach~ve par la d6monstration de l'existence des repr6sentations simples de poids fondamental donn~. Pour les alg~bres simples, cette existence avait 6t~ 6tablie par Cartan par des constructions directes dans ehaque cas partieulier. Weyl, lui, applique au groupe compact G~ la m6thode de d6composition de la , obtenue au moyen de la th~orie des 6quations int~grales suivant l'id~e que nous avons expos6e plus haut. Pour eonclure /~ partir de 1/~, il lui faut encore un lemme de nature plus technique, ~none6 seulement dans le m~moire de 1926, et dont Weyl n'a publi6 la d6monstration que

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'183

dans son cours de 1934-35 (paru h Princeton sous forme de notes mim6ographi6es, The structure and representations o/ continuous groups).

Beaucoup plus tard, Weyl revint sur la d6termination des repr6sentations des groupes semi-simples dans son ouvrage The classical groups, their in~ariants and representations. L ' e s p r i t de ee livre est assez diff6rent de celui du m6moire de 1926. L ' o b j e t de l ' a u t e u r est m a i n t e n a n t d'une p a r t de d6montrer par des m6thodes p u r e m e n t alg6briques les r6sultats d6j/~ obtenus au sujet des repr6sentations des groupes classiques (groupe lin6aire g6n6ral, groupe lin6aire sp6cial, groupe orthogonal et groupe symplectiqne), et d ' a n t r e p a r t de faire la synthSse entre ces r6sultats et ]a th6orie formelle des invariants qui s'6tait d6velopp6e sons l'influenee de Cayley et Sylvester au cours du x~x e si6cle. Esp6rait-il cette lois se laver d6finitivement du reproehe de S t u d y en r a m e n a n t h la vie cette th6orie qui 6tait sur le point de sombrer dans l'oubli ? I1 nons dit lui-m~me que la d6monstration par Hilbert du th6or6me g6n@al de finitude a v a i t t~presque tu6 le sujet ~>; on peut se d e m a n d e r si Weyl ne lui a u r a pas, en r6alit6, port6 le coup de grace. La situation dans laquelle on se trouve en th6orie des invariants est la suivante. On a uric ou plusieurs repr6sentations lin6aires p, p', ..., d'un groupe G, op6rant sur des espaces r e c t o rids V, V', ... On consid@e des fonetions F (a, x', ...) d6pendant d'un a r g u m e n t a dans V, d'un a r g u m e n t a' dans V', etc. et s'exprim a n t c o m m e polynSmes par r a p p o r t aux eoordonn6es de ces arguments, homog6nes par r a p p o r t aux coordonn6es de chacun d'eux. Une telle fonction s'appelle un i n v a r i a n t si, pour t o u t s dans G, on a F (s.x, s.x', ...) ~- F (x, x', ...) .

Si J~, ..., Jh sour des invariants, t o u t polyn6me en J1, ..., Jh en est un aussi p o u r v u qu'il satisfasse aux conditions d'homog6n6it6 impos6es. Le premier probl6me de ]a th6orie est de t r o u v e r des invariants J~, ..., Jh tels que t o u t autre invariant puisse s'6crire comme polyn6me en les J~; eela fait, on se propose 6galement

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C. C H E V A L L E Y E T A. W E I L

de d~terminer les relations alg~briques F (J1, ..., Jh) = 0, dites (~syzygies ~, qui lient entre eux les invariants q u ' o n a construits. Pla~ons-nous plus partieuli~rement dans le cas off G a ~t~ identifi0, au m o y e n d'une certaine representation ,ol, avee un sous-groupe du groupe lin~aire/~ n. variables, operant sur l'espace vectoriel V 1 ~- k n, off k est un corps de base q u ' o n suppose de caract~ristique 0. Consid0rons d ' a b o r d le cas off les repr~sentations ~, f , ..., coincident toutes avec 91; on dit alors q u ' o n eherche les invariants d'un certain n o m b r e de ~ vecteurs ~ (on entend par 1~ des vecteurs de V1). R e p r e n a n t sans grand changem e n t son travail de 1924 par lequel il a v a i t r~pondu ~ Study, Weyl montre alors, pour un groupe G unimodulaire, que la d~termination des invariants de vecteurs en n o m b r e quelconque peut se ramener, au m o y e n des identit~s de Capelli, au probl~me analogue pour n I vecteurs. Si G n ' e s t pas unimodulaire, ce r~sultat reste vrai pour les (~invariants relatifs ~ (polyn6mes se multipliant par une puissance du d~terminant de s quand on t r a n s f o r m e tons les vecteurs par s). Weyl d0duit de 1~ la solution des deux probl~mes ci-dessus pour le groupe unimodulaire et pour le groupe orthogonal; et il ~tend cette solution au cas des invariants d~pendant, non seulement d'un certain h o m b r e de vecteurs ~ cogr~dients ~ (se t r a n s f o r m a n t suivant p1), mais aussi d ' u n certain n o m b r e de veeteurs (~contragr~dients ~ (se transf o r m a n t c o m m e les formes lin~aires sur V1). Ensuite il passe aux invariants d~pendant de ~ quantit~s ~> x, x', ... a p p a r t e n a n t des espaces de representation quelconques du groupe ~tudi~; le cas off x, x', ... sont des formes homog~nes p a r r a p p o r t aux coordonn~es d ' u n vecteur ~contragr~dient ~ est celui dont t r a i t a i t plus particuli~rement la th~orie classique. Pour pouvoir aborder ]a question dans ce cadre g~n~ral, il faut a v a n t t o u t connaitre les representations simples du groupe; aussi une partie i m p o r t a n t e du livre est-elle consacr0e ~ la d~termination algabrique des representations (~tensorielles ~ des groupes classiques. Cela fair, Weyl m o n t r e que les i n v a r i a n t s d~pendant de plusieurs ~ quantit~s ~ d'esp~ee quelconque s ' e x p r i m e n t c o m m e polynSmes en un n o m b r e fini d'entre eux; il ~tend ce rSsultat, dans une certaine mesure, au groupe affine. Enfin, il emploie la m~thode d'int~gration pour d~montrer le r~sultat correspondant pour les

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repr6sentations quelconques d'un groupe compact, le corps de base 6rant cette lois le corps des r~els.

Pour f~ter son soixante-dixi~me anniversaire, les amis et 61~ves de Hermann Weyl publi@ent un volume de Selecta extraits de son oeuvre. I1 n'y a peut-6tre pas lieu de se f61iciter de cette mode des morceaux choisis destin6s '~ cS16brer la raise la retraite de math6maticiens 6minents. C'est trop pour les uns; ce n'est pas assez pour les autres. Du moins le volume en question contient-il une bibliographie complete de l'oeuvre de Hermann Weyl, ~tablie par ordre chronologique s, et dont nous avons naturellernent fair grand usage pour r6diger la pr6sente notice. Pour rem6dier en quelque mesure aux inSvitables lacunes de celle-ci, nous donnons ci-dessous une liste des mSmoires de Weyl, classSs par sujet; rien ne peut mieux, croyons-nous, en faire ressortir l'Stonnante vari6t6. Les num@os, bien entendu, renvoient h la liste des Selecta.

I. Analyse. a) Equations int6grales singuli@es: i, 3. b) Probl~mes de valeurs propres et dSveloppements fonctionnels associSs h des 6quations diff@entielles ou aux diff@ences finies: 6, 7, 8, 12, 103.

c) R6partition des va]eurs propres d'op@ateurs complStement continus en physique math6matique: 13, 16, 17, 18, 19, 22.

d) Espace de Hilbert: 4, 5. e)

Ph6nombne de Gibbs et analogues: 10, 11, 14.

/)

Equations diff@entielles li6es h des probl~mes physiques: 36-37 (d~veloppements asymptotiques, apparent6s au ph6nom~ne de Gibbs, au voisinage d'une discontinuit6 dans un

n convient de signaler qu'on n ' a pas fait figurer clans cette bibliographie les notes de cours, publi6es sous formc mim~ographi6e par l ' I n s t i t u t e for A d v a n c e d Study de Princeton, ct qui reproduisent plusieurs des cours qu'il y professa.

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WElL

probl~me d'61eetromagnOtisme), 123-t24-125 (6tude directe d'une Oquation diff6rentielle li6e /~ un probl6me de eouehe limite).

g) Problbmes elliptiques: 121 (principe de Dirichlet trait6 par la 0 , par son centre = quotient du groupe lin6aire sp6cial SL(R, n), it n variables, de d6terminant 1, par son centre), et du groupe discret F0, image dans G o du groupe multiplicatif F des matrices de d6terminant _+ 1 /l coefficients dans Z. Un sous-groupe compact maximal de Go est l'image K o dans Go du groupe orthogonal spdcial SO(R, n) (matrices orthogonales de ddterminant 1). Soit P l'espace des formes quadratiques positives non d6g6ndr6es fin variables: n

F(x) = ' x . A . x =

~

aijxixj,

'A = A.

i,j--I

(un vecteur x, en notation matricielle, sera toujours conqu c o m m e matrice ",i n lignes et 1 colonne). On 6crira A >> 0 pour exprimer que la matrice sym6trique A est 360

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Reduction des formes quadratiques, d'apres Minkowski et Siegel celle d'une forme positive non degendrde. Le groupe L(R, n) opere sur P par la loi (A, X) -~ tX 9A - X, off A ~ P e t X est un element de L(R, n) ecrit comme matrice. Toute forme de P s'ecrivant comme somme de n carrds (de formes lineaires en les x;), le groupe opere transitivement dans P: l'element 1, de P (c'est-fi-dire la forme "standard" ~ x~) etant invariant par le groupe orthogonal O(R, n), P s'identifie l'espace homogene L(R, n)/O(R, n). Dans l'espace vectoriel (de dimension n(n + 1)/2) de routes les formes quadratiques dans R" (ou, ces formes 6rant exprimdes au moyen de la base canonique de R", dans l'espace des matrices symdtriques fi n lignes et n colonnes), Pest ddfini par les inegalites P(x) > 0 (x r 0) et forme donc un c6ne convexe; on verifie immediatement que ce c6ne est ouvert; sa frontiere est l'ensemble des formes positives ddgenerees. Par passage au quotient par la relation d'equivalence dont les classes d'equivalence sont les rayons ("demi-droites") issus de 0, P determine une partie convexe Po d'un espace projectif de dimension n(n + 1)/2 - 1. Par passage au quotient, le groupe projectif Go = P L ( R , n) opere dans Po; il resulte de ce qui precede qu'il y opere transitivement, et que P0 s'identifie fi Go/Ko, espace "riemannien symetrique" associe/t Go. La theorie de Minkowski conduit/t la determination, dans Po, d'un domaine fondamental pour le groupe discret F o, qui est un polyedre convexe (plus exactement, la reunion de l'intdrieur d'un polyedre convexe et d'une pattie convenable de sa frontiere). Soit d'abord n = 2 ("formes binaires"); on ecrit F ( x ) = ax 2 + 2 b x y + cy2;

P e s t le c6ne determine par a > 0, ac - b 2 > 0 (intdrieur de l'une des nappcs d'un cene du second degr6 dans R3); Po est, dans le plan projectif, l'intdrieur de la conique ac - b 2 = 0. C o m m e X --. t X 9A 9X est une representation du groupe lineaire dans l'espace des formes symdtriques A, il s'ensuit, par passage au quotient, que les operations de Go dans Po sont des homographies ou automorphismes du plan projectif ambiant, conservant la conique frontiere de Po; Po 6rant pris comme "modele cayleyien" de la geometrie plane hyperbolique, G induit donc, sur P0, un sous-groupe du groupe des automorphismes de cette geometrie; on verifie sans peine que c'est marne exactement la pattie connexe de ce dernier groupe (c'est-fidire le "groupe des ddplacements non-euclidiens"). La correspondance entre le "modele cayleyien" et le demi-plan de Poincar6 s'obtient comme suit :/t toute forme F e P on fait correspondre, d'une part le point f qu'elle determine dans Po, d'autre part celle des racines de l'equation a z 2 + 2bz + c = 0 dont la partie imaginaire est > 0; la correspondance f ~ z e s t une correspondance biunivoque entre P0 et le demi-plan supdrieur de la variable z; les operations de Go dans ce demi-plan sont 6videmment celles du groupe homographique reel de determinant > 0 (groupe des ddplacements non-euclidiens dans le modele de Poincare). Pour determiner un point de P, on peut, au lieu de se donner une forme quadratique dans R", se donner une forme quadratique F, positive non degenerde, dans un espace vectoriel E de dimension n sur R, et une base (el . . . . . e,) de cet espace. Si (x, y ) est le produit scalaire dans E, associe fi F de la maniere habituelle, les donnees

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[1957c] Reduction des formes quadratiques, d'apres Minkowski et Siegel en question determinent la forme quadratique dans R" donnee par

Posons A = Hai~ll, air = (ei, ej~; soit X = ]lxijl] un 61ement du groupe lindaire L(R, n); par definition, la transformee de A par X est A' = tX. A - X , ce qui s'ecrit aussi n J A' = Ila'ijll avec aijt = (ei,/ ej), ei! =

xkiek.

2 k

1

Dire que X est ~ coefficients entiers de determinant _+ 1 6quivaut/t dire que les "lattices" (sous-groupes discrets de rang n de l'espace E) engendres par (el . . . . . e,) et par (e'l . . . . . e',) coincident. L'ensemble form6 par A et tous ses transformes par le groupe F est donc l'ensemble des matrices A' = (e~, e:i) lorsqu'on fair parcourir fi (e'1. . . . . e',) l'ensemble de t o u s l e s systemes de n gdndrateurs du lattice engendr6 par el . . . . . e,. Supposons qu'on ait choisi, dans P0, un systeme de reprdsentants Mo pour la relation d'dquivalence ddterminee par les operations du groupe F 0 sur Po; convenons momentandment de dire que la matrice A d'une forme quadratique dans R" est "rdduite" si le point qu'elle determine dans Po appartient /t M0, et aussi qu'une base (e~ . . . . , e,) de l'espace E est "reduite" pour une forme quadratique F dans E si la matrice A des (ei, e j) est reduite. I1 rdsulte de ce qui precede qu'dtant donnds une forme F et un lattice A darts E, il y a au moins un systeme de generateurs de A qui est une base reduite pour E, et que deux tels systemes determinent ndcessairement la m0me matrice A, donc ne different l'un de l'autre que par une transformation du groupe orthogonal de F. Choisir un systeme de representants M 0 revient donc "fi peu pres" fi enoncer une loi qui, /t tout couple form6 d'une forme quadratique F et d'un lattice A, permette d'associer, avec le moins d'ambigu'ite possible, un systeme de generateurs de A. On va, d'apres Minkowski, formuler une telle loi. Soit d'abord n = 2; on prendra pour el un vecteur va0 du lattice A dont la " l o n g u e u r " F(el) ~/2 soit la plus petite possible, puis pour ez un vecteur dont la longueur soit la plus petite possible parmi ceux qui ne sont pas de la forme le~ (t ~ R). I1 est clair qu'il n'y a qu'un hombre fini de vecteurs el satisfaisant/t la 1c condition, et qu'il y e n a au moins deux; des considerations geometriques 61ementaires (et evidentes) font voir qu'il y e n a exactement deux, saufdans les cas suivants : (a) lattice de Gauss (points fi coordonnees entieres dans le plan muni de la forme x2 ~_ y2); (b) lattice hexagonal (engendre par les vecteurs (1, 0) et (89 , / 3 / 2 ) d a r t s le plan muni de x 2 + y2). I1 s'ensuit que, dans tousles cas, les divers choix possibles de el se deduisent les uns des autres par une rotation laissant A invariant (rotation d'angle ~ dans le cas general, d'angle m~/2, resp. m7c/3, avec m entier, dans les cas (a), resp. (b)). Quant au second vecteur e2, on constate non moins aisement qu'il est determine d'une maniere unique au signe pres (une lois q u ' o n a choisi el) saul dans le cas d'un lattice engendre par deux vecteurs (1, 0) ct (2l, y), avec I):1 >x/3/2, dans le plan muni de x 2 + y2. O n levera l'indetermination de signe, autant

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Rdduction des formes quadratiques, d'apr6s Minkowski et Siegel que possible, en convenant de prendre e 2 tel que ( e l , e2) > 0; il se trouve que, lorsque cette r6gle laisse subsister une ambiguit6, le lattice A admet la droite 0el pour axe de symdtrie, et les choix possibles de e 2 sont symdtriques l'un de l'autre par rapport 5- cet axe. Une base (el, e2) du plan muni d'une forme quadratique F sera dite rOduite si elle poss6de les propridtds 6nonc6es ci-dessus, par rapport au lattice qu'elle engendre et 5- la forme F. Une forme quadratique F = a x 2 "4- 2 b x y + cy 2 dans R 2 sera dite rOduite si la base canonique de R 2, form6e des vecteurs (1, 0) et (0, 1), est r6duite par rapport 5. cette forme. Si on 6crit les indgalitds F((1, 0)) _< F((0, 1)) < F((_+ 1, 1)) on obtient imm6diatement les conditions a < c, 12b[ _< a, qui sont donc ndcessaires pour que F soit r6duite (avec bien entendu a > 0 puisque F doit 6tre positive non d6g6n6r6e). La condition subsidiaire imposde 5. ez s'6crit ici b _> 0. Doric: 0 0 pour 1 < i _< n - 1. I1 est clair d'ailleurs que la condition (i), jointe /t al 1 > 0 (resp. all -> O) suffit fi entrainer que F est positive non ddgdndrhe (resp. positive). Si on remarque qu'avec les notations ci-dessus on a e~ c E~ et ej-4-_ e~ ~ Ej chaque fois que i < j, on en conclut que (i) entraine les indgalitds: aii 0; si A a p p a r t i e n t 5- l'adhdrence de ce c6ne et cst d6g6n6r6e, on a u r a B'-IAB'-~A = 0. Ecrivant que A est dans l'adh6rence de S'(u) et est # 0, on trouve que A est de la forme

o) avec C non d6gdndrde; de B ' - I A B ' 1A = 0, on conclut alors que forme

(: 0)

B '-1

est de la

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[1957c] Groupes des formes quadratiques ind6finies et des formes bilin6aircs altern6es

d o n c a au moins un coefficient d i a g o n a l 0. P a r suite, B' 1, d o n c aussi B ' = tB' 9B ' - 1 9B', d o n c aussi B, " r e p r 6 s e n t e n t " 0 (ce qui veut dire que, si F est la forme de matrice B, F(x) = 0 a une solution rationnelle ~ 0). La r6ciproque s'ensuit de m6me. A u t r e m e n t dit, p o u r que G/F soit compact, il faut et il suffit que B ne repr6sente pas 0 (ce qui peut arriver p o u r n = 3 et p o u r n = 4; en revanche, un th6or6me classique de Meyer affirme que toute forme ind6finie fi n _> 5 variables, "fi coefficients entiers, " r e p r 6 s e n t e " 0).

3. Formes altern6es; groupe de Siegel. C'est le couplet suivant de la c h a n s o n ; il se chante sur le marne air. Soit F bilin6aire altern6e non d6g6n6r6e sur E; cela exige, bien entendu, que E soit de dimension paire 2n. Soit f l'application de E sur E' d6finie p a r F. Soit G le g r o u p e des a u t o m o r p h i s m e s de F ; soit K un sous-groupe c o m p a c t de G; il laissc invariante une q~ positive non d6g6n6r6e, "filaquelle a p p a r t i e n t une a p p l i c a t i o n q) de E sur E'. L'adjoint, p a r r a p p o r t fi ~, de l ' a u t o m o r p h i s m e i = q ~ - l f de E, est - ~ ; il s'ensuit que t est " s e m i - s i m p l e " (du p o i n t de vue matriciel, cela veut dire que z peut ~tre r6duit fi la forme diagonale, sinon sur R, en tout cas sur C), b, valeurs propres routes p u r e m e n t imaginaires. Si 1 a au moins deux valeurs propres distinctes et non imaginaires conjugu6es l'une de l'autre, E se d6compose en s o m m e directe de sous-espaces d o n t chacun est invariant p a r tout a u t o m o r p h i s m e de E qui c o m m u t e avec l; on en conclut,/t peu pr6s c o m m e au w qu'alors K ne peut 6tre maximal. P o u r que K soit maximal, il faut et il suttit que t n'ait que deux valeurs p r o p r e s distinctes, imaginaires conjugu6es l'une de l'autre; en multipliant qb p a r un facteur scalaire > 0, on peut supposer que ces valeurs propres sont + i , ce qui revient fi dire que t 2 = - 1. En ce cas, t d6termine une structure complexe sur E (on d6finira dans E la multiplication scalaire p a r les complexes au moyen de (~ + i~)x = :~x + [4lx); on 6crira E, p o u r E muni de cette structure; p o u r celle-ci, il est imm6diat que la forme bilin6aire fi valeurs complexes H = 9 + iF est hermitienne positive non d6g6n6r6e (N.B. lci, et dans ce qui suit, on note indiff6remment p a r ~ , p a r abus de langage, soit la forme q u a d r a t i q u e introduite ci-dessus, soit la forme bilin6aire associ6e.) C o m m e les 616ments de K c o m m u t e n t avec ~, ce sont des a u t o m o r p h i s m e s de E muni de sa structure complexe; il est clair alors que K est le g r o u p e unitaire d6termin6 par la forme hermitienne H. I1 contient donc toujours un centre non discret, form6 des multiples e"- 1 de l ' a u t o m o r p h i s m e identique (cela, au sens de la structure complexe). Dans le cas des formes q u a d r a t i q u e s ind6finies de signature (p, q), le centre du sous-groupe c o m p a c t m a x i m a l est non discret si p = 2 ou q = 2, et dans ce cas seulement. O n d6montre que l'existence d ' u n tel centre est n6cessaire et suffisante p o u r qu'il y air sur G/K une structure complexe invariante p a r G; on v a l e v6rifier dans le cas pr6sent. [N.B. La suftisance de la condition se justifie en g6n6ral c o m m e suit: K op6re darts G/K, avec un point fixe qui est le point de G/K qui c o r r e s p o n d fi K lui-m6me; il op6re donc sur l'espace des vecteurs tangents fi G/K en ce p o i n t ; dans cet espace, chacun des deux 616ments d ' o r d r e 4 du centre de K d6finit un a u t o m o r p h i s m e de cart6 - 1, et p e r m e t doric de d6finir une structure complexe, invariante p a r K. O n peut en faire a u t a n t en chaque point; on a ainsi une structure presque complexe;

[1957c]

373 Groupes des formcs quadratiques ind6finies et des formes bilineaires altern6es

reste ~, montrer qu'elle est int6grable. On peut le voir par exemple (d'apr6s Ehresmann) en remarquant qu'en g6n6ral, pour une structure presque complexe, le "d6faut d'int6grabilit6" s'exprime par un "tenseur mixte," celui qui donne les coefficients des co/~J~., dans l'expression des diff6rentielles d~)~ des formes ~i)~ de type (1, 0); en exprimant que ce tenseur est invariant par le centre de K, on trouve qu'il s'annule.] En d6finitive, on voit que G/K a 6t6 mis en correspondance biunivoque avec l'ensemble des structures complexes sur E pour lesquelles F est la partie imaginaire d'une forme hermitienne positive non d6gdndrde H = 4 ) + iF, et aussi avec l'ensemble V(F) des parties r6elles (I) de telles formes; comme au w V(F) est une sous-varidt6 du c6ne P des formes positives non ddgdn6rees sur E, et G/K V(F) est une bijection analytique rdelle de G/K sur V(F). Si on est dans R 2", et qu'on suppose F donn6e par une matrice fi coefficients entiers, on ddmontre, exactement c o m m e au w que le groupe des "unitds arithmdtiques" de F est minkowskien darts le groupe des automorphismes de F. Dans un expos6 ultdrieur, on ddfinira, d'une mani&e plus ou moins explicite, un ouvert U de G/K satisfaisant 5 la condition (M); ce sera fait, du moins, pour le " g r o u p e de Siegel" (ou " g r o u p e modulaire d'ordre n") proprement dit, qui est celui des unites si F est donnde dans R 2" par une matrice alternde de dOterminant 1. Si " o n " a du vice, " o n " ddfinira m~me, dans ce dernier cas, un " d o m a i n e fondamental" qui, avec ses transform6s, fournit un beau pavage de l'espace G/K. [N.B. I1 est connu que, par un choixtconvenable de 2n gdndrateurs pour le sous-groupe Z z" des vecteurs fi coordonndes enti6res dans R 2", toute formc alternde /t coefficients entiers peut s'6crire Y,i di(xiy,,+~ - x,+~yi), off les di sont des entiers, les "diviseurs 616mentaires", dont chacun est multiple du pr6c6dent. I1 n'y a donc pas besoin de la th6orie de la rdduction, dans ce cas, pour montrer qu'il n'y a, pour un d6terminant donn6, qu'un nombre fini de formes non 6quivalentes deux 5 deux. De plus, toutes ces formes sont 6quivalentes sur Q. Or il est facile de voir que les groupes d'unit6s arithm6tiques de deux formes 6quivalentes sur Q sont toujours commensurables; cela est vrai aussi, bien entendu, pour les formes quadratiques; mais ici on peut en conclure que les groupes de toutes les formes alterndes fi coefficients entiers sont commensurables au groupe de Siegel.] O n va s'occuper maintenant de structure complexe. Pour cela, on introduit le "complexifi6" de E, qu'on notera Ec (pour raison typographique, au lieu de la notation canonique Ec; c'est, c o m m e on sait, E | C muni de sa structure vectorielle sur C; on consid6re E c o m m e plong6 dedans de la mani6re 6vidente). Tout automorphisme t de E, de carr6 - 1 , se prolonge 5. Ec en un automorphisme analogue, qui d6termine une d6composition de Ec en somme directe des sousespaces Vi, V_~ formds des vecteurs propres relatifs aux valeurs propres i resp. - i de z; Vii, V_i sont sous-espaces de Ec sur C, donc sont espaces vectoriels sur C, de dimension n; on a d'ailleurs V_~ = ~, off, suivant l'usage, la barre denote l'imaginaire conjugu6 (d6fini dans Ee de la mani6re 6vidente). Si iE ddsigne l'ensemble des vecteurs "imaginaires purs" de E~. (image de E par x ~ ix), Ec = E @ iE est une somme directe; si ~ ("partie rdelle") est le projecteur de E~ sur E qu'elle d6termine, il est imm6diat que 9l induit sur Vii un isomorphisme de la structure complexe de Viisur la structure complexe de E qui est d6termin6e par l, c'est-',i-dire

374

[1957c] Groupes des formes quadratiques indefinies et des formes bilineaires alternees

sur celle de E,. D o n c ~est c o m p l e t e m e n t ddtermin6 p a r la d o n n e e de Vii.Rdciproquement, soit V~un sous-espace de Ec de d i m e n s i o n n sur C; p o u r que ~ induise sur V~une bijection de V, sur E, il faut et il suffit q u ' o n air l'une des relations 6quivalentes V~ r~ iE = {0}, Vi r~ E = {0}, Vi r~ V~ = {0}; lorsqu'il e n e s t ainsi, V~ p e r m e t doric de ddfinir sur E, p a r t r a n s p o r t de structure au moyen de N, une structure complexe, donc un a u t o m o r p h i s m e t de E de carr6 - 1 ; si alors on 6tend t it E,, V~sera l'espace des vecteurs p r o p r e s de l relatifs it la valeur p r o p r e i. Soit (5 la grassmannienne complexe des sous-espaces de Ec de dimension n sur C; d a n s (5, soit ,3 l'ensemble des sous-espaces d o n t l'intersection avec E se rdduit it {0} ; c'est un ouvert dans (5; d'apres ce qui precede, il s'identifie avec l'ensemble des structures complexes sur E. Si on s'est d o n n e c o m m e precedemment, sur E x E, une forme bilindaire alternde non degeneree F, celle-ci peut s'etendre it une forme Fc sur Ec • Ec; de m e m e p o u r l'extension q~c d'une forme symetrique qb. Si on a, sur E • E, qS(x, y) = F ( - tx, y) (ce qui equivaut it la relation z = q~- l f e c r i t e au debut de ce numero), la relation a n a l o g u e sera vraie p o u r les extensions de F, 0), t, it Ec; en particulier, sur V~, Fc et q5c induisent une forme alternee F ' et une forme symetrique qb' telles que l'on air @' = - iF', ce qui exige evidemment F ' = 0. A u t r e m e n t dit, V~est alors un espace isotrope maximal de F~ ( " i s o t r o p e " signifie j u s t e m e n t que F~ induit 0 sur Vii; on trouve alors, par exemple par le choix d ' u n e base convenable, que Vii, etant de d i m e n s i o n n, est isotrope maximal parce que F c e s t non ddgeneree). Ces espaces forment une sous-variet6 (analytique complexe) ~9 de la grassmannienne (g; on verifie sans difficulte que le g r o u p e des a u t o m o r p h i s m e s de E~ qui laissent F, invariante (le " c o m p l e x i f i e " du g r o u p e G) opere transitivement sur ~3. Reciproquement, soit V ~ r~ ~3; p u i s q u ' o n a E c = Vii| V i, on peut, quel que soit z~E~, 6crire z = u + v, u e Vii, v e V-i, et on a alors zz = iu - iv; si de meme on a z' = u' + v', avec u' ~ V/, v' e I/ i, on a u r a Fc(u, u') = 0 et Fc(v, v') = 0 parce que V/et p a r suite V~ = ~ sont isotropes p o u r F,.; cela p e r m e t de calculer F~(--lz, z') et de voir que cette forme bilineaire est symetrique. I1 faut exprimer de plus que @ est positive non degeneree sur E, ce qui equivaut b. F ( - tx, x) > 0 quel que soit x ~ 0 dans E. Or, l ' i s o m o r p h i s m e de E, sur V/, inverse de r i s o m o r p h i s m e de V/sur E, induit sur V~p a r 9~, s'ecrit x ~ z = x - i - t x (verification immediate); en tenant c o m p t e de ce que Vii, l / ~ = ~ sont isotropes p o u r Fr l'inegalite precddente s'ecrit encore iFc(z, ~) < 0 quel que soit z r 0 dans V~. I1 est clair que les V~satisfaisant it cette condition forment un ouvert f~ dans ~ ; ce qui precede implique que cet ouvert n'est pas vide (puisque tout point du riemannien symetrique G/K determine justement un point de f~). I1 est clair aussi que Vii ~ f~ implique Vii~ ~ ; sinon, en effet, il y aurait un z 4= 0 dans V~ r~ E, et on aurait z = ~, iF(z, z) < 0, ce qui est idiot. De ce qui precede resulte d o n c que G/K s'identifie it fL qui est une variete (analytique complexe) plongee dans (5. D'ofl la structure complexe de G/K. Le g r o u p e G des a u t o m o r p h i s m e s de F (i.e. des a u t o m o r p h i s m e s de E qui laissent F invariante) opere sur ~9 et sur f2 d'une maniere evidente, y laisse invariante la structure a n a l y t i q u e complexe, et sa maniere d ' o p e r e r sur fl est celle qui resulte, p a r t r a n s p o r t de structure, de son o p e r a t i o n sur G/K. Satisfaction gendrale. Profitant de l'absence de Dieudonne, on va traduire qa en matrices, p o u r faire le j o i n t avec Siegel et p o u r se mettre en etat de calculer q u a n d on ne peut pas faire

[1957c]

375 Groupes des formes quadratiques inddfinies et des formes bilin6aires altern6es

a u t r e m e n t (~a arrive encore quelquefois). O n prend une base (el . . . . R p o u r laquelle la matrice de F soit

,

e2n ) de E sur

soient E', E" les sous-espaces engendr6s sur R, respectivement, p a r (el . . . . . e,) et (e,+~ . . . . . e2,); ce sont des sous-espaces de E isotropes m a x i m a u x p o u r F. Si V~E f l , et que l, q), etc., aient le m6me sens que ci-dessus, @ sera positive non d6g~n6r6e sur E, d o n c sur U , ct on p o u r r a choisir dans E' n vecteurs o r t h o n o r m a u x p o u r q~; ils le seront alors aussi, dans E,, p o u r la forme hermitienne H = 9 + iF (puisque E' est isotrope p o u r F); ils formeront d o n c une base de E, sur C, ce qui entraine que E' et rE' sont suppldmentaires dans E. Alors Vii est transversal au complexifi6 E',, de E'; en effet, E'~ est l'ensemble des x' + iy', avec x' e E', y' c E'; si un tel p o i n t est dans Vii,on a y' = - zx' e E' r~ zE', donc x' = y' = 0. De marne Viiest t r a n s v e r s a l / t E',[. Choisissons darts Vi n vecteurs formant une base de Vi (sur C); 6crivons-les c o m m e " c o l o n n e s " (matrices ~ 2n lignes et 1 colonne) au moyen de (el . . . . , e z n ) pris c o m m e base de Ec sur C; cela d o n n e une matrice ~t 2n lignes et n colonnes (sur C), q u ' o n peut 6crire

V ' o/1 U, V sont deux matrices/t n lignes et n colonnes;

si on change les vecteurs de base choisis dans V~, cela revient/L multiplier U, V ~ droite p a r une m~me matrice carr6e inversible. Puisque V~est transversal/L E'~' la matrice U est de rang n, c'est-fi-dire inversible. Ecrivons que V~ est isotrope p o u r F ; cela s'exprime p a r la formule

:0, ou a u t r e m e n t dit ' U - V = tV. U. D e m~me, 6crivons que iFc(z, 2) < 0 p o u r tout z r 0 dans Vi; cela signifie (1/i)(~U 9V - ' V - U) > 0 (le premier m e m b r e est visiblement une matrice hermitienne). P o s o n s Z = V U - 1, matrice qui est ind6pendante de la base choisie dans Vii.La premi6re des relations ci-dessus s'6crit ' Z = Z ; Z est sym6trique. La seconde s'6crit (en m u l t i p l i a n t / t droite p a r U - ~, fi gauche p a r ' U - ~, ce qui ne modifie pas le fait que le premier m e m b r e est hermitien positif non d~g6n6r6) ( 1 / i ) ( Z - ~Z) > 0; a u t r e m e n t dit, si on 6crit Z = X + i Y avec X, Y sym6triques r6els, Y doit 6tre positive non ddg6ndr6e. O n a ainsi obtenu une bijection de fL donc en d6finitive de G/K, sur l'espace de Siegel ~ , form~ des matrices sym6triques Z = X + i Y sur C telles que Y > 0; ~ peut ~tre consid6r6 c o m m e un ouvert de C N, avec N = n(n + 1)/2, muni de la structure complexe induite p a r celle de C ~. L ' o p 6 r a t i o n de G sur ~ s'~crit imm6diatement. En effet, avec les notations ci-dessus, un 616ment de G s'6crira sous forme d ' u n e matrice carr6e

376

[1957c] Groupes des formes quadratiques inddfinies et des formes bilineaires altern6es

off A, B, C, D sont des matrices "5. n lignes et n colonnes sur R; cette matrice doit satisfaire fi la condition

qui exprime qu'elle laisse F invariante. Cette matrice op6re sur V~, d+finie par les matrices U, V, au moyen de la formule

ou autrement dit:

(U, V)--+ (AU + BV, CU + DV); elle opere donc sur Z par la formule

Z ~ (C + DZ). (A + B Z ) - I Enfin, on peut donner de l'op6ration sur ~ du groupe modulaire (sous-groupe de G form6 des matrices "a coeff• entiers) une interpr6tation int6ressante, et meme importante. En effet, F 6tant donnde dans E, les points de ~ sont, d'apr6s ce q u ' o n a vu, en correspondance biunivoque avec les structures complexes E, q u ' o n peut mettre sur E, pour lesquelles F est partie imaginaire d'une forme hermitienne H >> 0. Supposons donn6 en meme temps dans E un lattice A tel que F soit/t valeurs enti6res sur A x A ; E/A est alors un tore de dimension (r6elle) 2n, sur lequel F d6termine une classe de cohomologie enti6re de dimension (r6elle) 2. Toute structure complexe sur E d6termine sur E/A une structure de tore complexe (de dimension complexe n); pour que celui-ci soit une vari6t6 ab61ienne, il faut et il suffit qu'il existe dans E une forme hermitienne >>0 dont la partie imaginaire soit b. valeurs enti6res sur A x A; et, lorsqu'il e n e s t ainsi, il y a sur E/A un "diviseur positif" appartenant 'a la classe de cohomologie d6termin6e par cette partie imaginaire; muni de cette classe, E/A s'appelle alors une vari~t~ ab~lienne polaris~e. O n voit donc qu'/t tout point de ~ correspond sur E/A une structure de varidt6 ab61ienne polaris6e par F; pour qu'/t deux points corresponde la m6me structure, il faut et il suffit qu'ils se d6duisent l'un de l'autre par un automorphisme de E qui laisse invariants la forme F et le lattice A, donc un 616ment du groupe discret F des automorphismes de F qui sont fi coefficients entiers lorsqu'on prend pour base un syst6me de gdndrateurs de A. Autrement dit, les points de ~ / F (quotient de par la relation d'6quivalence d6finie dans | par le groupe discret F) sont en correspondance biunivoque avec les structures de vari6t6 ab61ienne polaris6e par F q u ' o n peut ddfinir sur E/A. Lorsque F est de d6terminant 1 sur A (c'est-fi-dire a tous ses diviseurs 616mentaires sur A 6gaux/~ 1), le groupe F qu'on obtient est le groupe modulaire de Siegel proprement dit; les vari6tds ab61iennes correspondantes sont dites "vari6t6s abdliennes polaris6es de la famille principale" (toute jacobienne est une telle variet6).

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377 Groupes des formes quadratiques inddfinies et des formes bilindaires altern6es

Bibliographie Minkowski, Hermann. Geometrie der Zahlen. Leipzig und Berlin, B. G. Teubner, 1910; New York, Chels,ea, 1953. Minkowski, Hermann. Diskontinuit~itsbereich ffir arithmetische Aquivalenz, J. ffir reine und angew. Math., t. 129, 1905, p. 220-274; Gesammelte Abhandlungen, Band 2, Berlin, B. G. Teubner, 1911, p. 53 100. Siegel, Carl Ludwig. Einffihrung in die Theorie der Modulfunktionen n-ten Grades, Math. Annalen, t. 116, 1939, p. 617-657. Siegel, Carl Ludwig. Einheiten quadratischer Formen, Abh. math. Sem. Hamburg Univ., t. 13, 1940, p. 209 239. Siegel, Carl Ludwig. Discontinuous groups, Annals of Math., t. 44, 1943, p. 674 689. Siegel, Carl Ludwig. Symplectic geometry, Amer. J. Math., t. 65, 1943, p. 1 86. Weyl, Hermann. Theory of reduction for arithmetical equivalence, 1., Trans. Amer. math. Soc., t. 48, 1940, p. 126 164; II., Trans. Amer. math. Soc., t. 51, 1942, p. 203-231. Weyl, Hermann. Fundamental domains for lattice groups in division algebras, Comment. Math Helvet., t. 17, 1944/45, p. 283-306.

[ 1958a] Introduction h l'6tude des vari6t6s k ihl6riennes (Pr6face) L a th6orie des vari6t6s k~ihl6riennes a pris un g r a n d essor depuis un q u a r t de sibele. Ces vari6t6s f u r e n t d6finies p o u r la p r e m i b r e fois, semble-t-il, dans une n o t e de K~ihler de 1933 (Hamb.Abh. 9, p. 173). Mais leur i m p o r t a n c e n ' a p p a r u t q u ' h la suite des p r e m i e r s t r a v a u x de H o d g e et s u r t o u t de l'expos6 d ' e n s e m b l e q u ' i l en d o n n a au c h a p i t r e V de son livre The theory and applications o[ harmonic integrals ( C a m b r i d g e 1941), off, i n d 6 p e n d a m m e n t de Kfihler, il e n t r e p r e n d l ' 6 t u d e s y s t d m a t i q u e de la m 6 t r i q u e (( kfihl6rienne )) q u ' o n p e u t d6finir sur t o u t e varidt6 alg~brique sans p o i n t m u l t i p l e plong6e dans un espace projeetif, et en tire des cons6quences tr+s i m p o r t a n t e s p o u r la g6omdtrie alg6brique. C'est p r i n c i p a l e m e n t dans la direction ainsi inaugurde p a r lui que les recherches se s o n t poursuivies depuis lots. On ne t r o u v e r a pas iei une m o n o g r a p h i e de ce sujet, mais, c o m m e l ' i n d i q u e le t i t r e , une simple i n t r o d u c t i o n , bas6e sur des cours professds h Chicago et h G S t t i n g e n d a n s les derni~res ann6es. Ce v o l u m e a u r a a t t e i n t son b u t s'il facilite au l e c t e u r l'~tude des t r a v a u x rdcents sur la question, et particuli~r e m e n t de ceux de K o d a i r a et ses dl~ves. J e n ' a i pu ndanmoins rdsister h la t e n t a t i o n d ' i n s d r e r un c h a p i t r e t r a i t a n t de la thdorie des fonctions t h ~ t a et des vari~t~s ab~liennes sur le corps des complexes. Cette th~orie p e u t ~tre considdrde c o m m e celle d ' u n t y p e p a r t i c u l i e r de s t r u c t u r e s kfihleriennes (les s t r u c t u r e s i n v a r i a n t e s sur un tQre), et c ' e s t m~me ce p o i n t de vue qui c o n d u i t h la d d m o n s t r a t i o n la plus n a t u r e l l e d ' u n des p r i n c i p a u x th~or+mes d'exist e n c e de la th~orie (le th~or~me d i t (( d ' A p p e l l - H u m b e r t )9. I1 n ' a p a s dt~ donnd de b i b l i o g r a p h i e ; on en t r o u v e de fort c o m p l e t e s dans plusieurs o u v r a g e s r~cents. J e me suis efforcd de rdduire au m i n i m u m la s o m m e de connaissances exigde du lecteur : quelques r~sultats dl~mentaires Reprinted from Introduction (~ l'~tude des variet~s kiihl~riennes, Hermann, Paris, by permission of Hermann, 6diteurs des sciences et des arts.

379

380

[1958a] l0

INTRODUCTION A L ' E T U D E DES VARIE,TES K,~HLERIENNES

d'analysc et de thdorie des fonctions ; quelques notions d'alg~bre et de topologie gdndrale (pour lesquelles il sera en gdndral renvoyd aux Eldmenls de N. Bourbaki) ; les d~finitions essentielles de la th~orie des vari~tds diff~rentiables (qu'on trouvera dans le volume de G. de R h a m , Varidlds di[]~rentiables, Paris, HerInann 1955, paru dans cette m~me collection) ; la ddfinition des op~rateurs . , 8 , A de la th~orie des formes harmoniques (expos~e dans le m~me volume) ; et, en quelques points, quelques notions sur la cohomologie enti~re. Encore ai-je, dans la mesure du possible, rappel~ les d~finitions et r~sultats dont il aura h ~tre fait usage. L'indication (( de Rham, w ~, renverra toujours au volume q u ' o n vient de citer ; les renvois Bourbaki seront faits sous la forme canonique. Des thdor~mes d'existence de la thdorie des formes harmoniques, qui jouent bien entendu un r61e essentiel dans le present volume, le lecteur n'a fi connaitre que l'existence des opdrateurs H e t G de de R h a m avec les propridtds formelles dnoncdes au n ~ 1 du Chapitre IV. Pour le cas particulier des tores, une ddmonstration directe de ces rdsultats, inddpendante de celle qui est donnde dans le volume de de R h a m pour le cas gdn~ral, est exposde au n ~ 2 du Chapitre IV, ce qui permettrait, si on le ddsirait, d'aborder la thdorie des fonctions th~ta sans s ' a p p u y e r sur la th~orie g~n~rale des formes harinoniques.

Paris, le 31 mai 1957.

[1958b] On the moduli of Riemann surfaces 1 TO E m i l Artin on his sixtieth birthday The p u r p o s e of the following pages is partly to clarify my own ideas on an interesting topic, at a stage when they are still unripe for publication, but chiefly to be present by p r o x y at Artin's b i r t h d a y celebration. In speaking of these ideas as " m y own", my intention is not to claim originality for them. They are little m o r e than a c o m b i n a t i o n of those of Teichmfiller with the ideas on the variation of complex structures, recently introduced by K o d a i r a , Spencer and others into the t h e o r y of moduli. The first concept to be elucidated is that of reinforcement of structure. Let S o be an oriented c o m p a c t surface of genus g > 1, given once for all. Let S be a Riem a n n surface of genus g, i.e. a surface of genus g, p r o v i d e d with a c o m p l e x - a n a l y t i c structure (or, what a m o u n t s to the same, an oriented surface of genus g with a conformal structure, or again an oriented surface of genus g with a class of conformally equivalent ds2). By a "class of m a p p i n g s " of S o into S, we shall u n d e r s t a n d a class of c o n t i n u o u s mappings, equivalent under h o m o t o p y ; a class will be called " a d m i s sible" if it contains at least one orientation-preserving h o m e o m o r p h i s m of S o onto S. A R i e m a n n surface S, p r o v i d e d with the a d d i t i o n a l structure defined on it by assigning one admissible class of m a p p i n g s of S o into S, will be called a Teichmtiller surface. P e r h a p s the most r e m a r k a b l e of Teichmfiller's results is the following: when p r o v i d e d with a rather obvious " n a t u r a l " topology, the set O of all classes of m u t u a l l y isomorphic Teichmtiller surfaces is h o m e o m o r p h i c to an open cell of real dimension 6g - 6. This global result will neither be used nor discussed in the following pages, the chief p u r p o s e of which is to consider the local p r o p e r t i e s of O and to define on it a " n a t u r a l " c o m p l e x - a n a l y t i c structure, of complex dimension 3g - 3, and a " n a t u r a l " H e r m i t i a n metric. The above "reinforcement of s t r u c t u r e " can be modified in various ways. Instead of admissible classes of mutually h o m o t o p i c mappings, one might wish to introduce classes of m u t u a l l y isotopic orientation-preserving h o m e o m o r p h i s m s of S o onto S; unless I am mistaken, k n o w n results in surface-theory imply that this w o u l d not actually differ from what we have done. O n the other hand, one can also consider a weaker type of reinforcement, in which two m a p p i n g s of S o into S are considered equivalent if they induce the same h o m o m o r p h i s m of the onedimensional h o m o l o g y group of S ~ H1(S~ into H i ( S ) ; as a t e m p o r a r y terminological prop, let us call a "Torelli surface" the surface S with the a d d i t i o n a l structure d e t e r m i n e d by an admissible class of m a p p i n g s for this wider concept of equivalence.

Part of this was done, I am somewhat ashamed to say, while on contract with the Air Force. The opinions of the Air Force do not necessarily coincide with mine.

381

382

[1958b1 On the moduli of Riemann surfaces

Call ~z~ the fundamental group of S ~ with an origin a ~ chosen once for all ; it is the group generated by 2g generators A ~ with the well-known relation, which we shall write as R(A ~. . . . . A~ = 1. Let f , f ' be two mappings ofS ~ into S; they induce h o m o m o r p h i s m s h, h' of ~o into ~(S,f(a~ and ~(S,f ,( a 0) ) , respectively. By considering the universal coverings of S o and S, one sees easily thatfandf' are homotopic if and only if h' can be derived from h by moving the origin of the fundamental group of S along a suitable path f r o m f ( a ~ t o f ' ( a ~ If, by an obvious "abuse of language", we agree to speak of " t h e " fundamental group ~(S) of S (which is then defined intrinsically only up to an inner automorphism), we may say that f a n d f ' are h o m o t o p i c if and only if they induce h o m o m o r p h i s m s of ~0 into ~(S) which are equivalent under inner automorphisms of ~z(S). In particular, an admissible class of mappings of S o into S will define a class of isomorphisms of ~o onto it(S), equivalent under inner automorphisms of ~(S); such a class will be called admissible; and the images of the A ~ under an isomorphism in such a class will be called an admissible set of generators of ~(S). Thus a Teichmfiller surface is defined by selecting on S an admissible set of generators of ~(S), provided two such choices are considered equivalent whenever they can be derived from each other by an inner automorphism. We shall denote by _S any Teichmtiller surface with the underlying Riemann surface S. Similarly, a Torelli surface _Swill be defined by selecting on S an admissible set of generators for the h o m o l o g y group Hi(S); here we have no equivalence relation between such sets. By the definition of an admissible set, the intersection-matrix for an admissible set of generators is

This implies in the well-known manner that we can define on the Torelli surface S_ a normalized set of differentials of the first kind, for which the period-matrix (for the given set of generators) has the form II1,Z(S)II, where Z(S) is a g x g matrix. More precisely, if ~ is the Siegel space of symmetric g x g matrices with positivedefinite imaginary part, Z(_S) is a point of ~ ; if S' is another Torelli surface with the same underlying Riemann surface S, Z(S_') will be a transform of Z(_S) by an element of Siegel's modular group. Thus we have a mapping _S ~ Z(_S) of the set Z of all classes of mutually isomorphic Torelli surfaces into the Siegel space ~. The set Z has an involutory a u t o m o r p h i s m S ~ h(S), viz. the one which leaves the underlying Riemann surface S of S unchanged and induces on Hi(S) the automorphism x --, - x (it is easily seen, e.g. by considering a hyperelliptic surface of genus g, that this automorphism changes one admissible set of generators of H1(S) into another). It is clear that Z(h(S_)) = Z(S), i.e. that _S and h(S) have the same image in the Siegel space. Moreover, Torelli's classical theorem asserts that two Torelli surfaces S, S' have the same image Z(S) = Z(S') in the Siegel space if and only if _S' = _S or S' = h(S), and that S = h(S) if and only if S is hyperelliptic. Call Z1 the image of Z by S_ --, Z(_S); we see that the inverse image of a point of Z 1 by that mapping consists of one or two points according as the corresponding

[1958b]

383 On the re|

of Riemann surl:aces

Riemann surface is hyperelliptic or not. Combining this mapping with the obvious " n a t u r a l " mapping of | onto E, we get a mapping of | onto El, which we write as =S-~ Z(S). Finally, if ~JJl is the set of all classes of mutually isomorphic Riemann surfaces of genus g, we have a natural mapping of 21 onto ~ . It will be seen that, when O is provided with its natural complex structure, S ~ Z(=S) is a holomorphic mapping of O into the Siegel space. Actually 21 is an analytic subvariety of ~, whose points are all simple except those corresponding to hyperelliptic Riemann surfaces (this, with an important additional statement concerning sets of local coordinates in the neighborhood of a simple point of Y~I, is Rauch's theorem). As to ~JJ~,there is virtually no doubt that it can be provided with a structure of algebraic variety (non-complete, of course, and with multiple points), the "variety of moduli", so that the natural mapping of | onto ~ is holomorphic. Let _S be a Teichmfiller surface; this consists of a Riemann surface S with a preferred choice of generators A1 . . . . . A2o for ~(S), these being defined only up to an inner automorphism. We can represent conformally the universal covering of S onto the upper half-plane H = { z l J ( z ) > 0}; it is well-defined up to a hyperbolic displacement. This determines a representation of ~r(S) as a discrete group F of hyperbolic motions, generated by 2g elements z--* aiz = (c~iz + fli)/(?iz + (~i), with :tg6i - fizT~ = 1 and R(al . . . . , a2o) = 1. An inner automorphism of ~(S), or a hyperbolic displacement in FI, will merely transform F by such a displacement; conversely, if two Teichmfiller surfaces determine two sets (al), (a'i) which differ merely by an inner automorphism of the hyperbolic group, they are isomorphic. As remarked by Siegel (Math. Ann. 133), one can select that automorphism in a unique manner so as to get f12o = 72g = 0, ~2g > l, fig = ~ O ; as the matrices for the a~ are determined only up to a factor + 1, one can further normalize them by taking fi~ > 0 for 1 < i _< 2g - 1. In that normalization, all the a~ are uniquely determined by the 69 - 6 numbers (ct~, fl~, 6i) for 1 _< i _< g - 1 and g < i _< 2g - 1 ; more precisely, the relations ~6~ - ~ i ~ i = 1, e(al, . . . , 0"20 ) = 1 together with those which express the normalization, determine the 7~ and the coefficients of o"g, cr2o by means of a set of algebraic equations with non-vanishing functional determinant in the neighborhood of any point of the coordinate space ~6r corresponding to a group such as F. Let O~ be the subset of N6~-6 consisting of the points which can be obtained in this manner; the above remarks show how to define a bijection of O onto 01. It will be seen that this bijection is a real-analytic isomorphism when | is provided with its natural complex-analytic structure; this implies that O1 is an open subset of ~ 6 g 6. Moreover, we shall give explicit formulas to determine the complex structure on O1 which can be derived from that of O by that bijection. In order to justify the statements that have been made so far, we shall make use of the Kodaira-Spencer technique of variation of complex structures. This can be introduced in an elementary manner in the case of complex dimension 1, which alone concerns us here; this, in fact, had already been done by Teichmfiller; but he had so mixed it up with his ideas concerning quasi-conformal mappings that much of its intrinsic simplicity got lost. Perhaps the worst feature of his treatment, in the eyes of the differential geometer, is that his extremal mappings are destructive

384

[1958b] On the moduli of Riemann surfaces of the differentiable structure; this corresponds to the fact that his metric on | is almost certainly not to be defined by a d s 2, even though it is presumably a Finsler metric. Let __So be a Teichmfiller surface. In order to avoid the use of coordinate neighborhoods on the underlying Riemann surface So, we represent So as l-I/F, where FI is the upper half-plane and F is a discrete subgroup of the hyperbolic group; _SO will then be defined by a preferred set of generators (~i) for F, which we may assume to be in normalized form. We consider a small "variation of structure", depending differentiably upon some real or complex parameters u; this can be defined as the conformal structure on H / F determined by a complex-valued differential form ~ = dz + # d~, where # is a complex-valued function of z and of the parameters u, such that any element a of the group F merely multiplies ~ by a scalar factor; this amounts to saying that # dS/dz is formally invariant under F, a property which we also express by saying that g is of type ( - 1, 1) for the usual complex structure of FI/F. When # is so given, the complex structure of the varied surface Su underlying the varied Teichmfiller surface S,, is the one for which ~ is a differential form of type (1, 0) at every point; for this to have a meaning, we must have ]#[ < 1 for all z e F I . We assume that # = 0 for u = 0, i.e. that, for u = 0, S u is no other than _So.The universal covering of S, is FI with the conformal structure determined by ~; for each u, this can be conformally mapped upon H with its natural conformal structure; call F , the differentiable h o m e o m o r p h i s m of H onto itself which realizes this conformal mapping, i.e. which is such that (for fixed u) dF,, differs from ~ only by a scalar factor, i.e. is of the form dF,, =j'~, w h e r e f i s a complex-valued function, everywhere r 0 in II. The conformal mapping F, can be normalized in various ways, e.g. by prescribing that a given point of I1 and a given direction through that point remain fixed under F,. A respectable firm of ellipticians, w h o m I consulted concerning the properties of F,, has assured me that it depends differentiably on the parameters u, and that it is real-analytic in the u's if this is assumed of #. N o w assume that u is a single real parameter; then (c~/~u),_ o is an "infinitesimal variation" in the sense of Kodaira-Spencer; this operator will also be denoted by a dot. It is determined by

which is again a complex-valued function of type ( - 1 , 1). The infinitesimal variation is trivial if and only if the varied structure can be obtained from the initial structure (that of So) by an infinitesimal deformation of the surface. The latter will be defined by a vector-field, i.e. by a complex-valued fuction ~ of type ( - 1, 0); and one finds at once that such a vector-field ~ determines the infinitesimal variation of structure given by v = ~/0~. Therefore we introduce the Kodaira-Spencer space for So, which we define as the quotient of the space of all functions v of type ( - 1, 1) by the space of functions v = 0~/~5 with ~ of type ( - i, 0), both being considered as vector-spaces over C. We shall denote by D = D(v) the element of that space determined by a given infinitesimal variation v = / i . Let ~o be a quadratic differential on So; this can be written as (~) = q. dz 2,

[1958b]

385

On the moduli of Riemann surfaces where q is holomorphic of type (2, 0). Notations being as above, consider the integral SS qv d5 dz, taken over So; this has a meaning, since the integrand is invariant under F. Stokes's theorem shows at once that this is 0 if v = ~?~/c~5with of type ( - 1, 0); therefore it depends only upon co and D = D(v), and m a y be written as ((~), D). For any v, one can solve the equation v = &p/(?~; the solution will be uniquely determined modulo a holomorphic function of z in Fl. If we again assume that v is of type ( - 1, 1), any solution ~0 of that equation will be such that, for every ~ e F, the function ~/J~ = (p -

dz

- ~o'~

(1)

d(~z)

(where (S stands for the function defined by (p~(z) = q)(az)) is holomorphic in H. These functions satisfy the relations dz

~'o, = 0 ; d ( ~

+ ~

(2)

for all a, z in F. Conversely, given a "cocycle" (~9,), i.e. a system of holomorphic functions ~ in 11, satisfying (2) for all a, z in F, let F operate on H x C by the formula 0 % t) =

o-z, d z

(t -

~(~))

;

then the quotient (11 • C)/F is a complex-analytic fibre-bundle with base So, the fibre being the plane C (with the group t ~ at + b). By a well-known theorem on fibre-bundles (which, in the present case, can also easily be verified directly), this has a differentiable cross-section t = ~0(z); this means that (1) has a solution ~0, which is a complex-valued, real-differentiable function. Another general theorem on fibre-bundles (Serre, GAGA), which could also, in this case, be verified directly without difficulty, tells us that (H x C)/F is an algebraic bundle over So; therefore it has a meromorphic cross-section, so that (1) has a meromorphic solution ~o'. We conclude that the Kodaira-Spencer space can be defined in any one of the following manners : (a) as above, as the quotient of the space of functions v of type ( - 1 , 1) by the space of functions v = c?~/c?z with ~ of type ( - 1, 0); (b) as the space of functions qo such that v = ~?q)/c~5is of type ( - 1, 1), divided by the sum of the space of all holomorphic functions and of the space of functions of type ( - 1, 0); (c) as the space of holomorphic cocycles ( ~ ) , i.e. of systems of holomorphic functions satisfying (2), divided by the space of trivial cocycles, i.e. of those for which (1) has a holomorphic solution (p; (d) as the space of meromorphic functions q/ such that all the functions ~o' - (dz/daz)tp '~ are holomorphic, divided by the sum of the space of holomorphic functions and of the space of meromorphic functions satisfying ~0' = (dz/daz)(p '~ for all a.

386

[1958b] On the moduli of Riemann surfaces

In (d), since we are taking the meromorphic functions modulo the holomorphic functions, it is only the principal parts (at the poles) that must be considered; the condition on q~' implies that these are determined once they are given in a fundamental domain. Now, notations being as above, we can write

(co, D)= f f q v d S d z =

~qqodz,

(3)

where ~ means the integral over S 0, or, what amounts to the same, over a fundamental domain for F in I1, and ~ means the integral over the positively oriented contour of such a fundamental domain. The latter may be taken as a polygon with 4(J sides, corresponding (in that order) to al, ao+ 1, ~ ~, a~+~l, a2 . . . . . ~2g~. It is clear that .~ q2 dz is 0 whenever 2 is a continuous function, defined only along the contour of integration, such that, formally, 2/dz takes the same values along corresponding sides of the contour; for then the same is true of q)~ dz, so that the integrals along corresponding sides cancel each other. In particular, we can take 2 = q) - q)', where ~0' is defined as above ; this shows that (co, D) can be written as qq/dz, i.e. that it is the sum of the residues of the meromorphic differential (~o'/dz)co over S 0. It is a well-known consequence of Riemann-Roch that this sum, for a given ~o', will be 0 for all quadratic differentials co if and only if there is on So a " m e r o m o r p h i c differential of degree - 1" q are all 0. More generally, we shall show that this is the case whenever # is a real-differentiable function of z and of complex parameters wi in any n u m b e r and is holomorphic in all the w~. It is obviously enough to consider the case of one complex parameter w. Assume therefore that # = #(z, w) depends real-differentiably upon z and w and that it is holomorphic in w. For a given value w* of w, consider the vector D in the Kodaira-Spencer space for Sw,, defined by (~?/c?~)w-w*. Notations being the

388

[]958b]

On the moduli of Riemann surfaces same as before, the conformal mapping of the universal covering of Sw. onto gl is given by z* = Fw.(z), where Fw. is a differentiable h o m e o m o r p h i s m of II onto itself, such that dz ~ = dFw. is of the form f(z) 9(dz + tl(z, w*)d~), with a scalar factor f ( z ) . In terms of the variable z*, the conformal structure of Sw for any w may be defined by a differential form ~* = dz* + #*(z*, w)d~*

which has to differ from dz + I~(z, w)d~ only by a scalar factor. Then, for each quadratic differential co* = q*(dz*) 2 on Sw., (co*, D) is given by:

aJ

\~/w=w.

But a trivial calculation shows that /t* is again holomorphic in w; therefore ~?/~*/c?~ is identically 0, and D is 0. The second method depends upon the consideration of the integrals of the first kind. We again call in the elliptical engineer, who tells us the following: since the conformal structure of S. can obviously be derived from a d s 2 which depends differentiably upon the parameters u (viz. d s 2 = y 2 Idz + ~ d z I 2 with y = ..~(z), which is invariant under F), the space of real-valued harmonic differentials on S. has a basis consisting of differentials which depend differentiably upon the parameters. Take again the case of a single real parameter u, and denote (O/c?u). = 0 by a dot, as before. The differentials of the first kind on S. are those linear combinations of the harmonic differentials with constant complex coefficients which differ from dz + # d2 only by a scalar factor; clearly the space of such differentials is generated by those which depend differentiably upon u. Let q. = h.(dz + # dz) be one of these; it must satisfy dr/. = 0 and is invariant under F; for u = 0, it reduces to a differential of the first kind r/o = ho dz for So. As before, put v = / i and call (0 a solution of v = (?~0/t?2; differentiating r/. with respect to u for u = 0, we get /l = h " dz + h o v d~ = r . dz + d(ho ~o),

where we have put r = h - 0(h0 ~o)/c~z. This, too, must be invariant under F and must satisfy dO = O. Call pi(u) the periods of r/. along the cycles crl for 1 _< i _< 29; put p~ = p~(0); /~i is then the period of 0 along a~, Let r/~ be any differential of the first kind for So, with the periods p'~. We may write r/; = dJ; w h e r e f i s a holomorphic function in gl. Then r/or/; is a quadratic differential for So; and, if we put D = D(v) as before, we have:

r

: ff ho.

ff o

- f

Combining integrals along corresponding sides of the fundamental polygon, we get : 9

(r/or/'o, D) = ~ (P'o+iPi -- P'il}o+i). i=1

In particular, call r/J, for 1 < j < g, the normalized differentials of the first kind for =S.; by definition, they are such that I p / ( u ) l , for 1 _< j < g, 1 < i < g, is the unit-

[1958b]

389

On the moduli of Riemann surfaces matrix; and then, by definition, we have:

z(__x.) = IIp~+i(u)ll

(1 _< j _< g;1 _< i < g).

This gives II~ill = 0, II~+gll = Z. Substituting r/~, r/~ for r/0, rl• in the above formula, we get:

2 = -II(rt~rl~, D)ll. This proves that S --. Z ( S ) is a holomorphic mapping of O into 6. More precisely, it shows, not only that the coefficients po~+kof Z(=S) are holomorphic on O, but also that any 3g - 3 of them will be local coordinates in the neighborhood of a given point of _So if and only if the corresponding quadratic differentials ~/~tl~ are linearly independent. This is Rauch's theorem. It implies that Z(_S0) is a simple point of the image Y~I of O by Z if and only if the products ~/~r/~ generate the space of quadratic differentials, i.e. if and only if S o is not hyperelliptic. Onc can prove quite similarly that the subset of 521 consisting of the points which correspond to hyperelliptic Riemann surfaces is a non-singular analytic subvariety of ~ of complex dimension 29 - 1. It is clear now that the almost complex structure defined on O is a complex structure in the neighborhood of all the points which do not correspond to a hyperelliptic Riemann surface; by using well-known general theorems on analytic varieties, one can then extend this result even to the points which correspond to hyperelliptic surfaces. Finally, in order to define a " n a t u r a l " Hermitian metric on O, it is only necessary to define a Hermitian structure on the Kodaira-Spencer space for each So, or, what amounts to the same, on the dual of that space, i.e. on the space of quadratic differentials for So. This is done by putting, for any two quadratic differentials co = q d z 2, o)' = q' dz 2 for So: ((~), u/) = [ [ - q~fy2 dz d5 o ,/ with y = J ( z ) . In fact, the integrand can be formally written as e)~)/dS, where dS = y 2 dz dz is the hyperbolic area-element in H; it is therefore invariant under F, and the integral has an intrinsic meaning. This raises the most interesting problems of the whole theory: is this a Kfihler metric? has it an everywhere negative curvature? is the space O, provided with its complex structure and with this metric, a homogeneous space? It would seem premature even to hazard any guess about the answers to these questions.

[1958c] Final report on contract AF 18(603)-57 Within the framework of the project originally submitted to AFOSR, I eventually decided to concentrate on two lines of investigation: (I) The classical problem of the moduli of algebraic curves over complex numbers; (II) A study of the K;,ihler varieties topologically identical with the nonsingular quartics in projective 3-space (henceforward called K3 surfaces). In both directions, my results are still very fragmentary and incomplete; and I have had to postpone the arithmetical considerations which provided the original motivation for the whole project, in order to deal first with the function-theoretic aspects of the above questions. In both problems, the ideas of Kodaira and Spencer on the variation of complex structures have proved fundamental. I have much benefited from repeated consultation with them during my stays in Princeton, in January and February and again in June. It also turned out that Professor L. Bers had been engaged in a parallel investigation of problem (I); consultation with him on this topic has been very fruitful. On the other hand, various aspects of problem (II) have recently engaged the attention of Professor L. Nirenberg, Professor A. Andreotti, and Dr. Atiyah; I have learned a great deal from communications, written and oral, from all of them. In order to give, in what follows, a coherent account of these topics, it will be necessary to include much of the work of my colleagues, and it would be unpractical to try to unravel in detail what may belong to me and what belongs to each one of them. It should be understood that they deserve a large share of the credit for the work described in this report.

I. Moduli o f algebraic curves We consider curves of a given genus 9 -> 1. One basic concept is that of a Teichmtiller structure. If S, S' are two oriented surfaces of genus 9, we say that a class of mappings of S into S' in the sense of homotopy (or, briefly, a class C(S, S')) is admissible if it contains at least one orientation-preserving homeomorphism of S onto S'. Let So be an oriented surface of genus g, given once for all. By a Teichmfiller surface, we understand a Riemann surface of genus 9 (i.e. a surface of genus g, provided with a complex-analytic structure and oriented accordingly), together with an admissible class C(So, S). Isomorphism being defined for these in the obvious manner, we introduce, with Teichmfiller, a space T (the "Teichmfiller space"), whose points correspond in one-to-one manner to all the classes of mutually isomorphic Teichmfiller surfaces of genus 9. Teichmtiller's chief contribution was

390

[1958c]

391 Final report on contract AF 18(603)-57

to define on T a certain topology, the " n a t u r a l " one in a sense described below, and then to prove that T, with this topology, is h o m e o m o r p h i c to an open cell of real dimension 69 - 6. So far, I have mainly been concerned with the local propperties of the Teichm/iller space and of its " n a t u r a l " complex-analytic structure. The definition of the latter depends upon ideas introduced by Teichmiiller himself, but which do not appear to have been fully understood until K o d a i r a and Spencer attacked similar problems for higher dimensions. In order to describe them, it is convenient to substitute for the above definition o f a Teichmiiller surface the following one. Let A1 . . . . , A20 be a set of generators, fixed once for all, of the fundamental group G o of So (with a given origin), satisfying the relation

R(A 1. . . . .

A2o ) = A1AzAllA2

1 ...

A2o~ = 1.

A set at . . . . . a20 of linear-fractional substitutions on z will be called admissible if it has the following properties: (a) it generates a discrete group G of hyperbolic substitutions acting on P; (b) S = PIG is a compact surface of genus 9; (c) there is an admissible class C(S o, S), mapping G o onto G considered as the fundamental group of S, which maps A i onto ai for 1 _< i _< 2 9. Each Teichmfiller surface S has a universal covering which can be mapped conformally onto P, and can therefore be represented as PIG ; the class C(So, S) which belongs to it defines an isomorphism of G o onto G; therefore, to each Teichmtiller surface, there corresponds at least one admissible set (o~ . . . . . 0029). Conversely, it is easy to see that two such sets will define isomorphic Teichmiiller surfaces if and only if they can be transformed into one another by a linear-fractional substitution. Using this fact, it is possible to normalize admissible sets in such a way that there is a one-to-one correspondence between classes of Teichmtiller surfaces (i.e. points of the space T) and normalized admissible sets (00i); this gives a one-to-one mapping of T onto a subset of the coordinate space R 6~ 6. It turns out that the latter subset is open, and that the mapping and its inverse are both indefinitely differentiable (and even, presumably, real-analytic) if T is provided with its " n a t u r a l " differentiable structure; the proof of the latter fact is due to L. Bers. N o w introduce a "variation of structure" as follows. Let/~ be any indefinitely differentiable complex-valued function in P such that I#l < 1; let Pu denote the upper half-plane with the modified complex structure for which dz + # dz is a differential form of type (1, 0); if the latter structure is invariant under G, i.e. if l~ dS/dz is formally invariant under G in an obvious sense, the complex structure of P~ can be projected onto a complex structure P~/G, making the latter into a Riemann surface S u, or rather a Teichmtiller surface if we keep the ai as the distinguished generators of G. Let F , , in that case, be the conformal mapping of P onto Pu; the Teichmfiller surface S~ is then the one defined by the admissible set 00i(/~) = Fj~-100IF~; S~ is isomorphic to S if and only if F , (which is defined only up to a fractional-linear substitution) can be chosen so that it commutes with all the 00~; in that case, the "variation of structure" defined by # will be called trivial. F r o m this, we get an "infinitesimal variation" if we take/~ to depend (differentiably) upon a real parameter t, so that/~ = 0 for t = 0; then v = (dlz/dt)t-o is called an infinitesimal variation of structure; it is clear that any function v in P,

392

[1958c]

Final report on contract AF 18(603)-57 such that v dUdz is formally invariant under G, defines such a variation. The variation v will be called trivial if the finite variation # is tangent, for t = 0, to a trivial one, i.e. if there is a trivial variation y , depending upon t, such that d#/dt = d#'/dt for t = 0. It is easily seen that a necessary and sufficient condition for this is that there should exist a function { in P such that v = c3{/c3~and that {/dz is formally invariant. The infinitesimal variations v can be considered as the elements of a vectorspace V (of infinite dimension) over the complex numbers; the trivial variations make up a subspace V' of V. Standard procedures in c o h o m o l o g y theory and the theory of fibre-bundles (which, in a case like this one, depend merely upon elementary facts such as Stokes's theorem and the theorem of Riemann-Roch) show that V/V' is of finite dimension 3g - 3 and can be "canonically" identified with the dual space to the space of quadratic differentials of the first kind on S. As to the latter assertion, let co = q dz 2 be such a quadratic differential; in other words, we take for q a holomorphic function in P, such that q dz 2 is formally invariant under G. Then, for v in V, vq dz d5 is formally invariant under G, so that we may integrate it over P/G; Stokes's theorem shows that the integral is 0 for v in V', i.e. it depends only upon the class D of v modulo V'; denoting it by (co, D), one finds that this bilinear form defines a duality between V/V' and the space of quadratic differentials, as asserted above. At this point, one must make use of the fact (proved by Bers) that, if # depends differentiably upon some real parameters, the same will be true of the mapping function Fu and hence also of the coefficients in the substitutions ai(#)- it is now easy to calculate the effect of an infinitesimal variation on the coefficients of the cri, i.e. to calculate dai(#)/dt for t = 0, in terms of v = (d#/dt),= o. One can also calculate the effect of a given infinitesimal variation on the periods of the normalized integrals of the first kind on S. The conclusions one can derive from this are as follows. It is possible to provide T with a complex-analytic structure, of complex dimension 39 - 3, such that, whenever # depends holomorphically upon some complex parameters w~, the point of T which corresponds to S u depends holomorphically upon the w~; this observation is due to Bers, who also found, more precisely, that, if we take # = ~ Wiy2qi, with y = Im(z) and (ql . . . . , q3o 3) such that qi dz2 are a basis of the space of quadratic differentials of the first kind on S, then the wi can be taken as local complex coordinates in T in a neighborhood of the point corresponding to S. Furthermore, the quadratic differentials of the first kind on S can be identified with the covectors on T at that point ; and Petersson's hermitian metric, in the space of those differentials (which is no other than the space of automorphic forms of degree - 4 for the group G) defines an intrinsic Hermitian metric on T, which turns out to be a K~ihler metric. The facts concerning the mapping of T into R 6g 6 by means of the coefficients of the ~ri have already been stated. Finally, the periods of the normalized integrals of the first kind on S define a mapping of T into the Siegel space of symmetric g x g matrices with positive-definite imaginary part; the image of T under that mapping is a complexanalytic variety W, whose singular points are those corresponding to hyperelliptic Riemann surfaces; and one obtains a new proof for Rauch's theorem, stating which of the periods of the normalized integrals of the first kind can be used as local coordinates in the neighborhood of any given simple point of W.

[1958c]

393 Final report on contract AF 18(603)-57

H. The K3 surfaces We may start here from the observation (made independently, I believe, by Atiyah and myself) that, when a non-singular surface S in projective 3-space acquires a node, i.e. a conical double point, and the latter is desingularized by a standard dilatation, this process gives a surface S' which is homeomorphic to S. It was easy to surmise that the same is true when a surface acquires any number of distinct nodes; this, in fact, or rather a much more precise theorem, was proved by Atiyah. It shows, in particular, that the non-singular quartic in 3-space, the double plane with a non-singular sextic branch curve, and the desingularized K u m m e r surface, are all homeomorphic. Such surfaces will be called K3 ; they had already occurred in the work of the Italian geometers, and, more recently, in that of Kodaira. The Italians, in fact, had discovered an infinite sequence of families F, (n = 1, 2 . . . . ) of regular surfaces with P0 = 1 ; F 1 consists of double planes with a sextic branch curve; F2, of quartics in 3-space; F, consists of surfaces of degree 2n in projective (n + D-space, whose hyperplane sections are canonical curves of genus n + 1. There are very plausible arguments to indicate that all such surfaces are of type K3, although no complete proof for this seems to have been given yet. For K3 surfaces, the intersection matrix of two-dimensional cycles has the signature (19, 3), the determinant - 1, and is even (i.e. the self-intersection of every cycle is even); hence there is exactly one double differential of the first kind ; if we call it tl, then we must have dq = O, qz = 0 and t/f/>_ 0; if we assume (as seems very likely) that all K3 varieties (algebraic or not) constitute only one connected family, then the canonical class must be 0, so that ~/ 4= 0 and ~/f/> 0 everywhere; this implies that the complex structure is entirely determined by q. Conversely, let there be given, on a differentiable manifold of that nature, a complex-valued differential form t/ of degree 2, satisfying drl = 0, q2 = 0, and ~/f/> 0 everywhere; this determines a complex structure. It seems very plausible (but not at all easy to prove) that two such forms with the same periods must determine complex structures which can be transformed into one another by a difl'erentiable homeomorphism, homotopic to the identity; that all such structures are K~hlerian; and that the periods of rl do not have to satisfy any other condition than those which are implicit in the relations f]2 = 0, /7/7/ • 0. These conjectures (which have also been made independently by Andreotti) may also be expressed as follows. Let S be a class of such structures, two structures being put into the same class if and only if they can be transformed into each other by a differentiable homeomorphism, homotopic to the identity. Let (al . . . . . a22 ) be a minimal set of generators for the two-dimensional homology group with integral coefficients; iD/is as described above, let the Pl be its periods corresponding to the cycles ai; let P be the point with the homogeneous coordinates (Pa . . . . . P 2 2 ) in the complex projective space of dimension 21. Let F(x, y) be the symmetric bilinear forms in x = (xl ..... ")r Y = (Yl . . . . . Y22) whose matrix is the intersection-matrix of the cycles ag. Then the conditions q2 = 0, r/F/ > 0 imply that P is in the open subset H of the quadric F(x, x) = 0 which is determined by the inequality F(x, ~) > 0; H is a homogeneous space of complex dimension 20 for the orthogonal group determined by the real form F(x, x). Now, if we assign to each

394

[1958c] Final report on contract AF 18(603)-57

class S of structures of the given type the point P ~ H, we have a mapping of the set of all such classes into H; and Nirenberg, by an argument combining the Kodaira Spencer technique with techniques derived from the theory of elliptic equations, has proved that the image of that set in H must be open. The conjectures stated above would mean that the mapping is a one-to-one mapping of that set onto H. Furthermore, by a fundamental theorem of Kodaira, a given K3 structure will define an algebraic variety if and only if it has a K~hler metric whose fundamental form has integral periods, i.e. belongs to an integral cycle a; we must have F(a, a) < 0. For such a structure, the point P defined above must belong to the linear variety L~ defined by F(a, x) = 0, and therefore to the set Ha = H c~ L,. It is easy to see that for each integral a such that F(a, a) < O, Ha can be identified with the Riemannian symmetric space belonging to the orthogonal group of the quadratic form F o of signature (19, 2) induced by F on La. Again, one is led to conjecture that this establishes a one-to-one correspondence between such structures and Ha. Now it may happen that two classes S, S' may be distinct and still define isomorphic structures; this will be so when structures belonging to these classes can be transformed into one another by a differentiable homeomorphism, not homotopic to the identity. The latter will induce an automorphism of the homology group, and therefore a unit U of the quadratic form F(x, x), i.e. a matrix with integral coefficients belonging to the orthogonal group of F; let G be the group of all the units of F which can be obtained in this manner. Assuming the truth of the conjectures stated above, we see that two points P, P' of H will determine isomorphic structures if and only if they are equivalent under the group G. A similar statement will hold for Ha; here G has to be replaced by the subgroup Go of G consisting of the elements which leave a invariant. This shows that it is important to determine G, and in particular to determine whether G coincides with the group G of all the units of F. My results on this, too, are still incomplete. However, by applying the theory of theta functions to the Kummer surface considered as a model for K3 surfaces, it has been possible to reduce part of the problem to a purely arithmetical question which has been recently solved by M. Kneser. The result is that G is, at any rate, of finite index in G. Now, since H is not the Riemannian symmetric space for the orthogonal group of F, it follows that G does not act upon H in a properly discontinuous manner; hence there can be no theory of the moduli, in the ordinary sense, for the postulated connected family of K3 surfaces. This is as expected, and is analogous to the wellknown fact that there is no theory of moduli for complex toruses, but only for "polarized" abelian varieties. The situation is quite similar here. For, if one restricts oneself to the family of algebraic K3 surfaces polarized by assigning a K~ihler form belonging to a given integral cycle a, then it follows from what we have said that Go acts in a properly discontinuous manner on Ho, and is of finite index in the group of all units of the quadratic form Fo, so that HjGo is of finite measure. It is therefore to be expected that the automorphic functions in Ha, for the group G o, make up an algebraic function-field, the field of the moduli for the K3 surfaces of the given family. One interesting feature here is the occurrence, in a problem of moduli, of the automorphic functions belonging to the group of units of a quadratic form of signature (n, 2) (with n = 19 in the present case). This is believed to be

[1958 c ]

3 95

Final report on contract AF 18(603)-57 the first time that such a group has appeared in such context. Of course, before this can be more thoroughly investigated, it will be necessary to obtain full proofs for the conjectures stated above. After that is done, analogies with the theory of abelian varieties and of their fields of moduli (given by Siegel's modular functions) will undoubtedly suggest a number of further problems, of a function-theoretic and also of a number-theoretic nature; most fascinating, perhaps, are the possibilities suggested by the theory of complex multiplication. But this is still too remote to be discussed here.

[1958d] Discontinuous subgroups of classical groups (Notes by A. Wallace)

Introduction The object of this course is to prepare the way for a study of certain types of discrete subgroups of the real classical groups and the corresponding quotient spaces. The classical groups will be constructed in a rather special way which actually yields all these groups with only a small number of exceptions. The method consists in taking a semi-simple algebra A over the rationals with an involution or, extending A to an algebra A R over the real numbers, and considering the connected component G of the group of automorphisms of A R which commute with or. G is, in a natural way, an algebraic matric group, and a subgroup Gz of matrices in G whose elements are rational integers is a discrete subgroup. Discrete subgroups obtained in this way are to form the main object of study. An illustration of the kind of theorem to be studied is given in w where conditions for the compacity of G/Gz are worked out. The method of study of G/Gz involves introducing a second involution on AR which is positive (definitions in w and studying the set P(A~) of positive symmetric elements of A~ with respect to this involution. It turns out that G/Gz can be related to a subset of these positive symmetric elements, and that a special type of set W in P(AR) (an M-domain) covers the image of G/Gz, in a certain sense, only a finite number of times. Attention can then be transferred to the set W, which is arithmetically defined. The study of the set W depends on a study of P(A~) which generalizes classical results of Minkowski on the theory of positive definite quadratic forms and their equivalence under transformation by integral matrices. w167 of these notes are concerned with this theory. The next two sections give a list of the classical groups which can be obtained as indicated above. In w some results are obtained on algebras with involutions, and in w these are applied, along with the earlier results, to the construction and study of M-domains. It may appear that the results obtained in this way will depend on the particular way in which the group G is written as a matric group, since the definition of Gz certainly depends on this. However, the properties which are to be of interest eventually are only those which are invariant under commensurability; this can be defined as follows: Two discontinuous subgroups F and F' of a group G are said to be commensurable if F r~ F' is of finite index in each of them. Now if the group G is an algebraic matric group overQ, and is represented as a matric group in two different ways, then the two subgroups F' and F" of integral matrices in these two representations are commensurable. To prove this let G', G"

396

[1958d]

397 Discontinuous subgroups of classical groups

be the two representations of G as matric groups and write x' = 1, + (x'ij) for an element of G', x" = 1,, + (x~,) for an element of G". The isomorphism between G' and G" is expressed by equations xij = Pi~(x~,,), x~, = Q,u(xlj) where the Pij and Q , , are rational functions and in fact can be taken to be polynomials over Q with zero constant terms. If N is a c o m m o n denominator for all the coefficients in the P;i and Qa~,, then, for x;{, -= 0 (mod N), the corresponding x'ii will be integral. Thus F' contains F~, the subgroup of F" consisting of matrices -= 1,, (mod N). Similarly F" D F;e. N o w F~ is of finite index in F", and so the larger group F' c~ F" is of finite index in F"; and similarly in F'. Thus F', F" are commensurable. The result shows that, as far as properties invariant under commensurability are concerned, no generality is lost by the special method used here of constructing Gz. In particular it is easy to see that the compacity of G/Gz discussed in w is such a property.

[1959a] Ad61es et groupes alg6briques

On d6signera toujours par k, soit un corps de nombres algdbriques, soit un corps de fonctions alg6briques de dimension 1 sur un corps de constantes fini. On d6signera par k~ le compl6t6 de k par rapport 5. une valuation v; si une valuation est discr6te, on la ddsignera le plus souvent par un symbole tel que p, et on ddsignera par rp l'anneau des entiers p-adiques dans kp. O n ddsignera par S tout ensemble fini de valuations de k, contenant l'ensemble So des valuations non discr6tes (pour lesquelles le compldt6 est R ou C); bien entendu So est vide si k est un corps de fonctions. Si V e s t une varidt6 alg6brique, ddfinie sur un corps k, on identifiera, suivant l'usage, V avec l'ensemble des points de V sur le domaine universel, et on notera Vk l'ensemble des points de V rationnels sur k. Cette convention s'appliquera notamment si V e s t une varidt6 de groupe, une varidt6 d'alg6bre, une vari6t6 de corps, etc.; on dira par exemple que V e s t une varidt6 d'alg6bre de dimension n, ddfinie sur k, si, en tant que varidt6, c'est un espace a n n e de dimension n, sur lequel on s'est donn6, outre la structure additive usuelle, une structure multiplicative, c'est-'~-dire une application bilindaire (toujours suppos6e associative) de V x V dans V, ddfinie sur k; Vk est alors une alg6bre sur k au sens usuel; si Vk est un corps (donc, au sens usuel, une extension de k de degr6 n), on dit que Vest une vari6t6 de corps de dimension n sur k. Cette mani6re de parler est conforme fi l'usage ancien de Kronecker, Hilbert, etc. (qui ne se g~naient pas pour parler de l'616ment g6n6rique d'un corps). 1. Soit V une varidt6 ddfinie sur k; Vk, peut ~tre munie, d'une mani6re 6vidente, d'une topologie qui la rend localement compacte, et compacte si Vest compl6te (on commence par le cas oh V e s t une vari6t6 a n n e , Vkv 6tant alors considdrde comme partie fermde d'un espace vectoriel de dimension finie sur k~; on passe de lfi d'une mani6re 6vidente au cas d'une vari6t6 abstraite quelconque). Soit p une valuation discr6te de k. Si Vest une varidt6 a n n e , on notera V~ l'ensemble des points de Vkp dont les coordonndes sont dans r~; il est immddiat que c'est une pattie compacte de Vkp. Plus gdndralement, soit V une varidt6 abstraite, d6finie sur k; on sait que V admet toujours un recouvrement fini par des ouverts isomorphes/t des vari6tds affines V (i), ddfinies aussi sur k; autrement dit, on peut 6crire V = Ui Ji(V(~ oh less sont des isomorphismes (ddfinis sur k) des vari6t6s affines V (i) sur des ouverts de V (au sens de la topologie de Zariski, bien entendu). Pour toute valuation discr6te p de k, posons: = U i

c'est 15. une pattie compacte de Vk~. Posons maintenant, pour tout ensemble fini S

398

[1959a]

399 Adeles et groupes algebriques

de valuations de k contenant l'ensemble S O des valuations non discretes:

Vs = I1 vk, X [I veS

pr

off le second produit est 6tendu fi toutes les valuations de k (necessairement discretes) qui n'appartiennent pas fi S; c'est l• une pattie localement eompaete de l ~ Vk,~,et on a Vs = Vs, pour S ~ S'. On d~signera par VAk la limite inductive des Vs, c'est-~-dire la reunion des Vs munie de la topologie pour laquelle un systeme fondamental d'ouverts est form6 par la reunion de l'ensemble des ouverts dans t o u s l e s Vs. Cette notion est justifOe par le Jait que VA~ est dbfinie d'une manikre intrinsbque, c'est-'~-dire ne depend pas de la maniere dont on a 6crit V comme reunion finie d'images isomorphes de varietes affines; la verification de cette assertion est 616mentaire. On appelera VA~ l'espace adOlique associ6 fi V. Au lieu de A , on 6crira souvent A quand aucune confusion n'est possible. L'espace adelique associ6 fila droite affine n'est autre que l'ensemble des adeles (dits aussi "repartitions" ou "valuation-vectors") du corps k, avee sa topologie usuelle: cet ensemble (muni de sa structure topologique, et de sa structure d'anneau) sere note Ak, OU simplement A. La notion d'espace adelique poss6de des proprietes fonctorielles raisonnables. S i f e s t une application partout definie d'une variet6 V darts une variet6 W, definie sur k, on en deduit d'une maniere evidente une application continue de VA dens WA, Si par exemple on s'est donne sur V une loi de groupe algebrique, on en deduit une loi de groupe sur VA, et VA s'appellera le groupe adelique associe fi V. Par exemple, si Vest le groupe multiplicatifGm ?2 une variable sur k, le groupe adelique correspondant est le groupe des idbles de k au sens usuel. Si l'application f de V dens W e s t surjective, il n'en est pas de meme, en general, de l'application de VA dans WA qui lui est associee. Cependant, si Vest un fibre de base W, localement trivial sur k, au sens de la geometrie algebrique, et s i f e s t la projection de V sur sa base W, alors l'applieation correspondante de VA darts WA est surjective. En particulier, si G est un groupe et g u n sous-groupe de G, tous deux definis sur k, et q u ' o n pose H = G/g, l'application de GA darts HA, associee 5. l'application canonique de G sur H, n'est pas surjective en general; mais, si G est (sur le corps de base k) fibrO localement trivial sur H = G/g , alors on peut identifier canoniquement H A avec G A/gA . 2. Soit K une extension separable de k, de degr6 fini d; soit V une varidte affine ou projective de dimension n, definie sur K; on va indiquer comment on peut associer /tces donnees une varidte W de dimension nd, definie sur k, et qui, sur le domaine universel, est isomorphe au produit de V e t de ses conjuguees sur k. P o u r fixer les iddes, considerons le cas affine. Alors Vest definie par des equations: P,(X~ . . . . , XN) = 0

(1 _< /~ < m),

5. coefficients dans K. Soit (a~ . . . . . ad) une base de K sur k; posons: d

x~ = y~ a~ Yj~ j

1

(1 _< i _< N)

400

[1959a] Ad61es et groupes alg6briques

les Yji 6tant de nouvelles ind6termin6es; alors les P~, deviennent des polyn6mes fi coefficients dans K par rapport aux Yji, et peuvent donc s'6crire: d

Pu(X1 . . . . . X u ) = ~ ah Qnu(Yll . . . . . Ydu), h=l

off les Qhu sont des polyn6mes ~ coefficients dans k. Dans ces conditions, W est la vari6t6 d6finie par les 6quations Qhu(Yll . . . . . YdN) = 0

(1 < h _< d, 1 _< # < m)

dans l'espace affine de dimension dN. En effet, une v6rification facile montre qu'apr6s un changement de coordonn6es linOaire, ~ coefficients dans le corps compos6 de K et de tous ses conjugu6s sur k, ces 6quations sont pr6cis6ment celles qui d6finissent le produit de Vet de ses conjugu6es sur k. C'est d'ailleurs seulement cette derni6re v6rification qui exige que K soit s6parable sur k; s'il ne l'6tait pas, les d6finitions ci-dessus auraient encore un sens, mais d6finiraient une op6ration ayant des propri6t6s assez diff6rentes de celles qui nous int6ressent ici. On dira, dans les circonstances ci-dessus, que W e s t d6duite de V par restriction du corps de base de K h k, et on 6crira W = ~ ( V ) . I1 est imm6diat qu'il y a correspondance biunivoque (canonique !) entre VK et Wk. O n v6rifie aussi, sans difficult6, qu'il y a correspondance biunivoque (non moins canonique) entre VA,, et WAk. Cela permet, par exemple, dans t o u s l e s probl6mes relatifs aux espaces ad61iques sur les corps de nombres, de se ramener, si l'on veut, au cas off le corps de base est Q. 3. Soit G une vari6t6 de groupe de dimension n, d6finie sur k. I1 existe sur G une forme diff6rentielle ~o de degr6 n, invariante ~ gauche, d6finie sur k, et ~o est unique un facteur pr6s (facteur qui doit 6tre dans k); si xl . . . . . x, sont des fonctions sur G, d6finies sur k, qui soient des coordonn6es locales dans un voisinage de l'616ment neutre e (et, pour fixer les id6es, nulles en e), on pourra 6crire o~ = y dxl 9 .. dx,, off y est une fonction sur G, d6finie sur k, et finie en e. O n va montrer comment, si vest une valuation quelconque de k, on peut associer ~ Cn une mesure de Haar bien d6termin6e sur Gk,,, mesure qu'on notera ]~o I,,. P o u r cela on distinguera trois cas: (a) k~ = R; alors les xi peuvent servir de coordonn6es locales sur Gk,,au voisinage de e; au voisinage de e, y est fonction analytique r6elle de x ~. . . . . x,; dans ces conditions, y d x ~ . . , dx. peut s'interpr6ter c o m m e une forme diff6rentielle au sens usuel sur Gko au voisinage de e; c o m m e il est clair qu'elle est invariante ~ gauche, elle d6finit par translation une mesure de Haar sur Gk,,, q u ' o n note [~o [,. (b) k~ = C: les xi sont alors des coordonn6es locales complexes sur Gk~ au voisinage de e, et y est fonction analytique complexe des x~ au voisinage de e; on prendra [~o[~ = i"y~ dxl dx, ... dx, dx,. '(c) Soit p une valuation discr6te; Gkp est une vari6t6 analytique sur le corps valu6 complet kp; c o m m e pr6c6demment, x~ . . . . , x, peuvent fitre consid6r6es c o m m e coordonn6es locales au voisinage de e sur Gkp, c'est-~-dire qu'elles d6terminent un h o m 6 o m o r p h i s m e d'un voisinage de e dans Gkp sur un voisinage de 0

[1959a]

401 Ad61es et groupes alg6briques

dans l'espace k~,. Convenons, sur le groupe additif kp, de noter Idx Ip la mesure de Haar, norm6e par la condition que rp (anneau des entiers p-adiques) soit de mesure n 1; la mesure de Haar dans kp, produit des mesures Idxlp sur les n facteurs, sera notde Ddxl ... dx, Ip- On posera alors, au voisinage de e: I~o1~ = lyl~[dx~ ... dx, l~, off lYl~ est la valeur absolue p-adique de la valeur de y au point considdr6 (y est, c o m m e pr~c6demment, fonction analytique de xl . . . . , x, au voisinage de e, ce qui veut dire qu'on peut l'6crire comme s6rie convergente de puissances de x 1. . . . . x,). Naturellement, la valeur absolue p-adique est norm6e de la mani6re usuelle, c'est-fi-dire de fagon que l'automorphisme x ---, ax du groupe additif de k, multiplie la mesure de Haar par le facteur l a Iv. Les d6finitions ci-dessus se justifient du fait qu'elles sont ind6pendantes du choix des coordonn6es locales xl . . . . . x, au voisinage de e; c'est 6vident darts les cas (a) et (b), en vertu de la formule de changement de variables dans les int6grales multiples en analyse classique, et cela rdsulte immddiatement, dans le cas (c), de la formule correspondante en analyse p-adique (qui se d6montre encore plus facilement qu'en analyse classique). De ce qui pr6c6de, on peut, dans certain cas, d6duire la d6finition d'une mesure de Haar sur le groups addlique GA attach6 ~t G. En se reportant au paragraphe 1, il est 6vident que cela est possible chaque fois que, sur le produit

O < X ~0r %

vsS

il existe une mesure produit des mesures [colv; car il est 6vident que les mesures ainsi ddfinies sur Gs et sur Gs,, pour S ~ S', coincident sur Gs. Or, pour que la mesure en question soit ddfinie, il faut et il suffit que le produit infini

6tendu g toutes les valuations discr6tes de k, soit absolument convergent. Lorsqu'il en est ainsi, on notera ~],. Ice Iv la mesure ainsi obtenue. On observera que celle-ci ne d@end pas du choix de co; en effet, si on remplace co par cco, avec c e k • elle se multiplie par 1Iv Icl~, qui est 1 ("formule du p r o d u i t " d'Artin; rappelons que celle-ci se ddmontre comme suit: x ~ cx est un automorphisme du groupe additif Ak, qui multiplie la mesure de Haar par [Iv Iclv, et qui d'autre part induit un automorphisme sur k consid6r6 c o m m e sous-groupe de Ak, donc, par passage au quotient, d6termine un automorphisme du groupe compact Ak/k et par suite laisse invariante toute mesure de H a a r sur ce dernier; k 6tant discret dans A k, A k et Ak/k sent localement isomorphes, donc les mesures de Haar y prennent le m~me facteur). O n dira que G a la propri~t~ de convergence si 1 ~ [co I~ Y est ddfini; le cas off le produit infini 6crit ci-dessus, sans ~tre absolument convergent, est convergent lorsque les p sent ordonnds "naturellement" (c'est-fi-dire par ordre de grandeur croissante des normes) est int6ressant aussi; lorsqu'il en est ainsi, on dira que G a la propri~tO de convergence relative. Les exemples assez vari6s qu'on a trait6s

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jusqu'ici rendent assez plausibles les conjectures suivantes: pour que G a i t la propri6t6 de convergence relative, il faut et il suffit que G n'admette pas d ' h o m o morphisme sur G,,, ddfini sur k; pour que G a i t la propri6t6 de convergence, il faut et il suffit que G n'admette pas d ' h o m o m o r p h i s m e sur Gin, ddfini sur le domaine universel. En particulier, t o u s l e s groupes unipotents, et t o u s l e s groupes semi-simples qu'on a pu traiter de ce point de vue, ont la propri6t6 de convergence. Le groupe Gm n'a pas la propridt6 de convergence relative. Le groupe addlique attach6 au groupe additif G, fi une variable n'est autre que le groupe additif de Ak; ce groupe a 6videmment la propri6t6 de convergence. Nous poserons : k/k

(on convient, une lois pour toutes, d'identifier d'une mani&e 6vidente une mesure de Haar sur un groupe localement compact F avec celle qu'elle d6termine par passage au quotient sur l'espace homog6ne F/7, lorsque y est un sous-groupe discret quelconque de F). On voit facilement que #k = IAI ~/2, o5 A est le discriminant de k, lorsque k est un corps de nombres alg6briques, et que #k = qO 1 si k est un corps de fonctions de genre .q sur un corps de constantes "~ q 616ments. Soit alors G u n groupe de dimension n sur k, ayant la propridt6 de convergence; nous poserons:

nG=

H Io 1 , L'

et nous dirons que c'est la mesure de Tamayawa sur G A. I1 r6sulte de ce qui pr6c6de que cette mesure est d6termin6e d'une mani6re unique (elle ne d~pend pas du choix de co). Soit K une extension s6parable de k, de degr6 rink Soit G u n groupe d6fini sur K ; soit G' le groupe ~K.k(G), d6duit de G par restriction du corps de base de K k; on a d6j/t observ6 que GAK s'identifie canoniquement/t G~k; on v6rifie facilement que Get G' ont simultan6ment la propri6t6 de convergence, et que les mesures de T a m a g a w a ~2G (sur GA,~) et ~2~, (sur G~,~) co'fncident lorsqu'on identifie ces groupes. C'est, entre autres, pour qu'il en soit ainsi qu'on a introduit le facteur ps dans la d6finition de ~G. 4. Toujours avec les m6mes notations, Gk s'identifie d'une mani&e ~vidente avec un sous-groupe discret de GA; on dit que c'est le groupe des ad61es principaux de G. O n s'est aper~;u, dans ces derniers temps, qu'une bonne partie des rdsultats les plus importants de l'arithm6tique classique pouvait s'exprimer en 6 n o n , a n t des propri6t6s de GA/Gk pour des groupes alg6briques G convenables; cette observation, faite d ' a b o r d par Chevalley, c o m m e chacun sait, pour le cas particulier du groupe des id~les (groupe Gin), est d'une importance capitale; le m6rite semble en revenir principalement "~ O n o et Tamagawa. Le cas off G est commutatif est celui de la th6orie classique des corps de nombres alg6briques; celui off G est le groupe orthogonal est celui de la th6orie des formes quadratiques; il semble donc qu'on touche au m o m e n t off ces deux th6ories, confondues/t leurs d6buts (la th6orie des formes quadratiques binaires ne diff6rant pas de celle des corps quadratiques),

[1959a]

403 Ad&les et groupes alg6briques

vont enfin se fondre de nouveau en une seule, 5. savoir la thdorie arithm6tique des groupes alg6briques. I1 n'est pas difficile de ddmontrer (cf. O n o [1]) que GA/Gk est compact lorsque G est unipotent. I1 en est de marne, d'autre part, pour certains groupes semisimples; en voici deux cas typiques: a. G est le groupe des 616ments de norme 1 sur le centre darts une varidt6 de corps non commutatif; b. G est le groupe orthogonal d'une forme quadratique ne repr6sentant pas 0. Marne lorsque GA/Gk n'est pas compact, il peut arriver que cet espace soit de mesure finie (pour une mesure de Haar quelconque sur GA); it semble plausible qu'il en soit ainsi pour les m~mes groupes dont on a conjectur6 plus haut qu'ils ont la propri6t6 de convergence relative. En tout cas, au moyen de la r6duction des formes quadratiques, on a pu ddmontrer cette propri6t6 pour un grand nombre de groupes semi-simples, et Ramanathan a annonc6 qu'il poss6dait une d6monstration s'appliquant fi tous les groupes semi-simples "classiques". Pla~ons-nous dans le cas off G a la propri6t6 de convergence; nous poserons: r(G) = ~

flc,,

d GA/Gk

et nous appellerons z(G), lorsqu'il est fini, le nombre de Tamagawa de G. Le m6rite essentiel de T a m a g a w a consiste 5- avoir ddfini T(G), d'abord dans le cas particulier ot~ G est le groupe orthogonal d'une forme quadratique (sur le corps des rationnels, puis sur un corps de nombres alg6briques), et 5- avoir reconnu les fairs suivants: a. P o u r ce groupe (plus pr6cisdment, pour la composante connexe de l'616ment neutre darts ce groupe), on a r(G) = 2; b. La formule T(G) = 2 est enti6rement 6quivalente 5. l'ensemble des r6sultats des trois c616bres m6moires de Siegel sur les formes quadratiques (C. L. Siegel, Uber die analytische Theorie der quadratischen F o r m e n [2], soit 194 pages, qui ne contiennent m~me pas une d6monstration compl6te pour le cas g6n6ral des formes de signature quelconque sur un corps quelconque). En fait, une lois qu'on a eu l'idde d'exprimer les choses dans ce langage, l'6quivalence de la formule z(G) = 2 avec les r6sultats de Siegel n'est pas trop difficile 5. vdrifier. Naturellement, T a m a g a w a ne s'est pas arr6t6 l~t. I1 a d ' a b o r d cherch6 5. r6diger, darts ce m~me langage, la d6monstration m~me du th6or6me de Siegel (l'id6e de traduire celle-ci dans le langage des id61es 6tait ddjs- venue 5- M. Kneser il y a quelques ann6es, mais celui-ci n'avait rien publi6 sur ce sujet). Un avantage essentiel de la nouvelle mdthode consiste en ce que le thdor6me 5-d6montrer est, du fair marne de son 6nonc6, birationnellement invariant, alors que chez Siegel (et, avant lui, chez Minkowski) il apparaissait comme lid 5. un "genre" de formes quadratiques. En particulier, en vertu des isomorphismes bien connus entre groupes classiques, les cas n = 3 et n = 4 se ram6nent ainsi ~t des questions analogues sur les alg6bres de quaternions, qu'on sait traiter directement; or c'6tait justement les cas qui donnaient le plus de difficult6 chez Siegel. Ces cas 6tant acquis, tout le reste de la d6monstration peut maintenant se pr6senter fort simplement, et presque sans calculs. D'autre part, on a commenc6 "aexaminer d'autres groupes semi-simples: groupe

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spin (Tamagawa), groupes lin6aire sp6cial et projectif sur une alg6bre simple, etc. D a n s t o u s l e s cas q u ' o n a su traiter, on a trouv6 que ~(G) est un entier, et qu'il est dgal ~ 1 lorsque G est un groupe semi-simple "simplement connexe" (au sens alg6brique, c'est-g,-dire que tout g r o u p e isog6ne ~ G est une image h o m o m o r p h e de G). I1 en est bien ainsi, p a r exemple, si G est le g r o u p e des 616ments de n o r m e 1 sur le centre dans une vari6t6 d'alg6bre simple sur un corps de n o m b r e s ou bien sur un corps de fonctions.

Bibliographie 1. Ono, Takashi. Sur une propri6t6 arithm6tique des groupes alg+briques commutatifs, Bull. Soc. math. France, t. 85, 1957, p. 307 323. 2. Siegel, Carl Ludwig. Ober die analytische Theorie der quadratischen Formen, I : Annals of Math., t. 36, 1935, p. 527 606; II: t. 37, 1936, p. 230 263; III: t. 38, 1937, p. 212 291. 3. Tamagawa, Tsuneo, M6moire ',i para~tre aux Annals of Mathematics.

[1959b] Y. Taniyama (lettre d'Andr~ Weil) II est impossible, pour un math6maticien fran~ais de m o n ~ge, d'6crire sur T a n i y a m a sans songer aussit6t "~ Herbrand. Celui-ci aussi restera dans la m6moire de ceux qui l'ont connu c o m m e l'une des plus fortes personnalit6s parmi les math6maticiens de sa g6n6ration. Herbrand est mort h 23 ans dans un accident de montagne; au dire de ses compagnons, la prudence n'6tait pas en montagne sa qualit6 dominante; il semble que la vie n'importe plus beaucoup h ceux qui ont franchi certaines fronti6res de l'intelligence. Pour quiconque a connu Herbrand, il est difficile de croire que, s'il avait v6cu, notre math6matique, et particuli6rement notre th6orie des nombres, aurait tout h fait l'aspect qu'elle pr6sente aujourd'hui; il 6tait de ceux dont on attend, non seulement qu'ils r6solvent tel ou tel probl6me avant les autres, mais qu'ils enrichissent la science d'id6es que d'autres n'auraient point. T a n i y a m a aussi 6tait de ceux-l~. Sa personnalit6 rut pour moi l'une de celles qui domin6rent le colloque de T o k y o - N i k k o en 1955, et dont je conservai la plus forte impression a la suite du s6jour au Japon que je fis ~ cette 6poque; s'il 6tait clair qu'il s'y m61ait des 616ments dissonants, en conflit les uns avec les autres et avec le m o n d e ext6rieur, il n'y avait pas de raison de voir 1~ autre chose que le bouillonnement d'un temp6rament jeune qui n'a pas encore r6alis6 son harmonie interne; et j'attendais beaucoup, pour moi au moins autant que pour lui, du s6jour qu'il avait 6t6 invit6 ~ faire "~ Princeton. Je n'ai pas besoin de dire ici m o n 6motion et m o n chagrin quand j'ai appris queje ne devais plus le revoir. Du moins, plus heureux q u ' H e r b r a n d (dont le nom, en dehors de son oeuvre logique, reste attach6 seulement ~ deux ou trois r6sultats tr6s fins, tr6s en avance sur leur 6poque, et qui n'ont trouv6 leur explication que r6cemment), T a n i y a m a nous a laiss6 un travail de premier ordre, compl6tement achev6 dans le cadre qu'il s'6tait fix6, mais qui ouvre de vastes perspectives sur les plus importants probl6mes de l'arithm6tique moderne; c'est bien entendu de son m6moire de 1957 sur les fonctions L, paru darts le Journal of the Mathematical Society of Japan, que j'entends parler; trop modestement, il dit qu'il y suit "rues m6thodes," alors qu'en r6alit6, prenant pour point de d6part quelques observations que j'avais eu l'occasion de faire, et les joignant "~ ses propres r6sultats, il y d6veloppe des m6thodes enti6rement neuves et d'une grande port6e. Sans entrer ici dans des commentaires d6taill6s, j'observerai seulement que, pour bien apercevoir les id6es directrices du m6moire, il convient de commencer par le dernier chapitre. Lh il est montr6, au moyen de la r6duction modulo p, que, si A est une vari6t6 ab61ienne d6finie sur un corps de hombres alg6briques k, les repr6sentations l-adiques du groupe de Galois de k sur k (off/~ est la cl6ture alg6brique de k), d6termin6es par les groupes de points d'ordre / N sur A, ne sont pas ind6pendantes les unes des autres, mais sont li6es entre elles et avec la fonction z6ta de A par des relations tr6s pr6cises; et l'id6e peut-6tre la plus originale de T a n i y a m a a 6t6 de voir que ces relations, telles qu'il les formule, m6ritent d'fitre 6tudi6es en elles-m~mes, ind6pendamment de la vari6t6 A d'ofi

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[1959b] Y. Taniyama on les a tirees. C'est cette etude qu'il a menee ~t bien, par une analyse des plus fines et ingenieuses, dans le cas "abdlien" off les representations/-adiques en question engendrent des algebres commutatives (cas qui se presente justement dans la multiplication complexe); ses resultats comprennent comme cas particulier ses thdoremes anterieurs sur les fonctions zeta des varietes abeliennes fi multiplication complexe; en meme temps, il 6claire par lfi d'un jour tout nouveau la theorie des caracteres "de type (A0)" dont j'avais seulement signal6 l'existence au cours du colloque de Tokyo-Nikko. Mais c'est bien entendu le cas non abelien dont l'etude l'attirait, sans qu'il air pu, semble-t-il, rien entreprendre ~ ce sujet; il est inutile d'ajouter que ce probleme ne semble abordable par aucune des methodes dont nous disposons actuellement. Pendant longtemps encore, sans doute, ce travail fournira des sujets de meditation aux arithmeticiens. J'ignore, au m o m e n t off j'dcris, s'il s'est retrouv6 parmi les papiers de T a n i y a m a des traces de recherches plus recentes. Mais, quand meme il n'en serait pas ainsi, son memoire de 1957 suffirait fi lui assurer dans l'histoire de notre science une place durable. A ceux qui l'ont connu personnellement, il laisse, avec un inoubliable souvenir, l'amer regret de n'avoir rien su ou rien pu faire pour le retenir parmi eux.

[ 1960a] De la m&aphysique aux math6matiques (h propos d' un colloque re'cent)

Les math6maticiens du XVlIIe si~cle avaient coutume de parler de la (( m6taphysique du calcul infinit6simal )), de la (( mdtaphysique de la thdorie des ~quations )). Ils entendaient par lh u n ensemble d'analogies vagues, difficilement saisissables et difficilement formulables, qui n~anmoins leur semblaient jouer u n r61e i m p o r t a n t hun m o m e n t donn6 dans la recherche et la d~couverte math~matiques. Calomniaient-ils la (( vraie )) mdtaphysique en e m p r u n t a n t son n o m pour d6signer ce qui, dans leur science, 6tait le moins clair? Je ne chercherai pas h 61ucider ce point. E n tout cas, le mot devra 6tre e n t e n d u ici en leur sens; h la (( vraie )) mdtaphysique, je me garderai bien de toucher. Rien n'est plus f6cond, tousles mathdmaticiens le savcnt, que ces obscures analogies, ces troubles reflets d ' u n e th6orie h une autre, ces furtives caresses, ces brouilleries inexplicables; rien aussi ne donne plus de plaisir au chercheur. U n jour v i e n t off l'illusion se dissipe; le pressentiment se change en certitude; les th6ories jumelles r6v~lent leur source commune a v a n t de disparaitre; comme l'enseigne la G~tgt on a t t e i n t h la connaissance et h l'indiffdrence en m~me temps. La m6taphysique est devenue math~matique, prate h former la mati~re d ' u n trait6 dont la beaut6 froidc nc saurait plus nous 6mouvoir. Ainsi nous savons, nous, ce que cherchait ~ deviner Lagrange, q u a n d il parlait de mdtaphysique h propos de ses t r a v a u x d'alg~bre; c'est la thdorie de Galois, qu'il touche presquc du doigt, h travers u n 6cran qu'il n'arrive pas k percer. Lh off Lagrange voyait des analogies, nous voyons des th6orbmes. Mais ceux-ci ne p e u v e n t s'dnoncer q u ' a u moyen de notions et de (c structures )) qui pour Lagrange n'dtaient pas encore des objets math6matiques : groupes, corps, isomorphismes, automorphismes, tout cela avait besoin d'&tre con~u et d6fini. T a n t que Lagrange ne fair que pressentir ces notions, r a n t qu'il s'efforce en vain d'atteindre h leur unit6 substantielle h travers la multiplicitd de leurs incarnations changeantes, 52

Reprinted by permissionof Hermann, 6diteurs des sciences et des arts

4O8

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il reste pris dans la m6taphysique. Du moins y trouve-t-il le fil eondueteur qui lui permet de passer d ' u n probl~me ~ l'autre, d'amener les mat6risux k pied d'oeuvre, de tout mettre en ordre en vue de la th6orie g6n6rale future. Grace h la notion d6eisive de groupe, tout eela devient math6matique ehez Galois. De m~me eneore, nous voyons les analogies entre le ealeul des diff6renees finies et le ealeul diff6rentiel servir de guide h Leibniz, h Taylor, h Euler, au eours de la p6riode h6roique durant laquelle Berkeley pouvait dire, avee autant d'humour que d'h-propos, que les ~ eroyants ~ du ealeul infinit6simal 6taient peu qualifi6s pour eritiquer l'obseurit6 des myst~res de la religion ehr6tierme, eelui-l~ 6t~nt pour le moins aussi plein de myst~res que eelle-ei. Un peu plus t~rd, d'Alembert, ennemi de toute m6taphysique en math6nmtique eomme ailleurs, soutint dans ses articles de l'Encyclop~die que la vraie m6taphysique du ealeul infinitesimal n'6tait pas autre chose que la notion de limite. S'il ne tira pas lui-m~me de eette id6e tout le patti dont elle 6tait susceptible, les d6veloppements du si~ele suivant devaient lui donner raison; et rien ne saurait 8tre plus elair aujourd'hui, ni, il faut bien le dire, plus ermuyeux, qu'un expos6 correct des 616ments du ealeul diff6rentiel et int6gral. Heureusement pour les ehereheurs, h mesure que les brouillards se dissipent sur un point, e'est pour se reformer sur un autre. Une grande pattie du eolloque de Tokyo s'est d6roul6e sous le signe des analogies entre la th6orie des nombres et la th6orie des fonetions alg6briques. Lh, nous sommes eneore en pleine m6taphysique. C'est de ees analogies, paree que j'en ai quelque exp6rienee personnelle, que je voudrais parler iei, avee l'espoir, vain peut-~tre, de donner aux leeteurs ~ honnStes gens ,~ de eette revue quelque id6e des m6thodes de travail en math6matique. D~s l'enseignement 616mentaire, on fait voir aux 61~ves que la division des polynSmes (~ une variable) ressemble beaueoup h la division des entiers et eonduit des lois toutes semblables. Pour les uns eomme pour les autres, il y a un plus grand eommun diviseur, dont la d6termination se fait par division sueeessive. A la d6eomposition des hombres entiers en faeteurs premiers correspond la d6eomposition des polynSmes en faeteurs irr6duetibles; aux nombres rationnels correspondent les fonetions rationnelles, qui, elles aussi, peuvent toujours se mettre sous forme de fractions irr6duetibles; eeUes-ei s'ajoutent par r6duetion au plus petit eommun d6nominateur, ete. I1 est done tout naturel de penser qu'il y a analogie entre les hombres alg~brlques (raeines d'6quations dont les eoeffieients sont des nombres entiers) et les ]onctions algdbriques d'une variable (raeines d'6quations dont les eoeffieients sont des polynSmes ~ une variable). Le fondateur de la doute 6t6 Galois s'fl qu'on trouve sur ee mort, d'ofl on peut

th6orie des fonetions alg6briques d'une variable aurait sans avait v6eu; e'est ee que permettent de penser les indications sujet dans sa e61~bre lettre-testament, 6erite k la veille de sa eonelure qu'il touehait d6jk h quelques-unes des prineipales 511

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d~eouvertes de Riemann. Peut-6tre aurait-il donnd h cette thdorie une allure algdbrique, conforme ~ l'esprit des t r a v a u x contemporains d ' A b e l et de ses propres reeherches d'alg~bre pure. A u eontraire, Riemann, l'un des moins alg6bristes sans doute parmi les grands math6maticiens du xIxe si~ele, m i t la th6orie sous le signe du (( t r a n s c e n d a n t , (mot qui, pour le math6matieien, s'oppose t~ (calg6brique ,, et ddsigne t o u t ee qui a p p a r t i e n t en propre au continu). Les mdthodes tr~s puissantes mises en oeuvre p a r R i e m a n n amen~rent presque du premier coup la th6orie un degrd d'aeh~vement qui n ' a gu~re 6t6 d6pass6. Mais elles ne tiennent aucun eompte des analogies avee les hombres alg6briques, et ne p e u v e n t 6tre transposdes telles quelles en vue de l'6tude de eeux-ci, 6rude qui relive traditionnellement de l'arithmdtique ou th6orie des hombres, et qui, du v i v a n t ddjh de Riemann, 6tait en voie de d6veloppement rapide. C'est Dedekind, ami intime de Riemann, mais algdbriste eonsomm6, qui d e v a i t le premier tirer p a r t i des analogies en question et en faire un instrument de recherche. I1 appliqua avec succ~s, aux probl~mes trait6s p a r R i e m a n n p a r voie transeendante, les mdthodes qu'il a v a i t lui-m~me cr6~es et raises au point en vue de l'6tude arithm6tique des nombres alg6briques; et il fit voir qu'on peut retrouver ainsi la partie p r o p r e m e a t alg6brique de l'oeuvre de Riemann. A premiere vue, les analogies ainsi mises en 6videnee restaient superficielles, et ne paraissaient pas pouvoir porter sur les probl~mes les plus profonds de l'une ni de l ' a u t r e thdorie. H i l b e r t alia plus loin dans cette voie, ~ ce qu'il semble; mais, s'il est probable que ses 61~ves subirent l'influence de ses iddes sur ce sujet, il n'en est rest~ quelque trace que dans un eompte rendu obscur qui n'a m~me pas dt6 reproduit dans ses (Euvres complktes. Les lois non dcrites de la math6matique moderne interdisent en effet de publier des r u e s mdtaphysiques de eette esp~ce. Sans doute est-ce mieux ainsi; a u t r e m e n t on serait accabl6 d'articles encore plus stupides, sinon plus inutiles, que tous ceux qui encombrent h prdsent nos pdriodiques. Mais il est dommage que les iddes de H i l b e r t n'aient 6t6 ddveloppdes par lui nulle part. I1 y a v a i t loin encore, cependant, de l'arithm6tique, off r~gne le discontinu, h la th~orie des fonctions au sens classique. Or, en disant que les fonetions algdbriques sont raeines d'dquations dont les coefficients sont des polyn6mes, j ' a i volontairem e n t omis un point i m p o r t a n t : ces polynSmes eux-m6mes ont des coefficients; mais eeux-ci, quels sont-ils? Lorsqu'on traite de la division des polyn6mes darts l'enseignement 616mentaire, il v a sans dire que les coefficients sont des ((nombres ,~ : nombres (( rdels ,~ (rationnels ou non, mais donnds en tout eas, si l'on veut, p a r un d6veloppement d~eimal), ou, h u n niveau un peu plus 61ev6, nombres(( r6els ou imaginaires ,, ou, comme on dit, ((nombres complexes ~. C'est exclusivement de nombres complexes qu'il s'agit dans la th6orie riemannienne. Mais, du point de vue de l'alg6briste pur, t o u t ee qu'on demande a u x (( hombres ~ en question, c'est qu'ils se laissent combiner entre eux au moyen des quatre op6rations (ce que l'algdbriste exprime en disant qu'ils forment un ,~ corps ,~). Si on 54

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e

s

n ' e n suppose pas plus sur leur eompte, on obtient une thdorie des fonetions alg6briques, fort riche ddj~t (comme en t6moigne le volume r6cent et ddj~ classique q u ' a publi6 Chevalley sur ce sujet), mais qui ne l'est pas assez pour que les analogies avec les nombres algdbriques puissent btre poursuivies j u s q u ' a u bout. Heureusement il s'est trouv6 u n domaine intermddiaire entre l'arithmdtique et la thdorie riemannienne, et qui possbde, avec chacune de ces deux derni~res thdories, des ressemblances beaucoup plus 6troites qu'elles n ' e n ont entre elles; il s'agit des fonctions algdbriques (( sur u n corps fini )). Comme on le savait depuis Gauss, s'il ne s'agit que de pouvoir faire les quatre opdrations, il suffit d ' u n nombre fini d'dldments. I1 suffit par exemple d'en avoir deux, q u ' o n nommera 0 et 1, et pour lesquels on posera par convention la table d'addition et la table de multiplication que voici : 0+0=0, 0+1=1+0=1, 1+1=0; 0 • 0=0, 0 • 1~1 • O~ 1 X 1=1. Quelque paradoxale que puisse paraltre au profane la rSgle 1 -4- 1 = 0, quelque t e n t a n t qu'il soit de dire que c'est 1~ u n pur jeu de l'esprit qui ne rdpond ~ aueune (( rdalitd ,, u n tel syst$me est monnaie eourante pour le mathdmaticien; et Galois en ~tendit beaucoup l'usage en construisant les ~( imaginaires de Galois ,. P r e n a n t done les coefficients de nos polynSmes dans u n (~ corps de Galois ~, on construit des fonetions alg6briques dont la thdorie remonte ~ Dedekind mais s'est particuli~rement d~velopp6e depuis la th~se d'Artin. Pour dire en quoi elle consiste, il faudrait entrer darts des d~tails beaucoup trop techniques qui n ' a u r a i e n t pas leur place icio Mais on peut, je erois, en donner une idde imagde en disant que le mathdmatieien qui 6tudie ces probl~mes a l'impression de d~chiffrer une inscription trilingue. Darts la premiere colonne se trouve la thdorie riemannienne des fonctions alggbriques au sens classique. La troisi&me colonne, c'est la th$orie arithmdtique des nombres algdbriques. La eolonne du milieu est celle dont la ddeouverte est la plus rdeente; elle eontient la thdorie des fonetions alg6briques sur u n corps de Galois. Ces textes sont l'unique source de nos connaissances sur les langues dans lesquels ils sont dcrits; de ehaque colonne, nous n'avons bien entendu que des fragments; la plus complete et celle que nous lisons le mieux, encore ~ prgsent, c'est la premi&re. Nous savons qu'il y a de grandes diff6rences de sens d ' u n e colonne ~ l'autre, mais rien ne nous en avertit ~ l'avance. A l'usage, on se fait des bouts de dictionnaire, qui permettent de passer assez souvent d'une colonne ~ la colonne voisine. C'est ainsi q u ' o n avait ddchiffrd depuis longtemps, dans la derni~re colonne, le ddbut d ' u n paragraphe intituld (~ fonetion z6ta ~). Vers la fin de ee paragraphe, on croit lire une phrase tr~s mystdrieuse; elle dit que t o u s l e s zdros de la fonction se t r o u v e n t sur une certaine droite. Jamais on n ' a pu savoir s'il en est bien ainsi, ou s'll y a eu erreur de lecture. C'est le c~l~bre probl~me de 1' (( hypoth~se de R i e m a n n )), qui dans quelques mois sera tout juste centenaire. 55

412

[1960a1 ~

at

h ~ m a ~

q

ue~

La prineipale ddcouverte d'Artin, dans sa thtse, c'est qu'il y a, dans la seconde colonne, un paragraphe intittd6 aussi (( fonetion z6ta ~,, et qui est t~ peu de chose pros une traduction de celui qu'on connaissait d6jh; notre dietionnaire s'en est trouv6 beaueoup enrichi. Artin apergut aussi, dans eette colonne, la phrase sttr l'hypoth~se de Riemann; elle lui parut tout aussi mystArieuse que rautre. Ce nouveau probl~me, t~ premi$re rue, ne semblait pas plus faerie que le pr6e~dent. Ign rfiahtd, nous savons maintenant que la premitre colonne contenait ddjh tous les 616ments de sa solution. II n'dtait que de traduire, d'abord en th~orie (( abstraite ~ des fonctions algdbriques, puis dans le langage (( galoisien ~ de la seconde eolonne, des rfisultats obtenus depuis longtemps par Hurwitz en (( riemarmien ,,, et que les g~om~tres italiens avaient ensuite traduits dans leur propre langage. Mais les meilleurs sp6eiahstes des th6ories arithm~tique et (( galoisienne ~ ne savaient plus lire le riemannien, ni h plus forte raison l'italien; et il fallut vingt arts de recherches avant que la traduction f6t rnise au point et que la d6monstration de l'hypoth~se de Riemann dans la seeonde eolonne ffit complttement ddehiffr6e. Si notre dietionnaire 6tait suftisamment c0mplet, nous passerions aussitx3t de I~ la troisi~me colonne, et l'hypoth~se de Riemann, la vraie, se trouverait d~montrde, ere aussi. Mais nos cormaissanees n'atteignent pas jusque lh; bien des d~ehiffrements patients seront encore ndcessaires avant que la traduction puisse 6tre faite. Au tours du colloque auquel il a $t6 fair allusion plus haut, il a 6t6 beaucoup diseut6 de (( m~taphysique ~ tr propos de ces probltmes; un jour eelle-ci fera place h une thdorie mathdmatique dans le cadre de laquelle ils trouveront leur solution. Peutgtre, comme c'6tait le cas pour Lagrange, ne nous manque-t-il, pour franehir ee pas d6cisif, qu'une notion, un concept, une (( structure ~. D'ing6nieux philologues ont bien trouv~ le secret des archives de Nestor et de celles de Minos. Combien de temps faudra-t-il encore pour que notre pierre de Rosette, h nous autres arithm~tieiens, rencontre son Champollion?

56

[ 1960b] Algebras with involutions and the classical groups

IT has been knowr~ for a long time t h a t there is a close connection between semisimple algebras with involutions and the classical semisimple Lie groups and Lie algebras. B u t the precise degree of generality of this relationship does not seem to have been ascertained anywhere, at least explicitly, in the printed literature on this subject. I n the first part of the present paper, it will be shown how this can be done, at a n y rate over a groundfield of characteristic 0, b y borrowing some elementary techniques from m o d e r n algebraic geometry. Then, taking for our groundfield the field R of real numbers, we shall give, for the l~iemannian s y m m e t r i c spaces a t t a c h e d to the classical groups, an i n t e r p r e t a t i o n which rests u p o n the use of algebras with involution. This was already implicit in Siegel's f u n d a m e n t a l work on discontinuous groups, and it is hoped t h a t our results will help to achieve a b e t t e r u n d e r s t a n d i n g of t h a t work and to clarify also some of its arithmetical aspects.

PART

I

~EMISlMPLE GROUPS OVER A FIELD OF CHARACTERISTIC 0. 1.

I n P a r t I, all spaces, varieties, groups are to ba understood

in the sense of algebraic geometry; b y this we mean t h a t t h e y are all allowed to have points in the universal domain, and not only in their field of definition. I f V is a v a r i e t y (for instance a group), t Work.supported (in part) by the O.U.I%.P.A.F. The author is greatly indebted to his colleague Mr. M. for a counterexample and other helpful remarks, to Mr. P. (the famous winner of many cocycle races) for the main idea of Part I, to others for conversations totally unrelated to the problems considered here, and to the Indian iV~atheraatieal Society for kindly allowing this paper to rest for over two years in their editorial officesbefore letting it take its flight into the world. Reprinted by permission of the editors of J.

Ind. Math. Soc.

413

414

[1960b] 5'90

ANDRE WEIL

d e f n e d over a field k, we shall denote b y Vk the set of points of V with coordinates in k (in group-theory, this differs from the usage of Chevalley, according to which a group G, defined over an infinite field/~, is always identified with the set which we call G~). We assume the universal domain to be of characteristic 0 ; without restricting the generality, one could take it to be the field of complex numbers. Within the universal domain, we select a groundfield k and a normal algebraic extension K of k, of finite degree d; we call g the Galois group of K over /~, and we write ~ for the imago of an element ~ of K u n d e r an automorphism a ~ g ; we have (~a), = ~a~ for all a, ~ in g. I f V is a vector-space of dimension n (over the universal domain), defined over k, Vk and VK are vector-spaces of dimension n over k and over K, respectively. We have Vk a VK, and we m a y identify VK with the tensor-product V~ | K t a k e n over k; g operates in an obvious m a n n e r on VK, and Vk consists of the elements of VK which are invariant under g. I f V' is the dual space of V (over the universal domain), and if T is a n y tensor-product (V | V | ...) | | ...), over the universal domain, of f~etors identical either with V or with V', t h e n T~ is the tensor-product similarly built up from V~ and its dual V~ over/~; and a similar s t a t e m e n t holds for T K. B y a c o c y d e , we shall u n d e r s t a n d a mapping a - + $'~ of g into the group of automorphisms of the vector-space V, such t h a t : (a) for each a E g, Fo is defined over K ; (b) for all a, ~ in ~, we have Fa~ = ( F o y o F~. Let (Fo) be such a cocycle; for e v e r y x ~ VK, p u t :

O n e s e e s at once t h a t x [~] = (xt~])[~], which means t h a t the group g can also be made to operate on VK b y (x, a ) - + x E~ These operations are k-linear, b u t not K-linear; we have (~x) t~] = ~~ t~] for x e VK and ~ e K. F r o m this, it follows t h a t the set W of those elements of VK which are invariant under all operations x - + x ["], i.e. which satisfy x ~' = F o ( x ) for all a, is a vector-space over k. I f ql . . . . . a~ are linearly independent over k in W, it is easy to see, b y induction on m, t h a t t h e y are linearly independent over K in

[1960b]

415 ALGEBRAS W I T H

INVOLUTIONS

59l

VK; for otherwise, because of the induction assumption, there would be a relation E ~ i a ~ = 0 , with the ~ in K and not all in k, a n d ~m = 1 ; a p p l y i n g to this the operation x - + x [~], we get Z i ~ a~ = 0, hence Z i ( ~ - - ~) a~ = 0, and therefore, because of the induction assumption, ~ = a s s u m p t i o n on the a~. Now k; t a k e a basis (~1 . . . . . ~ ) elements Z~ ~ x [~l are in W ~

~ x [~]

c:~B

~i for all i a n d ~, which contradicts the t a k e for al, ..., a m a basis for W o v e r of K over }; for e v e r y x e V~, the for i = 1 . . . . . d, so t h a t we can write : ~

(1~i

d)

~=1

with % ~ k. I t is w e l l - k n o w n t h a t the d e t e r m i n a n t [ ~ [ is not O, and therefore these equations can be solved for the x ["1, yielding expressions for t h e m as linear combinations of the a with coefficients in K ; in particular, x itself is such a linear combination. We h a v e t h u s shown t h a t a~ . . . . , am is a basis for IrK o v e r K , so t h a t in particular we m u s t h a v e m = n. T a k e now a basis b1.... , b, of Vk over k ; let (I) be the a u t o m o r p h i s m of V which m a p s bi onto a i for 1 ~< i ~< n ; it is defined over K. Combining the relations a i = O(b~), a~ : F~(ai), b~ = bi, we get $'~(O(bi) ) = O~(b~) for all i, hence F~ o 9 : 9 ~ or F~ = (I)~ r (which can be expressed b y saying t h a t "all eocycles are trivial " ). N o w lot t, t', ... be elements of " t e n s o r - s p a c e s " T, T ' , .... i.e. of tensor-products built u p f r o m factors identical with V or V'. More precisely, a s s u m e that t e T k, t' ~ T ' k, . . . , and that all the teasers t, t', ... are i n v a r i a n t u n d e r every one o f the a u t o m o r p h i s m s $',; b y this we m e a n of course t h a t t is i n v a r i a n t under the canonical extension of F , to T, etc. Similarly, write (I) for the canonical extension of ~) to T, T ' , ... and p u t tl = (I)-l(t), etc. We have: t~ = ( r

= r

= tl

for all a, hence t 1 e T~, and similarly t' 1 e T'~, etc. The m a i n application which we have in view concerns the case of algebras a n d of algebras with involution. B y an algebra A , we u n d e r s t a n d a vector-space V with the additional structure deter-

416

[1960b] 592

ANDRE W E I L

mined on it b y a bilinear mapping of V • V into V, or, what amounts to the same, b y an element t of the tensor-space T = V' | V' | V; if, in addition to this, we prescribe an endomorphism ~ of V, or, what amounts to the same, an element t' of T ' ~- V' | V, such t h a t is an i n v o l u t o r y antiautomorphism (or, as we shall say more briefly, an involution) of the algebra A, t h e n we speak of V, with the structure determined on it b y t and t', as the algebra with involution (A, ,). All algebras will be assumed to be associative and to have a unit-element, usually denoted b y 1. We say t h a t the algebra A, with the underlying vector-space V and the multiplicative structure determined b y the element t of V' | V' | V, is defined over k if V and t are defined over k; t h e n A k is an algebra over k in the usual sense, and A K is the algebra derived from A k b y extending the groundfield from k to K. The same holds for algebras with involution. As our universal domain is assumed to be of characteristic 0, it is known t h a t A, is somisimple if and only if A (as an algebra over the universal domain) is so. The center Z of a semisimple algebra A, defined over k, is a commutative semisimple algebra, also defined over k; Z k is then the center of A k. We say t h a t A is absolutely simple if it is simple as an algebra over the universal domain ; then it is isomorphic to a m a t r i x algebra Mn over the universal domain. On the other hand, we say t h a t a semisimple algebra A, defined over k, is simple over k if it is not a direct sum of subalgebras of A, all defined over k; this will be so ff and only ff A k is simple as art algebra over k, or also if and only if the center Z of A is simple over k, or again ff and only if Z k is a field. I t is clear t h a t the groups of automorphisms of algebras and of algebras with involution are algebraic groups. As a special case of the results p r o v e d above, we have now the following theorem : THEOREM 1. Let A be an algebra (resp. an algebra with involution) defined over a field k; let K be a Galois extension of k with the Galois group g. Let (2'0) be a cocycle of fi, consisting of automorphisms of A. Then there is an algebra (rosp. an algebra with involution) A 1, defined

[]960b]

417

ALGEBRAS WITH INVOLUTIONS

593

over k, and an isomorphism r of A 1 onto A , defined over K , such that F a ---- r o Cp-:for all (~ Gg.

In,fact, let V be the underlying vector-space of A; and let t be the tensor (resp. let t, t' be the pair of tensors) on V which defines the structure of A. Define t: and (I) (resp. t:, t~ and (I)) as above by means of the cocycle (F~); and define A 1 as the algebra (resp. the algebra with involution) defined on V by t: (resp. by tl, t~). These will satisfy all the conditions in our theorem. 2. It is our purpose to show, by means of Theorem 1, that, with few exceptions, the classical semisimple groups over any field of characteristic 0 can be represented as groups of automorphisms of semisimple algebras with involution ; in fact, it will turn out that there is almost a one-to-one correspondence between these two classes of objects. We begin by discussing the classical simple groups over the universal domain. Consider first the projective linear group PL(n) in n variables, i.e. the factor-group of GL(n) by its center (to be consistent with our notation, we omit any mention of the underlying field when the latter is the universal domain). Let A be the direct sum of two algebras, both isomorphic to the matrix algebra M~ of order n (i.e. consisting of all n • n matrices); on A, consider the involution defined by (x, Y) - + (t y, tx) ' where X , Y are two matrices of order n, and tX, t y are their transposes. Using the classical theorem of Skolem-2qoethor, one sees immediately that the automorphisms of the algebra with involution A (i.e., the automorphisms of the algebra A which commute with the given involution) make up an algebraic group G, consisting of two connected components G0, G:; GO consists of all the automorphisms of A which leave each component M , of A invariant, and more precisely of the automorphisms: (X, Y ) ---+ (X, y),(M) = ( M - 1 X M ,

tM. Y. t M - 1 )

where M is an arbitrary invertible matrix ; the mapping M - + r is then a homomorphism of GL(n) onto Go whose kernel is

418

[1960b] 594

ANDRE WEIL

the center of GL(n), so that GO may be identified with P L ( n ) . As to (71, it consists of those automorphisms of A which exchange its two components, and may also be defined as the coset of (7o in (7 which contains the automorphism (X, Y ) - + (Y, X). The inner automorphisms of G induce automorphisms on Go ~hieh are either inner automorphisms of Go or products of such automorphisms with the automorphism induced on G o by (X, Y ) - + ( Y , X); the latter may be written as r r and it is easy to see that, for n > 3, this is not an inner automorphism It is well known that these are all the automorphisms of G0 = PL(n). As our calculation also shows that no automorphism of A, other than the identity, induces the identical isomorphism on Go for n > 3, it follows that, for n > 3, every automorphism ofG 0 can be derived, in one and only one way, from an automorphism of A. This implies that, for n > 3, if A ' is an algebra with involution, isomorphic to A, and G' o is the connected component of the identity in the group of automorphisms of A', every isomorphism of G0 onto G'0 can be derived, in one and only one way, from an isomorphism of A onto A'. In order to obtain in a similar manner the orthogonal and sympleotio groups, or rather their quotients by their centers, take A = Mn; sine0 X - + t X is an involution on A, the most general antiautomorphism of A is of the form X - - + . F - I . t X . . F , which is involutory if and only if t F = h F , with ~ in the center; this implies ~9.= 1, so that our involution is given by X - + F -1. t X . . F with F invertible and t.F - 4- .F. An automorphism X - + M - 1 X M of the algebra A commutes with that involution if and only if .F = t M . . F . M ; let G be the group consisting of such automorphisms. I f t $ , = _ F, n must be even, and the matrices M satisfying .F -= t . M . F . M are of determinant 1 (as follows from the consideration of the pfaffian) and make up a connected algebraic group, the symplectic group Sp(n) ; G is the quotient PSp(n) of that group by its center. It is known that in that case (7 has only inner automorphisms ; and we see, just as above, that every automorphism of G

[1960b]

419 ALGEBRAS W I T H I N V O L U T I O N S

595

can be derived in one a n d only one w a y f r o m an a u t o m o r p h i s m of the given algebra with involution. T a k e now the case tF ~ F ; as our underlying field is the universal domain, it would be no restriction to t a k e for F the u n i t - m a t r i x I n. The matrices M for which F = t M . F . M m a k e u p the o r t h o g o n a l group O(n), with two connected c o m p o n e n t s O + ( n ) , O - ( n ) consisting of the matrices M in the group with the d e t e r m i n a n t + I a n d - - 1 , respectively; a n d the connected c o m p o n e n t GO of the identity in 0 is the quotient PO+(n) of O+(n) b y its center. [f n is odd and ~ 3, the center of O(n), consisting of :t: 1~, h a s one element in each one of those components; therefore G is connected v~nd m a y be identified with O+(n). I t is k n o w n that, also in this c~se, G has only inner a u t o m o r p h i s m s , a n d our f u r t h e r conclusions are the same as before. Finally, t a k e the case in which n is even a n d ~ 4. T h e n the center of O(n), consisting again of 4- ln, is contained in O + ( n ) ; therefore G has two connected components. Hero it is k n o w n t h a t the g r o u p of inner a u t o m o r p h i s m s of G o is of index 2 in the g r o u p of all a u t o m o r p h i s m s of Go, e x c e p t for n =: 8, in which case it is of index 6. On the o t h e r hand, it is easily seen t h a t the inner a u t o m o r p h i s m s of (7 induced by elements of G~ determine on Go a u t o m o r p h i s m s which are not inner ones of G0. Therefore, if we leave aside the exceptional ease n = 8, we can again conclude t h a t e v e r y a u t o m o r p h i s m of G O can be derived in one a n d only one w a y f r o m an a u t o m o r p h i s m of the algebra w i t h involution A. N o w observe t h a t e v e r y semisimple algebra A o v e r the universal d o m a i n is a direct sum of m a t r i x algebras, a n d t h a t e v e r y involution of A m u s t either leave a c o m p o n e n t of A i n v a r i a n t or interchange it with a n o t h e r one. Thus, o v e r the universal d o m a i n , e v e r y semisimple algebra with involution is, in an obvious senso~ the direct s u m of algebras with involution of one o f the t y p e s discussed above. On the o t h e r hand, ff Go is the connected c o m p o n e n t O f the i d e n t i t y in the g r o u p of a u t o m o r p h i s m s o f the algebra with involution A, it is clear t h a t the a u t o m o r p h i s m s in Go m u s t t r a n s f o r m each c o m p o n e n t of A into itself. F r o m this

420

[1960b] 596

ANDRE WEIL

it follows immediately t h a t GO must be a direct product of groups of the various types considered above, and t h a t conversely e v e r y such p r o d u c t can be obtained in this manner. Some groups, however, are obtained in this m a n n e r more t h a n once, because of the well-known isomorphisms between groups of the various families ; those are as follows : (a)

M2 admits an inv~176

with J = ( --10 ~ )

which is invariant under all automorphisms and antiautomorphisms of M 2 ; this follows for instance from the fact that, for any invertible m a t r i x M in M2, we have

J-l.tM.J

----dot(M). M -1.

Therefore SL(2) is identical with Sp(2), hence PL(2) with PSp(2). (b)

P O + ( 3 ) is isomorphic with PSp(2).

(c)

PO +(4) is isomorphic with the product of itself.

(d)

PO +(5)

(e) PO +(6)

PO +(3)

with

is isomorphic with PSp(4). is isomorphic with PL(4).

In view of these circumstances, let us restrict our list of groups and algebras to the following: (I) (~roups: all semisimple groups, with center reduced to the neutral element, which, when decomposed into a direct product of simple groups, contain no factor isomorphic either to one of the exceptional groups or to P0+(8). (II) Algebras with involution: all semisimple algebras with involution which, when decomposed into a direct sum, consist of summands isomorphic to one of the following : (a) M~, $ M~ for n ~ 3, with an involution exchanging the two summands ; (b) M2n for n >~ 1, with the involution X - + j - a . eX. j determined b y an invertible alternating m a t r i x J ; (c) M s with the involution X - + tX, f e r n :=- 7 or n ~ 9.

[1960b]

421 ALGEBRAS WITH INVOLUTIONS

597

Then it follows from what we have p r o v e d t h a t each elm of the groups in ore. list is isomorphic to the connected component of the identity in the group of automorphisms of one of our algebras with involution, and that, if A and A' are two such algebras and G and G' are the corresponding groups, a n y isomorphism between G and G' is induced b y a uniquely determined isomorphism between A and A'. The latter s t a t e m e n t holds in particular for A == A', (7 -- (7'. 3. L e t (7 be a connected algebraic group, defined over the groundfield 1~. Lot us assume t h a t (7 is semisimplo, has a center reduced to the neutral element, and that, when G is decomposed into a p r o d u c t of simple factors over the universal domain, none of these factors is isomorphic to one of the five exceptional groups or to PO+(8). F r o m the results ill w2 it follows t h a t there is an algebra with involution A0, defined over the prime field, such t h a t the connected component of the identity Go in the group of automorphisms o f A 0 is isomorphic to G over the universal domain. Let f be an isomorphism of (7 onto Go; take for K a normal algebraic extension of/c, of finite degree, over which f is defined. Notations being now the same as in w1, f~ o f - 1 is an automorphism o f Go, defined over K. B y the results of w2, there is a uniquely determined automorphism $'~ of A o which induces the automorphism f " o f - 1 on Go; it is therefore invariant u n d e r all automorphisms of the universal domain over K ; the characteristic being 0, this implies t h a t it is defined over K. I t is obvious t h a t (F~) is a cocycle. Therefore, b y T h e o r e m 1, there is an algebra with involution A d e f n e d over b, and an isomorphism O, defined over K, of A onto A 0, such t h a t F , = (Pr (I)-1 for all a. Let G' be the connected component of tlm identity in the group of automorphisms of A ; r determines an isomorphism, which we again call (I), of G' onto G o ; t h e n ~ ----(I)-1 o / i s an isomorphism, defined over K, of (7 onto G'. Moreover, we have (I)~ o r ___f~ o f - 1 for all a. This can also be written as ~ = ~ . Therefore ~ is defined over b, and we m a y use it to identify G with G'. We have thus proved t h a t any group G

of the given type can be represented as the connected component of the

422

[1960b] 598

ANDRE W E I L

identity i~ the group of automorphisms of a 8emisimple algebra with involution defined over k. 4. We shall now determine when the connected component Go of the identity in the group of automorphisms of a semisimple algebra with involution (A, ~) is not a semisimple group. Over the universal domain, this is immediately a p p a r e n t from the the results of w3. I n fact, P L ( n ) is simple except for n = 1, in which ease it is reduced to 1; PSp(n) is defined only if n is even, and is always simple; and P O + (n) is semisimple except for n - - 1 and n = 2. Therefore Go is semisimple, and has a center reduced to the identity, provided no component of (A, ~) is isomorphic to M~ with the involution X - + tX. We shall say t h a t a semisimple algebra with involution is non-degenerate if it has no such component and if at the same time it has no c o m m u t a t i v e component. Then we have the following t h e o r e m : THEOREM 2. Let (A, ~) be a non-degenerate semisimple algebra with involution; let Go be the connected component of the identity in its group of automorphisms, and let Uo be the connected component of 1 in the multiplicative group of the elements u of A such that u'u----1. Then Go is a semisimple group, isomorphic to the quotient of Uo by its center; and the center of Go is reduced to the identity. We have already shown t h a t Go is semisimple a n d has a center reduced to the identity. As the group of automorphisms of the center Z of A is finite, e v e r y element o f Go must induce the i d e n t i t y o n Z and is therefore (by the classical theorem of Skolom-Noothor) a n inner automorphism x--+ v - 1 xv, with an invertible v in A. C_,allj(v) this automorphism ; ff we write t h a t it commutes with ~, we find t h a t z = v'v m u s t be in Z; it must then be in the multiplicative group H consisting of those invertible elements o f Z which are even for ~. Call V the group of those elements v o f A for which v'v is in H, and U the group of those elements u of A for which u'u = 1. Consider the homomorphism (h, u) -+ hu o f / / • U into V; one sees at once t h a t it maps H • U onto the group of those elements v of A for which v~v is i n / / 3 ; but, over the universal domain,

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423 ALGEBRAS W I T H

INVOLUTIONS

599

we h a v e H ~ --- H (since H is c o m m u t a t i v e a n d the characteristic is not 2), a n d therefore t h a t h o m o m o r p h i s m m a p s H • U onto V. F r o m this, it follows t h a t U a n d Y h a v e the same image j(U) = j ( V ) u n d e r j. As we h a v e seen t h a t G o is contained in j ( V ) , our conclusion follows. Incidentally, we observe t h a t the classical Cayley t r a n s f o r m a t i o n m a y be used to s t u d y the g r o u p U o of T h e o r e m 2. L e t us say t h a t an e l e m e n t x of an algebra with involution (A, ~) is even (:for ~) if x' = x, and odd (for t)if x ' = x; as the characteristic is not 2, A is the direct s u m of the spaces A +, A - of even a n d odd elements for e. N o w let u be a generic element of U 0 over a field of definition k for (A, ~) ; write w = (1 - - u). (1 + u ) - l ; t h e n w is an odd element of A for ,. Conversely, if w is a generic element of A - over ]r the formula u = (1 - - w).(1 ~- w) -1 defines a generic element o f U 0 o v e r k. Thus these formulas define a birational correspondence between U 0 and A S. E v e r y algebra with involution o v e r a field ]c can be w r i t t e n as (A~, ~), where (A, ~) is, in the sense explained above, a n algebra with involution, defined o v e r It; we h a v e already observed t h a t , if the latter is semisimple, the f o r m e r is semisimple, a n d conversely. We shall say t h a t the f o r m e r is non-degenerate if the latter is so. T h e relation between the structures of these two algebras will now be briefly discussed, particularly in order to find when (A k, ~) is non-degenerate. To decompose the semisimple algebra A~ over ]c into simple c o m p o n e n t s is the s a m e as decomposing A o v e r k, i.e. writing it a s a direct s u m o f sub-algebras, defined o v e r k, in such a w a y t h a t none o f the s u m m a n d s can be split a n y f u r t h e r in the same manner. E a c h s u m m a n d is t h e n t r a n s f o r m e d into itseff or interchanged with a n o t h e r one b y the involution L. T h u s it is enough to consider the case in which (Ak, ~) is simple, which m e a n s t h a t A~ is either simple or the direct sum of two simple algebras Bk, Uk interchanged

424

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by e. I n the latter case, let K be the center of Bk; call d the degree of K over/c; if we consider B k as a vector-space over K, its dimension must be of the form n 2. Then A splits into the direct sum of two algebras B, C, which m a y be regarded as the tensor-products o f B~ and C~ with the universal domain over ]c, and which are interchanged b y ~. Over the universal domain, B is the direct sum of d algebras isomorphic to M s. Degeneracy can only occur for n -- 1, i.e. when A~ is commutative. Next, assume t h a t A k is simple; let K be its center, and K + the set of all even elements of K for ~; K + is a field; let d be its degree over ]c. I f K is not the same as K +, it is an extension of K + of degree 2; call n 2 the dimension of A~ as a vector-space over K. Then A is, over the universal domain, the direct sum of 2d algebras isomorphic to Ms; b y considering the center of A, one sees at once t h a t none of these 2d components is invariant u n d e r ~. Therefore degeneracy occurs only for n = 1, i.e. again when A~ is commutative. Finally, assume t h a t A k is simple and t h a t ~ induces the identity on the center K of A k, so that, with the above notation, we have K = K +. Let d be the degree of K over ]c, and n ~" the dimension of A~ as a vector-space over K ; t h e n A is the direct sum of d algebras isomorphie to M~, each of which is invariant under ~; it is nondegenerate whenever n ~ 3. I f n = 1, A k is c o m m u t a t i v e and degenerate. I f n = 2, A k is a quaternion algebra over K (which m a y be isomorphic to M s (K)); let 8 be the dimension of A / a s a vectorspace over K. Then one sees at once that, in each one of the d simple components of A over the universal domain, the odd elements for e make up a vector-space of dimension 3; comparing this with the results of w2, we find t h a t we have 3 = 3 or 3 = 1 according as c, in each one of these components, is of the t y p e X ' ~ j - a . tX. j with t j = _ j or of the t y p e X ' = tX. Thus degeneracy occurs if and only if ~ : 1. One verifies easily t h a t this is so if and only if A~, as an algebra over K, can be generated b y an e v e n element u and an odd

element v such t h a t

uv =

-- vu, u ~ ~ K,

v 2 ~ K.

Thus a necessary and sufficient condition for the non-degeneracy of a simple algebra with involution (Ak, e) over /c is t h a t it should

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425 ALGEBRAS WITH INVOLUTIONS

601

not belong to the t y p e just described and t h a t A~ should not be commutative. A necessary and sufficient condition for the nondegeneracy of a semisimple algebra with involution (Ak, ,) over/c is then t h a t none of its simple components should be of either one of these types. II.

ALGEBRAS WITH INVOLUTION OVER THE REAL FIELD

6. F r o m now on, we shall not need a universal domain, as we shall be operating with algebras over a fixed groundfield, mostly the field R of real numbers. I f A is an algebra over any ground field/r we denote b y ' t r ' the trace of the regular representation of A. In other words, if L u is the ondomorphism x - + ux of the underlying vector-space to A, tr(u) is the trace of L u. Then tr(xy) is a symmetric bilinoar form on A • A ; according to a well-known criterion, it is non-degenerate if and only if A is absolutely semisimplo (i.e. if it is semisimple and remains so under a n y extension of the groundfiold). The trace tr(x) is invariant u n d e r all automorphisms of A ; if A is semisimplo, the right-hand and left-hand regular representations are equivalent, and t h e n the trace is also invariant under all antiautomorphisms o f A, or, as we shall say more briefly, all antimorphisms of A. I f u is a n y element of A, its " minimal polynomial " P is the polynomial o f smallest degree, with coefficients in the groundfield k, such t h a t P ( u ) = 0; P is also the minimal polynomial for the endomorphism L~ of the underlying vector-space to A. The " s p e c t r u m " Su of u is the set of all the distinct roots of P in the algebraic closure of k. L e t us say t h a t u is semisimple if L is so (or, what a m o u n t s to the same, if u generates an absolutely semisimple subalgebra of A) ; this will be the ease if and only if all roots of P are simple. Assume t h a t u is semisimple, and t h a t its spectrum S, is contained in k; let f be a n y k-valued function, defined on k or on a subset o f k containing S u. T h e n there is a polynomial Q, with coefficients in k, coinciding with f on S~; moreover, Q is uniquely determined modulo P ; therefore the element Q(u) o f A does not depend upon the choice of Q ; this will be denoted by f(u) ; f --+ f(u)

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is clearly an isomorphism of the ring of k-valued functions on Su onto the subalgebra of A g e n e r a t e d b y u. I f f is a k-valued function, defined on Su, the s p e c t r u m of f(u) is f(Su); and, ff g is a k-valued function, defined on f(Su), we h a v e g(f(u)) = h(u), with h == g o f. The t r a n s f o r m u ~ of art element u of A b y an a u t o m o r p h i s m or a n t i m o r p h i s m A of A has the same s p e c t r u m S u a n d the same m i n i m a l polynomial as u, and is somisimplo if u is so. Therefore, if S u is contained in k a n d f is a k-valued function on S u, we h a v e f ( u ~) = f(u) ~. L o t u be a n invertible element of an algebra A. We shall denote b y j(u) the inner a u t o m o r p h i s m determined b y u, i.e. defined b y the formula X ~

X j(u) "-- % - - l x u .

I f u, v are b o t h iuvertible, we h a v e j(uv) = j(u)j(v). I f A is a n y a u t o m o r p h i s m or a a t i m o r p h i s m of A, A-lj(u)A is the inner autom o r p h i s m j(u') with u ' = u ~ if ~ is an a u t o m o r p h i s m a n d u'---(u~) -1 if it is an a n t i m o r p h i s m . We shall write e(2) = 1 w h e n e v e r is an a u t o m o r p h i s m , e()t) = - - 1 whenever it is an a n t i m o r p h i s m , a n d define a s y m b o l u [~] b y the formula

so t h a t we have, in all cases

A-1 j(U) ~ =j(u[;~]). L e t A be a semisimple algebra, a n d let ~ be an a n t i m o r p h i s m of A. T h e n tr(x ~ y) is a non-degenerate bilinear f o r m on A x A. We h a v e (x ~ y)~ = y~ x ~, and hence tr(x ~ y) = tr(y ~ x~), which shows t h a t the bilinoar f o r m tr(x ~ y) is s y m m e t r i c if a n d only if ;~2 __ 1, i.e. if a n d only ff ~ is an involution. More generally, let A be a n a n t i m o r p h i s m of A; t a k e a e A, b e A, a n d consider the bilinear f o r m tr(ax ~ by). Obviously, it is non-degenerate if a n d only ff a, b are invortible, i.e. if t h e y are not zero-divisors. Assume now t h a t ~ is an involution, and t h a t a a n d b are invortiblo ; the formula

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427 ALGEBRAS WITH

I/~/VOLUTIONS

603

tr(ax ~ by) ~ tr(x "~bya) = tr(y ~ b~ xa ~) shows t h a t the f o r m tr(ax ~ by) is s y m m e t r i c if a n d only if, for all y, we h a v e bya = b ~ ya ~, i.e. y = b -1 b ~ ya ~ a -1 ; this is so if a n d only if we h a v e b ~ ~- bz - i , a ~ = az, where z is an invertible element in the center Z of A; as a~-----az implies a = z~ a ~, z m u s t satisfy z z ~ - 1 . L e t , be an involution of the semisimple algebra A. As tr(x' y) is a. non-degenerate s y m m e t r i c bilinoar f o r m on A • A, we can use it to a t t a c h an " a d j o i n t " L ' to e v e r y e n d o m o r p h i s m L of the underlying v e c t o r - s p a c e ; this is defined, as usual, b y the f o r m u l a : tr((Lx)' y) : t r ( x ' ( L ' y)). I n particular, the formula tr((ux) ~y) = tr(x~(u ~y)) shows t h e n t h a t the adjoint o f L~ is L ~ , a n d in particular t h a t L~ is self-adjoint if and only if u = u '. LEMMA 1. Let (A, ~) be a semisim21e algebra with involution over a field k of characteristic O. Let A +, A - be the subspaces of even and of odd elements of A f o r ~. T h e u A + and A - are the orthogonal complements of each other for each one of the symmetric bilinear f o r m s tr(xy) and tr(x' y). The formula t r ( x y ) = tr(y ~x ~) : tr(x' y~) shows at once t h a t , if one of the two elements x, y is in A + a n d the o t h e r in A - , 2tr(xy) a n d therefore tr(xy) m u s t be 0. Conversely, assume for instance t h a t x is orthogonal, with respect to tr(xy), to all vectors y such t h a t y~-=- 9 where e is + 1 or - - 1. T h e n it is orthogonal to y + e y' for all y ~ A, so t h a t we h a v e 0 = tr(x(y + 9 y~)) ----tr((x ~- e x~)y) for all y c A , a n d therefore x + 9 0 since tr(xy) is non-degenerate. The p r o o f for tr(x ~y) is quite similar. Of course L e m m a 1 remains valid for every groundfield of characteristic other t h a n 2, provided tr(xy) is non-degenerate, i.e. provided A is assumed to be absolutely somisimple.

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7. F r o m now on, we shall deal exclusively with semisimple algebras o v e r R; such a n algebra is isomorphic to the direct s u m of m a t r i x algebras o v e r R, o v e r C (the field of complex numbers) and over K (the division-algebra of quaternions). I f A is a n y algebra o v e r R, we say t h a t a n involution ~ on A is positive if tr(x~x) > 0 for all x other t h a n 0 in A ; tho existence of such an involution implies t h a t tr(x~y) is non-degenerate and therefore t h a t A is semisimple. PI~O1,OSITION 1.

Let A be a semisimple algebra over R; then there exists at least one positive involution on A; and all positive involutions on A coincide on the center Z of A. W r i t e A as a direct s u m of simple c o m p o n e n t s A i. A positive involution :r on A m u s t t r a n s f o r m each A i into itseff; for, if it t r a n s f o r m e d Ai, say, into Aj with j ~=i, x~x would be 0 for all x in A s. F r o m this one concludes a t once t h a t it is enough to p r o v e our proposition in the case o f a simple algebra A, which we can write as a m a t r i x algebra Mn(D ) over a division algebra D which can b e R , C o r K . I f D i s R o r K , the center Z i s R , a n d e v e r y a u t o m o r p h i s m or a n t i m o r p h i s m of A induces the identity on Z. I f D = C, we h a v e Z = D, a n d e v e r y a u t o m o r p h i s m or a n t i m o p r h i s m of A m u s t induce on Z either the identity or the a u t o m o r p h i s m z - + ~ , where ~ is the i m a g i n a r y conjugate of z; as we h a v e tr(z) = n 2 (z-]-z) for e v e r y z e Z, a n y positive involution :r on A m u s t induce z - + z on Z, fur, if it induced the identity, one would h a v e tr(z~z) = n~(z ~ + ~ ) , a n d this is < 0 for z = i. This proves the second assertion in our proposition. As to the first one, observe t h a t x~ > 0 for all x :/: 0 in D i f w e write x for x if D ----R, for the i m a g i n a r y conjugate of x if D = C, a n d for the q u a t e r n i o n conjugate of x if D = K ; also, in all three cases, x - + ~ is an involution on D, and X - + t 2 ~ is as usual, denotes the transpose p u t r(X)----Z~xi~; if, as always, regular r e p r e s e n t a t i o n of M~(D) seen t h a t we h a v e

an of we as

involution on M~(D) ff t~, X. F o r X----(x~j) in M~(D), denote b y t r the trace of the a n algebra over R, it is easily

[1960b]

429 ALGEBRAS ~VITH I N V O L U T I O N S

605

tr(X) = ~, [~-(x) + ~(x)]

with ~ equal to n/2 i f D = R , to n i f D = C , Therefore, if X -- (a0), we have

and t o 2 n i f D = K .

tr(t~. X) = 2y . ~ xox~, which shows that X--+tX is a positive involution on Mn(D ). This completes the proof. COROLLARY. I f ~, fl are two positive involutions on an al9ebra A over R, or is an inner automorphism of A. The assumption implies that A is semisimplo and that a-1fl is an automorphism of A; by prop. 1, it induces the identity on th0 center; b y the Skolem-Noethor theorem, it must therefore be an inner automorphism. If A is any semisimplo algebra over R, the automorphism of the center Z of A induced on it by all positive involutions of A will be denoted by z - + ~ ; if Z~ is any simpl0 component of Z, z --~ ~ induces the identity on it if it is isomorphic to R, and the imaginary conjugate if it is isomorphic to C. Let ~ be a positive involution on the algebra A over R. We have seen above that, ff u----u =, L~ is seff-adjoint for tr(x~y), i.e. for the quadratic form tr(x~x); as the latter is positive, it follows from well-known theorems that u is then semisimplo and has a real spectrum. Now assume that an element a of A is such that the bilinear form tr(x=ay) is symmetric, non-degenerate and positive; as it is symmetric and non-degenerate, a must be invertible and even for ~; that being assumed, the formula

tr(x=ay) = tr((L~x)~y) shows that the positivity of the bilinear form tr(x~ay), i.e. that of the quadratic form tr(x=ax), is the same thing as the positivity of the seff-adjoint operator L~ for the form tr(x~x); and this is positive if and only if all the characteristic roots of La(i.o., all the elements of its spectrum S~) are > 0. When that is so, we say that a is positive for ~; and we denote by P(~) the set of all such

430

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ANDRE WE.IL

elements. F r o m the results of no. 6, it follows t h a t , if f is a n y real-valued function, defined on the set of real n u m b e r s > 0, f(a) is defined for all aEP(~), a n d satisfies f(a) ~ = f(a); this will be so, in particular, for f(t) = t p, for a n y p c R, a n d for f(t).= log t. I f f(t) > 0 for all t > 0, ~hen f(a) is in P(~) for all a eP(~), since the s p e c t r u m of f(a) is the imago u n d e r f of the s p e c t r u m of a. Thus, for e v e r y p ~ R , we h a v e a m a p p i n g a - + a p of P(~) into itself. F o r all p , p ' i n R , wohaveaP+P'=aPa r a n d (aP) r = a pC. I n particular, we note that, for a eP(~), the only b ~P(a) such t h a t b 9" = a i s

b

=

a 1/2.

PROPOSITION 2. Let o~ be a positive involution on an algebra A over R. Then the set P(a) of positive elements for r162is a convex open subset of the vector-space of even elements for r162 ; and the group A* of invertible elements of A operates on P(~) by (x,a) ~ x~ax. I t is clear t h a t P(~) is convex. T a k e a eP(~) ; if p is the smallest element o f the s p e c t r u m S~ of a, i.e. the smallest charactreistie value of La(with respect to the quadratic form tr(x~x)), we h a v e p > 0 and

tr(x~ax) ~ P tr(x~x) for all x E A. Now, in the space of even elements o f ~ in A, i.e. of elements u of A such t h a t u ~ = u, t a k e a neighborhood U of 0 such t h a t , for e v e r y u E U, all characteristic values of L,~ are > - - p a n d < p ; then, for e v e r y u ~ U, a q- u is in P(~); this shows t h a t P(~) is open in the space of even elements. The last assertion in our proposition is obvious. P~OPOSITION 3. Let ~ be a positive involution on an algebra A over R ; let ~ be any automorphism or antimorphism of A. Then ~-1 ~ is a positive involution on A; and ~ maps P(cr onto P(h-I~2). The first assertion is obvious. F u r t h e r m o r e , we h a v e a e P(a) if a n d only if tr(x ~ ay) is s y m m e t r i c and positive. As the trace is i n v a r i a n t u n d e r 2, this is the same as to say, if ~ is an a u t o m o r p h i s m , t h a t tr(x ~ a ~ y~) is s y m m e t r i c a n d positive ; ~his p r o p e r t y will be unaltered if we replace x, y b y x ~-1, y~-l, a n d is therefore equivalent

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431 ALGEBRAS V~ITH INVOLUTIONS

607

to the s y m m e t r y and positivity of tr(x ~-1~ a ~ y), which proves our proposition in this ease. Similarly, if ;~ is an antimorphism, a 9 P(r162 is equivMent to the s y m m e t r y and positivity o f tr(y a a a x~); replacing x, y b y y~-l~, x~-l~, we get our conclusion as before. 8. We can now determine as follo~s the set of all positive involutions on a semisimple algebra A over R. PROPOSITION 4. Let cr be a positive involution on an algebra A over R. Then, for every a e P(r162 the formula fl

:

~j(a)

:

j(a -1/~) (zj(a 1/2)

determines a positive involution fl on A . Conversely, i f fl is any positive involution on A , it can be expressed by that formula with an a 9 P(~). For every a e P(:r the element b = a 11~is also in P(r162 and the formulas of no. 6 give j(a) = j ( b ~) and r162j(b-1) :r = j ( b ) ; this gives at once ~j(a) = j ( b -1) r162 ; as this is the transform of ~ by the automorphism j(b), it is a positive invohttion. The p r o o f of the converse will be derived from the following lemma : Lv,~MA 2. On every semisimple algebra A over R, there is a positive involution ~ such that, i f a and b are any two elements of A , the following properties are equivalent: (i) the bilinear form tr(ax ~ by) in x, y is no~-dege~erate, symmetric, and positive ; (ii) there is an element z of the center Z of A such that z~z = l, z a e P ( ~ ) , z - t b 9 P(r162

One can see at once t h a t (ii) implies (i) whenever ~ is a positive involution on A. I n fact, as tr(ax: by) does not change if we replace a, b b y za, z -1 b with z E Z, we m a y replace the assumption (ii) by the assumption t h a t a, b are in P(~). We have already seen

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ANDRE W E I L

in no. 6 t h a t tr(ax ~ by) m u s t t h e n be n o n - d e g e n e r a t e a n d s y m m e t r i c ; p u t t i n g n o w u ---- a 1/9, v = b 1/2, we h a v e t r ( a x ~ bx) = t r ( u 2 x ~ v 2 x) = t r ( ( v x u ) ~ v x u ) • 0 since u ~ : u, v ~ = v, w h i c h p r o v e s t h e positivity. Now, in order t o c o n s t r u c t a n i n v o l u t i o n for w h i c h t h e c o n v e r s e is true, it is clearly e n o u g h to consider the case in w h i c h A is simple, i.e. of the

f o r m M n ( D ) with D = R ,

the involution X-+tX

C or K ; we shall s h o w t h a t

has t h e n t h e required p r o p e r t y .

I n fact,

a s s u m e t h a t A, B are t w o m a t r i c e s in Mn(D), such t h a t t r ( A t X JBY) is n o n - d e g e n e r a t e , s y m m e t r i c a n d positive; as we h a v e seen in no. 6, t h e first t w o a s s u m p t i o n s i m p l y t h a t A, B are invertible a n d t h a t t ~ = z - 1 1~, A = z A , where z is an invertible e l e m e n t o f the center, i.e. w h e r e z is a n o n - z e r o scalar, a n d a real one if D is R or K. Moreover, z m u s t satisfy z.~ -- 1, w h i c h implies z = 4- 1 if D = R or K. I f D ~ C, t a k e ~ e C such t h a t ~2 = z, a n d p u t A 1 = CA, B1 = ~ - I B ; t h e n A 1, B 1 also satisfy (i), a n d we h a v e t ~ 1 = A I , tJB1 = B1; therefore, after so m o d i f y i n g A, B in t h e case D----C if necessary, we m a y a s s u m e t h a t t ~ __ 9 A, t/~ = 9 B, with 9 = 4- 1 for D = R or K a n d 9 = 1 for D ----C. N o w t h e p o s i t i v i t y a s s u m p t i o n in (i) m e a n s t h a t t r ( A t X B X ) :> 0 for all X :/= 0; p u t t i n g t = r ( A t X B X ) , w h e r e v, as above, d e n o t e s t h e usual t r a c e o f a m a t r i x , this can be w r i t t e n as t + t > 0. B u t , p u t t i n g W = A t X B X , t = ~(tW) = T(tX. 9 B . X .

we h a v e

9 A ) = t,

so t h a t o u r p o s i t i v i t y a s s u m p t i o n a m o u n t s t o v ( A t X B X ) > 0 fo X ~ 0. T a k i n g t w o c o l u m n - v e c t o r u = (u~), v = (vi), p u t :

f ( u ) = q~Au, g(v) = ~vBv ; these are m a t r i c e s o f o r d e r 1, i.e. scalars (in D). T h e p o s i t i v i t y a s s u m p t i o n , applied to t h e m a t r i x X ----v. tu ---- (vi ~j), gives

9 (A ~ X B X ) = ~(f(u) g(v)) =Au) g(v)> 0 for all u ~: 0, v r 0. I f D = R a n d 9 -- - - 1, we h a v e f ( u ) = O, g(v) = 0 for all u, v; therefore this case c a n n o t occur. I f 9 ---- 1, f ( u ) a n d g(v) are real ; t h e y m u s t h a v e t h e s a m e sign for all u r 0, v ~ 0, so t h a t ,

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after replacing A, B b y -- A , - - B if necessary, we can write our assumption as f ( u ) > 0 for all u ~ O, g(v) > 0 for all v ~ 0; as one sees immediately, this implies t h a t A, B are positive for the involution X --+ *X. Finally, if D = K and r ------ 1, p u t q = f ( u ) , r = g(v); t h e n we have q = -- q va 0, 7 ---- -- r ~ 0; applying the positivity assumption to u, 0v with 0 e K, we see t h a t we must have q 0r 0 > 0 for a l l 0 va 0; as one can always find 0 such t h a t 0 r this is impossible.

0------q,

We can now complete the proof of Prop. 4 for the ease of an involution 0r having the p r o p e r t y stated in L o m m a 2. L e t fl be a n y positive involution on A. B y the corollary of Prop. 1, r162 fl is an inner a u t o m o r p h i s m j ( a ) of A, so t h a t we m a y write fl ~ ~j(a) with an invertible a. As we have seen in no. 6, the antimorphism fl = ~j(a) is an involution if and only if tr(x~y) is symmetric ; therefore it is a positive involution if and only if the bilinear form tr(x~y) = t r ( a - a x~,ay) is non-degenerate, symmetric and positive, i.e. if (a -a, a) satisfies condition (i) of the lemma. B u t then, b y our assumption on a, there is z e Z such t h a t z - a a is in P(:r Replacing a b y z - a a does not change j(a) ; we have therefore proved t h a t fl is of the form ~j(a) with a e P(~), hence also of the form j(b) - 1 ccj(b) w i t h b = a 11~. This proves Prop. 4 for the particular :r which we have been considering. I t also proves t h a t all positive involutions on A are transforms of this involution ~ b y inner automorphisms. Since the p r o p e r t y of expressed b y Prop. 4 is obviously invariant under automorphisms, our proof is thus complete. One m a y observe t h a t the p r o p e r t y of ~ expressed in L e m m a 2 is also invariant u n d e r automorphisms ; therefore (i) and ( i i ) a r e equivalent whenever ~ is a positive involution. 9. B y Prop. 4, if ~ is a positive involution on A, the mapping a - + ~j(a) maps P(~) onto the set of all positive involutions on A ; this shows in particular t h a t the latter set is connected if it is provided with its natural topology as a d o s e d subset of the space of all endomorphisms of the underlying vector-space to A.

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One can also define as follows a one-to-one mapping, and more precisely a h o m e o m o r p h i s m , of a closed subset of P(cr onto the set of positive involutions on A. We observe first t h a t , if ~, fl are two positive involutions, a n d if we write fi as the t r a n s f o r m of ~ b y a n inner a u t o m o r p h i s m j ( b ) of A, then, b y Prop. 3, P(fl) is the imago of P(~) u n d e r j(b). I n particular, if Z is the center of A, and if we write P ( Z ) = Z c~ P(~), P ( Z ) is independent of the choice o f ~. I f A is the direct sum of the simple algebras Ai, and if, for each i, ei is the unit-element of A i, it is easily seen t h a t P ( Z ) consists of the elements Z~ ti ei where all the ti are real a n d > 0.

Let ~r be a positive involution on the algebra A over R; let a, a' be two elements of P(~r Then we have j(a) -~j(a') if and only if a - la' is in P(Z). LEMMA 3.

P u t b = a 1/2 a n d z---- a - 1 a'. I f z is in Z, we h a v e az-= b~zb ; b y Prop. 2, if this is in P(a), z m u s t be in P(~), hence in P(Z). The converse is obvious. Now, calling again A i the simple components of A, a n d ei the unit-element of A i for each i, write N~ for the n o r m of the regular r e p r e s e n t a t i o n of A i o v e r R ; if d~ is the dimension of A i o v e r R, we have, for t e R , Ni(tei) -- t% F o r e v e r y x i e A i , write vi(xi)-([Ni(xi) ll/di)ei; and, for e v e r y x = Y , i x i i n A, with x i e A i for all i, write v(x) : Z i vi(xi). Then v is a m a p p i n g of A into P(Z), inducing the identity on P ( Z ) a n d such t h a t v(xy) = v(x)v(y) for all x, y in A. Therefore, for e v e r y invertible x in A, there is one a n d only one element z of P ( Z ) such t h a t v(zx) = 1, viz. z = v(x) -1. F u r t h e r m o r e , if a is a semisimple elemenr of A with a positive s p e c t r u m (e.g. if a is in P(~) for some positive involution cr we h a v e v(a~) : v(a) p for all p e R ; for this ,is t r u e if p is a n integer, hence if it is rational, a n d therefore b y continuity in the general case. I f ~ is a n y a u t o m o r p h i s m or a n t i m o r p h i s m of A, we h a v e v(x ~) = v(x) ~ for all x e A. We shall denote b y PI(~.) the sot of all elements a of P(~) such t h a t v(a) ---- 1. Combining Prop. 4 with L e m m a 3 a n d some trivial topological considerations, we get t h e following 9

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I~OPOSITIO~ 5.

Let :r [J be two positive involutions on the algebra A over R. Then there is one and only one element a of PI(~) such that fl = r162 ; and an element a' of P(~) is such that fl = :r i f and only i f it is of the form az with z e P ( Z ) . Moreover, the mapping a----~:cj(a) induces on P1(:r a homeomorphism of PI(U) onto the set of all positive involutions on A .

10. W e shall d e n o t e b y ~ t h e g r o u p o f all a u t o m o r p h i s m s a n d a n t i m o r p h i s m s o f the semisimple a l g e b r a A o v e r R. F o r )~ e ~ , t h e n o t a t i o n e(•), u [~] will be used in t h e sense explained in no. 6. L~MA 4. Let r162 be a positive involution on A ; let ~ be an element of f~ commuting with ~. Then, for a ~ P(~), the involution ~j(a) commutes with ~ i f and only i f a TM ----za with z ~ P ( Z ) ; for a E Pi(~), ~ j(a) commutes with ~ i f and only if a TM = a. Clearly, aj(a) c o m m u t e s w i t h ~ if a n d o n l y i f j ( a ) does so, i.e., b y t h e f o r m u l a s o f no. 6, ff a n d o n l y if j ( a TM) is t h e same as j(a) ; this a m o u n t s t o s a y i n g t h a t a TM m u s t be o f t h e f o r m za w i t h z E Z. B u t , as ~ c o m m u t e s w i t h ~, it t r a n s f o r m s P(~) into itseff (by P r o p . 3), so t h a t a TMis in P(~) w h e n e v e r a is in P(m). A p p l y i n g L e m m a 3, we get t h e first assertion in o u r l e m m a . I f v(a) = 1, we h a v e v(a TM) = 1, hence u(z) = 1 for a TM -~ za ; for z ~ P(Z), this implies z = 1. LEMMA 5.

I f tWO positive involutions ~, fl commute with each other,

they coincide. I n L e m m a 4, t a k e )~ = ~, a n d w r i t e fl as ~j(a) with a E P I ( ~ ) ; w e get a -1 --~ a, i.e. a ~ -~ 1, hence a ---- 1. I t is clear t h a t e v e r y element )~ o f fr t r a n s f o r m s a positive i n v o l u tion ~ into a positive i n v o l u t i o n 2 ~ - ~ , to w h i c h we can a p p l y t h e results p r o v e d a b o v e ; in particular, we can write it as ~j(a(h)) w i t h a (~) e PI(~). W e shall n e e d t h e following p r o p e r t y o f t h e m a p p i n g - + a(%) :

Let ~ be a positive involution on A . For every ~ ~ ~ , let a(h) be the element of P~(a) such that %:r = =j(a(%)). LEMMA 6.

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Then, for all ~, ix in ~, we have a(A~) = aC~). a(~)t~-11. Call a ' t h e r i g h t - h a n d side o f t h e f o r m u l a to be p r o v e d . B y the definition o f a(~), a(~), we h a v e

~(galx-1) A-1 = }t~}t-~. A j(a(~)) Z -1 = ~j(a(A)) j(a(/x) t~-ll) = aj(a') so t h a t we m u s t h a v e a(Atx) = za', w i t h z in t h e c e n t e r Z. particular, if we replace t L b y A- I , we see t h a t we m u s t h a v e

In

a(~ -~) = z~ a(A)-E~l with z, ~ Z. P u t a = a(A), b = al/2; mj(a) is t h e same as j(b) -~ o:j(b), a n d therefore t h e definition o f a = a(~) can be expressed b y s a y i n g t h a t j(b)A c o m m u t e s w i t h ~ ; therefore it t r a n s f o r m s P(~) into itself. I n p a r t i c u l a r , a j(b)~ must" be in P(~) ; this is no o t h e r t h a n a *. F r o m this it follows a t once t h a t a -[~l is in P~(~) ; in t h e a b o v e f o r m u l a , z, m u s t therefore be equal to 1, a n d we h a v e : a(A -1) = a()l)-[~], which is n o t h i n g else t h a n t h e special case /x = A-1 o f t h e f o r m u l a to be p r o v e d . N o w we go b a c k to t h e g e n e r a l case. W e h a v e s h o w n t h a t a(A/x) : za', w i t h z ~ Z. L e t us w r i t e t h a t za' is in P(~) ; this a m o u n t s to s a y i n g t h a t t h e bilinear f o r m F(x, y) = tr(x ~ za()t) a(/x) [~-11 y) = ~r(a(2)x ~ - 1

za(/x) [~-11 y)

is s y m m e t r i c , n o n - d e g e n e r a t e a n d positive. As t h e t r a c e is i n v a r i a n t b y 2, F(x, y) can be w r i t t e n , if )t is a n a u t o m o r p h i s m , as

F(x, y) = tr(a(A) ~ x ~ z ~ a(/x) y~). B u t in t h a t case a(A) ~ is t h e s a m e as a(~l-1) -1 a n d is therefore in P(~) ; w r i t i n g it as c * w i t h c e P(~), we get

F(x, y) -= tr((x * c) ~ z ~ a(/x) (y~ c)) ; t o s a y t h a t this is s y m m e t r i c , n o n - d e g e n e r a t e a n d positive is to s a y t h a t z ~ a(/~) is in P(~) ; as a(/x) is in P ( a ) , z ~ m u s t therefore be in

P(Z). Similarly, if )t is a n a n t i m o r p h i s m , we use t h e fact t h a t a(2) ~

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is the same as a()t -1) in order to write it as c 2, with c e P(~), and t h e n write F as F ( x , y) = t r ((y~ e) ~ a(/i) -1 z ~ (x ~ c)), again with the same conclusion as before, viz. z ~ E P ( Z ) , i.e. z ~ P ( Z ) . Since obviously v(z) is 1, this implies z = 1, which completes our proof. 11. T~EORE~ 3. Let A be a semisimple algebra over R ; let K be a compact subgroup of the group of all automorphisms and antimorphisms of A . T h e n there is at least one positive involution on A which is invariant under K , i.e. which commutes with every element

elK. Assume first t h a t K is a group of automorphisms of A ; and choose on A a positive involution ~. As in L e m m a 6, we call a()t), for )~ ~ K, the elemen~ of Pi(a) such t h a t Ag~t- i -- ~ j(a(A)).

B y Prop. 5, this is uniquely defined, and A - + .a(2) induces a continuous mapping of K into P(~) ; b y L e m m a 6, this satisfies the relation a()l/i) = a (2). a (/i) ~-i for all A,/~ in K. Now p u t a : [ a(/~) d/z, ,1

K

where d/~ denotes the H a a r measure on K. As P(~) is open and convex in the vector-space of symmetric elements for a, a is in P(a). If, in the integral which defines a, we replace /z b y A/~ with a fixed A e K, and apply the above formula for a ()t/~), we get a -~ a (~). a ~'-1. On the other hand, we have ,~:r

A- i -~ r162

) )~j(a) )~-i : ~j(a(h)a a-i) ;

these two formulas, taken together, show t h a t the positive involution ~j(a) commutes with ~. As A is a r b i t r a r y in K, this proves our conclusion in the present case.

438

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N o w assume t h a t K contains a t least one a n t i m o r p h i s m p; then, if K o is the group of all a u t o m o r p h i s m s of A contained in K , we h a v e K = K 0 u p K 0, p 2 ~ K o a n d K o p - - p K o. B y w h a t we h a v e p r o v e d above, we can choose a positive involution ~ w h i c h ' c o m m u t e s with e v e r y element of K-o. I f we write, as before, ~ - 1 in the form aj(a(2) ), with a()t) e Pl(a), for e v e r y A c K , we h a v e a(2) = 1 for A e K 0. P u t a = a ( p ) ; b y L e m m a 6, we h a v e a ( p A ) = a for all A E K 0 , i . e . a ( / ~ ) = a for all / z r 0. As we h a v e p K 0 = K op, this gives, for ~ ~ K0 :

a "= a(~ p) = a(,~), a(p) ~-1 = a ~-1, a n d therefore a = a ~. Similarly, again b y L e m m a 6, we have 1 = a(p 2) = a.(aP) - 1,

a n d therefore a -- a p. Now p u t b -- a 1/2 and fl = aj(b). B~ L e m m a 4, f~ o o m m u t e s with all elements of K 0. Moreover, we h a v e

p fl p - 1 = ~j(a). p j(b) p - ~ = aj(a) j(b[p-ll). As a ==-a p, we h a v e b = bp ; the r i g h t - h a n d side of the last formula is therefore equal to ~j(b), i.e. to ft. This shows t h a t fl c o m m u t e s with all elements of K . CogencY. Let (A, t ) b e a 8emisimple algebra with involution over R. Let K be a compact subgroup of the group of automorphisms of (A, t). Then there is a positive involution on A which commutes with t and with all elements of K. Our a s s u m p t i o n s i m p l y t h a t K U t K is a c o m p a c t subgroup o f the g r o u p of all a u t o m o r p h i s m s and a n t i m o r p h i s m s of A ; our assertion is therefore an i m m e d i a t e consequence of T h e o r e m 3. 12. F r o m now on, we shall deal with a somisimplo algebra with involution (A, ~) o v e r R ; a n d we shall denote b y G its g r o u p of a u t o m o r p h i s m s . B y T h e o r e m 2 of P a r t I, the connected c o m p o n e n t o f the i d e n t i t y in G is semisimple if (A, t) is non-degenerate; and it h a s been explained in P a r t I hi w h a t sense one m a y say t h a t " a l m o s t all " somisimple real groups can be obtained in this manner.

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The corollary of Theorem 3, applied to the case in which K is reduced to the identity, shows t h a t there is at least one positive involution on A which commutes with ~. F r o m the results proved above, we can also deduce at once the following : PROPOSI~ON 6. Let (A, ~) be a semisimple algebra with involution over R. Let ~, fl be two positive involutions on A, both commuting with ~. Then there is one and only one element a of PI(~) such that fl = ~j(a), and it is such that a ' ~ - a - l ; conversely, for every such element a, aj(a) is a positive involution commuting with ~. Furthermore, we have fl-~j(a-1/~):cj(al/~); and the mapping p-+j(aP), for p E R~ is an isomorphism of the additive group R onto a one-parameter group of automorphisms of (A, ~). This is in fact an immediate consequence of Prop. 5 and L e m m a 5. The set of all positive involutions of A commuting with ~, provided with its " n a t u r a l " topology (as explained in no. 9) will be denoted b y R ; the group (7 operates on it b y the law (~,~)-+~-~.~

()~e(7, ~ e R ) ,

and Prop. 6 shows t h a t it operates on it transitively. We shall denote b y K(~) the subgroup of G consisting of the elements of G which commute with ~; R is therefore isomorphic to G/K(~). I t will be shown t h a t K(~) is a maximal compact subgroup of G, so t h a t R is essentially the Riemannian symmetric space attached to G. We first show t h a t R is homcomorphie to an open cell. This will be done b y defining a " t r a c e operator ", corresponding to the " n o r m o p e r a t o r " v defined in no. 9. Write once more A as the direct sum of the simple algebras A i ; call Z~ the center o f A i and ni 2 the dimension of A i as a vector-space over Z i ; denote b y S i the trace in A t over Z~, i.e. the trace of the regular representation of A i considered as an algebra over Z i. Lot x be a n y element of A ; write it as x --- Zixi, with x, e A i for all i; we p u t :

a ( x ) : a ( ~ . , x i ) = ~ n~-~Si(xi) ;

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a is then a linear mapping of A into Z, which induces the identity on Z ; and, if A is a n y a u t o m o r p h i s m or antimorphism of A, we have a(x ~) ~- a(x) ~ for all x in A. Furthermore, we write

~0(x) = ~[~(x) + ~(x)] where z - + ~ is as defined in no. 7. I f u is a n y element of A, and if e~ is defined as Z u~/n !, (ro(u) is nothing t h a n log v(e u) ; we need only a special case of this, which we formulate as a lemma :

LEMMA 7. Let u be a semisimple element of A with real spectrum ; then we have a o (u) ----log v(e~). I t is clearly enough to consider the case in which A is simple ; its center Z can be identified with R or (2. Let d be the dimension of A over R ; d. a 0 is then the trace of the regular representation of A over R, while, for a n y x e A, d. log ,(x) is the same as log I N (x)[, where N is the n o r m o f the regular representation of A over R. I f u is a semisimple element of A with real spectrum, we can, b y choosing a suitable basis for A, write L~ as a diagonal m a t r i x ; let r 1, ..., r a be its diagonal coefficients. Then, if v = e~, L~ is the diagonal m a t r i x with the diagonal elements e% I n the regular representation of A over R, the trace of u is Zr~ and the n o r m of v is IIe% This proves the lemma. I n particular, let ~ be a positive involution on A ; e v e r y even element u for ~ is semisimple with real spectrum, and, if we put, for such an element, v = e~, v is in P(~) ; conversely, u is given in terms of v by u = log v. Then L e m m a 7 shows that, for such a pair u, v, the relations % ( u ) = 0, v(v) : 1 are equivalent. On the other hand, if u, v is such a pair, and if ~ is a n y involution on A, the relations u ' = - - u , v ~ - - v -1 are equivalent. Combining this with Prop. 6, wo got the following "decomposition t h e o r e m " : THEOREM 4. .Let G be the group of automorphisms of a semisimple algebra with involution (A, ~) over R . Let ~ be a positive involution on A , commuting with e ; let K(r162 the subgroup of G consisting of the elements of G which commute with r162; let W be the vector-space of the elements w of A such that w ~ ~ w, %(w) -~ O, w' -=- -- w. T h e n

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the mapping (w, ~) -+j(ew)~,for w E W, ,~ ~ K(~), is a homeomorphism of W • K(a) onto G. F o r w e W, p u t c -- ew ; f r o m w h a t we h a v e seen above, it follows t h a t t h e m a p p i n g w - + c = ew is a h o m e o m o r p h i s m o f W o n t o t h e set o f all elements c o f Pa(~) such t h a t c r = e -1. Then, b y L e m m a 4, j(e w) is in G, so t h a t j(ew)2 is in G for e v e r y )~ in K(~). L e t n o w /Z be a n y e l e m e n t o f G ; t h e n /Z g/z- 1 is a positive involution c o m m u t i n g w i t h ~, a n d can t h e r e f o r e be w r i t t e n as ~j(a) with a e PI(~), a ' = a - ~. P u t t i n g c = a - ~/2, we g e t

/Z ~ /Z-1 : o~j(a) =: j(e):cj(c) - ~, w h i c h s h o w s t h a t ~ = j(e)-I/Z c o m m u t e s with g ; as /Z a n d j(c) b o t h c o m m u t e w i t h ,, 2 is t h u s in K(~). P u t t i n g w = log c, we get /Z =j(ew)2. This d e c o m p o s i t i o n o f /Z is u n i q u e ; f o r , if we h a v e / Z ----j(c') ?t' with c' e PI(~), c ' ' : - c ' - 1 , ~' e K(~), then, w r i t i n g t h a t )~' c o m m u t e s w i t h ~, we get

/z ~ / z - ~ = j ( ~ ' ) ~j(c')-~ =

~j(c '-~)

which, c o m p a r e d with t h e f o r m u l a w r i t t e n above, gives c ' - * = a a n d t h e r e f o r e c' = a-1/2, b y P r o p . 5. W e h a v e t h u s p r o v e d t h a t t h e m a p p i n g in o u r p r o p o s i t i o n is bijective. Trivial topological considerations will t h e n show t h a t it is bicontinuous. COrOLLArY. Let (A, e) be a semisimple algebra with involution over R. Let R be the set of all the positive involutions on A which commute with ~. Then R, provided with its natural topology, is homeomo~Thic to an open cell. As we h a v e o b s e r v e d above, P r o p . 6 implies t h a t identified w i t h G/K(:r

R can be

if ~. is a n y positive i n v o l u t i o n c o m m u t i n g

w i t h ~ on A. L e t f be t h e canonical m a p p i n g o f G o n t o G/K(a). T h e o r e m 4 s h o w s t h a t t h e m a p p i n g w-+f(j(ew)) is a h o m e o m o r p h i s m o f W o n t o G/K(a). I f the i n v o l u t i o n , induces the i d e n t i t y on the center Z o f A, it is e a s y t o see t h a t t h e relation a ' - - - - a - a , for a n e l e m e n t a o f P(~), implies v ( a ) = 1; a n d similarly t h e relation w ' - - - - - - w, for

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an even element w for ~, implies ao(W) -~ O. I n t h a t case, it is frequently advantageous, instead of the transcendental mapping w = log c, to use the Cayley transformation t -- (1 -

~). (1 + c) - 1 ,

c -

(1 -

t ) . (1 + 0 - 1 ,

where t satisfies t = t ~ = - - t ' and the inequalities t tr(x ~ tx) i < tr(x~x) for all x r 0 in A ; the set T of these elements t of A is a convex open subset of the space determined b y t = t ~ = - t', and one sees, just as in the proof of Theorem 4 and its corollary, t h a t the formulas written above determine a homeomorphism between T a n d R. 13. W h e n dealing with algebraic groups over R, one must be careful to distinguish between connected components in the algebraic and in the topological sense ; for the example of GL(n, R) shows t h a t the group of points with real coordinates in an algebraic group, defined over R, which is irreducible and therefore connected in the algebraic sense, need not be connected in the topological sense. F o r a similar example where the group is the group of automorphisms G o f a semisimple algebra with involution over R, one m a y take the algebra M ~ + z(R) with the involution X ' = S - ~. tX.S, S being the m a t r i x S----

0

--

?)

'

the algebraic connected component of the i d e n t i t y in G is PO + (S), and it is easy to see t h a t PO+(S, R) is not topologically connected. On the other hand, it is well-known t h a t the set of real points on a n y algebraic variety, defined over R, consists at most of a finite n u m b e r of connected components in the topological sense; this must t h e n be the case, in particular, for all the groups which we have considered so far. As above, let G be the group of automorphisms of a semisimple algebra with involution (A, ~) over R ; for e v e r y positive involution commuting with ~ on A, denote again b y K(a) the group of the elements of G which commute with ~. We shall denote b y G' and b y

[1960b]

443 ALGEBRAS W I T H

INVOLUTIONS

619

K'(a), respectively, the connected components of the identity in G and in K(~) in the topological sense.

Notations being as above, G'/K'(~) is isomorphic to G/K(~), and we have K'(~) -= G' n K(~). LEMMA 8.

P u t K" = G' n K(~) ; call f the canonical mapping of G onto G/K(:r Theorem 4 shows t h a t f(G') is G/K(~); therefore it is simply connccted. But trivial topological considerations show t h a t f(G') is the same as G'/K". On the other hand, K'(~) is obviously the topological connected component of the identity in K". Therefore G'/K'(~) is a covering space of G'/K" ; as the latter is simply connected, this implios t h a t K'(~) = K". T~EOREM 5. Let (A, L) be a semisimple algebra with involution over R ; let G be its group of automorphisms, and G' the topological connected component of the identity in G; let R be the space of all positive involutions commuting with ~ on A ; and let G act on R by (~,~)-+A-l~fori~G,~ ~ R. For each ~ R , let K(~) be the group of those elements of G which commute with ~, and let K'(:r be the connected component of the identity in K(~). Then, for each ~ R, K(g) (resp. K'(~)) is a maximal compact subgroup of G (resp. of G') ; conversely, all maximal compact subgroups of G (resp. of G') are of that form, and are transforms of one another by inner automorphisms of G (resp. of G'). Moreover, R is isomorphic both to G/K(~) and to G'/K'(~) for each ~ ~ R. We have Mready seen that R is isomorphic to G/K(:r ; therefore, by L e m m a 8, it is also isomorphic to G'/K'(~.). I t is clear t h a t K(~) and K'(~) are closed subgroups of the group of all automorphisms of the vector-space underlying A ; as the elements of K(~) and K'(~) commute with ~, t h e y leave invariant the positive quadratic form tr(x ~ x ) ; therefore these groups are compact. Assume t h a t K'(~) is not a maximal compact subgroup of G' ; then it is properly contained in a compact subgroup K of G'. B y the corollary of Theorem 3, K must be contained in K(fi) for some fi ~ R, and therefore also in K'(fl), so that K'(~) is properly contained in K'(fi). By Prop. 6, fi is the transform of ~ b y some inner automorphisra

444

[1960b] 620

ANDRE W E I L

j(b) of A, belonging to G' ; this implies at once t h a t K'(fl) is the image of K'(~) under the inner automorphism of G' determined by j(b), and therefore that these two groups have the same dimension ; as t h e y are topologically connected Lie groups, this shows t h a t K'(~) cannot be properly contained in K'(fi). Similarly, assume t h a t K(~) is not a maximal compact subgroup of G ; then, just as before, we see t h a t K(g) must be properly contained in some K(fi), which implies t h a t K'(~) is contained in K'(fl) ; therefore, as shown above, K'(~) must be the same as K'(fi). As before, we see t h a t K(fl) is the image of K(~) under an inner automorphism of G ; therefore these groups have the same n u m b e r of connected components, this n u m b e r being finite as we have seen before. Since the connected component of the identity in K(~.) and in K(fl) is the same, viz. K'(~), this shows that K(~) cannot be properly contained in K(fi). On the other hand, the corollary of Theorem 3 shows t h a t e v e r y compact subgroup of G is contained in some group K(~) ; therefore, if it is maximal, it must be of the form K(~) ; and a similar proof holds for G'. 14. In order to prove our last result, we have to consider the Lie algebras of the groups discussed above. With the notations of Theorem 5, the Lie algebras of G and of K(~) will be denoted b y g and b y ~(~), respectively. I f Z is the center of the semisimple algebra A, the group of automorphisms of Z is finite ; this implies t h a t any automorphism of A, sufficiently close to the identity, induces the identity on Z and is therefore an inner automorphism. F r o m this, one easily deduces the well-known fact t h a t the Lie algebra of the group of automorphisms of A consists of the inner derivations

for all u e A. I f u and v are in A, we have D~ = D~ if and only if u--v is in Z ; from this, it follows t h a t e v e r y inner derivation can be written in one and only one way as D u with a(u) = 0, a being as defined in no. 12 ; one m a y therefore identify the Lie algebra of the

[1960b]

445 ALGEBRAS ~VITH I N V O L U T I O N S

621

g r o u p of a u t o m o r p h i s m s of A with the s u b s p a c e of A d e t e r m i n e d b y a(u) = O. I f • is a n y a u t o m o r p h i s m or a n t i m o r p h i s m of A, a n d if e()~) h a s the s a m e m e a n i n g as in no. 6, one sees a t once t h a t 2 - 1 Du 2 is the inner d e r i v a t i o n Du, with u' = e()t) u ~. I n p a r t i c u l a r , .Du c o m m u t e s with ~ if a n d only if z - - u - - e ( ) t ) u z i s in Z ; w h e n t h a t is so, we h a v e z ~ ~(z) = ~ ( u ) -

e(~) ~ ( u ) ~,

which shows t h a t z m u s t t h e n be 0 if a(u) -- 0, or also if ~ is a n inner a u t o m o r p h i s m o f A ; in b o t h t h e s e eases, therefore, D , c o m m u t e s w i t h u if a n d o n l y if u = e(2) u z.

L~MMA 9. A s s u m e that (A, ~) is non-degenerate, and let ~, fl be two elements of R . T h e n ~(~) = t(fl) implies ~ = ft. F r o m the results p r o v e d a b o v e , it follows t h a t t h e Lie a l g e b r a g of G consists of t h e inner d e r i v a t i o n s D~ for a(u) = 0, u' = - - u; let U be t h e v e c t o r - s p a c e d e t e r m i n e d b y t h e l a t t e r conditions. L e t V(~), W(~) be t h e s u b s p a c e s of U consisting of the e l e m e n t s o f U which a r e o d d for ~ a n d e v e n for ~, r e s p e c t i v e l y , a n d let V(fl), W(fi) be defined similarly. T h e n ~(cr ~(fl) consist o f t h e inner d e r i v a t i o n s D v for v e V(~) a n d for v ~ V(fl), r e s p e c t i v e l y . B y o u r a s s u m p t i o n , D~ c o m m u t e s w i t h fl for e v e r y v E V(~.) ; as a(v) -~ 0 for v e V(~), this implies t h a t v ~ = v a n d t h e r e f o r e v e V(fl). T h e r e f o r e our a s s u m p t i o n can be e x p r e s s e d as V ( ~ ) = V(fl). N o w t a k e w e W(~) ; as w is e v e n for ~r L e m m a 1 of no. 6 shows t h a t , w i t h r e s p e c t to t h e bilinear f o r m tr(xy), w is o r t h o g o n a l to all t h e o d d e l e m e n t s for ~, a n d in p a r t i c u l a r to all the e l e m e n t s o f V(~), i.e. of V(fl). On t h e o t h e r h a n d , as w is o d d for ~, it is o r t h o g o n a l to all e v e n e l e m e n t s for ~. N o w let B be t h e space of o d d e l e m e n t s for fl ; for a n y b e B, w r i t e : b0 = 89 -~- b~), b 1 = 89 - - b'), b~. = ~(bl), b~ = b 1 - - b 2. T h e n we h a v e b = b o -}- b~ + b 8 ; b 0 is e v e n for ~. As fl c o m m u t e s with ~, b 1 is o d d for fl ; t h e r e f o r e b~ is in V(fl). Moreover, b~ is o d d for fl a n d is in t h e center, so t h a t it is o d d for ~ (by P r o p . 1 of no 7).

446

[1960b] 622

ANDRE WEIL

Thus bo, b~ a n d b a are all orthogonal to w. This proves t h a t w is orthogonal to e v e r y b ~ B ; b y L e m m a 1 of no. 6, w m u s t therefore be even for ft. Thus W(~) is contained in W(fl); exchanging with fl, we see t h a t W(cr = W(fl). As the odd a n d the even elements for u a n d for fi in U are the same, it follows t h a t ~ and fl coincide on U, or in other words t h a t ~ - 1 fi induces the identity on the Lie algebra g of G. Now, b y Prop. 6, ~ - 1 fl is an inner a u t o m o r p h i s m of G' ; since it induces the identity on the Lie algebra of G, it m u s t therefore be in the center of the algebraic connected component of the identity G Oin G. B u t we h a v e a s s u m e d t h a t (A, c) is non-degenerate; and it follows f r o m T h e o r e m 2 of P a r t I that, when t h a t is so, G o is semisimple with a center r~duced to the identity. This completes our proof.

Notations and assumptions being as in Theorem 5, assume also that (A, ~) is non-degenerate. Then, for ~ e R, K'(~) is its own normalizer in G' ; K(cr is the normalizer of K'(~) and is its own normalizer in G; and the mappings ~ - + K'(cr ~-+ K(~) are bijeetions of .R onto the sets of maximal compact subgroups of G' and of G, respectively. T~EOREM 6.

The latter s t a t e m e n t follows at once f r o m L e m m a 9. Now, for e R a n d A ~ G, p u t fi --- A-1 :cA ; A t r a n s f o r m s K ' ( a ) into K'(fl) ; therefore, if A t r a n s f o r m s K'(~) into itself, L e m m a 9 shows t h a t we m u s t have ~ ~ fi, i.e. t h a t A m u s t be in K(~). This proves the theorem. 15. I f (A, ~), when decomposed into simple components, has no s u m m a n d on which ~ induces a positive involution, the group G' has no c o m p a c t factor, so t h a t the space R -- G'/K'(~) is the R i e m a n n i a n s y m m e t r i c space a t t a c h e d to the g r o u p G'. I n fact, it follows f r o m the results of P a r t I t h a t one can obtain in this m a n n e r all the R i e m a n n i a n s y m m e t r i c spaces a t t a c h e d to the semisimple groups which h a v e no c o m p a c t factor and no factor isomorphic to a n exceptional Lie group. Unfortunately, the latter are (for the t i m e being, a t least) still beyond the scope of the m e t h o d discussed in this paper.

[1960b]

447 ALGEBRAS W I T H I N V O L U T I O N S

623

Finally, we o b s e r v e that, if a positive i n v o l u t i o n ~ is chosen in t h e space R, a n d all o t h e r p o i n t s o f R are w r i t t e n as fl = ~j(a) w i t h a e PI(~), a' = a -1, the i n v a r i a n t m e t r i c in R is e x p r e s s e d b y

ds ~ = t r ( a - 1 da. a - 1 da), a n d t h e geodesic joining ~ to fl = ~j(a) ~ j ( a p) for p e R. The Institute for Advanced Study Princeton, :N'ow Jersey

consists o f t h e p o i n t s

[ 1960c] On discrete subgroups of Lie groups

1. L e t G be a topological g r o u p and F an a r b i t r a r y group; one m a y t h i n k o f F as being provided w i t h the discrete topology. Consider the space G (r) of all m a p p i n g s of F into G; this is the same as the product lIve~Gv, w h e r e Gv is the same as G for e v e r y 7 e F, and will be provided w i t h t h e usual p r o d u c t topology. The set .~ = ~ ( F , G) of all r e p r e s e n t a tions of F into G m a y be described as the subset of G (r~, consisting of all t h e m a p p i n g s r of F into G which s a t i s f y r(77') ~ r(~/)r(7 ') for e v e r y pair 7, 7' of e l e m e n t s of F; this is a closed subset of G (F) and will be provided w i t h the topology induced on it by t h a t of G(~); w i t h t h a t topology (the so-called " t o p o l o g y of pointwise c o n v e r g e n c e " ) , ~ will be called the space of r e p r e s e n t a t i o n s of F into G. If F is g e n e r a t e d by a family of e l e m e n t s (%)~ea, indexed by a set A, a r e p r e s e n t a t i o n r of F into G is uniquely det e r m i n e d by the e l e m e n t s r(%), so t h a t t h e r e is a one-to-one correspondence b e t w e e n .9t and a c e r t a i n subset of the set G (A) of all mappings of A into G. More precisely, F is t h e n a homomorphic image of the f r e e group F' w i t h the g e n e r a t o r s (7")~ea; let ~ be the homomorphism of F' onto F which maps 7" onto % for e v e r y a e A; let A' be the kernel of q~; the elem e n t s of A', being e l e m e n t s of F', are " w o r d s " w(7') in the 7", and we have then, for e v e r y such " w o r d " , w(7) = e, w h e r e z is the n e u t r a l elem e n t of F; these are the " r e l a t i o n s b e t w e e n the g e n e r a t o r s % of F " . The space 9~' = ~ ( F ' , G) is t h e n in an obvious one-to-one correspondence with G (A), and it is a trivial m a t t e r to v e r i f y t h a t this is a homeomorphism w h e n G (~) is provided w i t h t h e p r o d u c t topology in a m a n n e r similar to t h a t described above. T h e n ~ is in an obvious one-to-one correspondence w i t h t h e closed subset ~, ef ~', consisting of all the r e p r e s e n t a t i o n s of F' into G which m a p A' into the n e u t r a l e l e m e n t e of G; and it is again a trivial m a t t e r to v e r i f y t h a t this is a homeomorphism. We will i d e n t i f y .~' w i t h G du corps Q des nombres rationnels, ensemble qu'on peut considdrer comme ayant pour 616ments les nombres premiers rationnels et le symbole Go; enfin, pour chaque v, av(S, T) d6signe, en un certain sens, la ) de 1'ensemble des solutions de l'6quation S[X] = T dans le comp16t6 Qv de Q pour la place v (compl6t6 qui est le corps des r6els s i v = 0% et le corps Qv des nombres p-adiques siv est le nombre premier p), et ne d6pend que du genre de S et de celui de T.

9

471

472

[1962a] Dans les m6moires suivants de la m~me s6rie, Siegel 6tendit la formule (I) aux formes quadratiques, d6finies ou non, sur un corps de nombres alg6briques; dans tous les cas, la structure de la formule qui exprime son th6or6me fondamental reste la m~me, mais les nombres fz(Si, T) qui y figurent ne peuvent se d6finir en g6n6ral au moyen des nombres de solutions enti~res de certaines 6quations; on les d6finit au moyen des volumes de ~<domaines fondamentaux>> pour certains groupes discontinus. Ce qui nous int6resse pour l'instant, c'est qu'on aboutit dans tous les cas une formule du type (1), et en particulier que le premier membre comporte toujours une sommation sur routes les classes du genre de S. Dans un travail ult6rieur ([5]), Siegel, en appliquant aux formes inddfinies une m6thode nouvelle, obtint pour celles-ci une formule plus pr6cise, qui donne la valeur de #(S, T); compte tenu de (1), ce nouveau r6sultat peut s'exprimer en disant que t o u s l e s termes du premier membre de (1) ont m~me valeur. Cependant, ce r6sultat n'est d6montr6 que moyennant une restriction suppl~mentaire sur le nombre n de variables de la forme T; supposant que S e t T sont ~t coefficients entiers rationnels et que S est 6quivalente, sur R, h une forme h p carr6s positifs et m -- p carr6s n6gatifs, Siegel est oblig6, pour appliquer sa m6thode, de supposer que

(2)

_~__3) n ~ inf (p, m -- p,

I m>2n+2 "

r >1 n

I1 6tait naturel de se demander si ces derniers r6sultats de Siegel sont vraiment ind6pendants des pr6c6dents, et si l'in6galit6 (2) est n6cessaire pour assurer leur validit6. Le but du pr6sent travail est de r6pondre h ces questions par la n6gative. Nous prendrons pour point de d6part l'id6e f6conde de Tamagawa, qui consiste ~t introduire dans la th6orie des groupes alg6briques le langage des >. I1 6tait d6jh connu, d'apr6s Tamagawa, que le (<nombre de Tamagawa>> du groupe orthogonal (d6fini au moyen d'une forme quadratique non d6g6n6r6e sur un corps de nombres alg6briques) est toujours 6gal ~t 2, et que ce r6sultat est 6quivalent h la formule (1) pour le cas T = S. Le calcul du nombre de Tamagawa du groupe orthogonal est d'ailleurs effectu6 compl6tement dans mon cours de Princeton ([6]), par une m6thode qui, si elle est en substance assez proche (au langage pr6s) de la d6monstration de la formule (1) par Siegel, en est logiquement 10

[1962a]

473 ind6pendante. Ce r6sultat 6tant suppos6 acquis, nous nous proposons ici de d6montrer ce qui suit : (A) La formule (1), p o u r S et T non-d6g6n6r6es et n ~< m -- 3 (la m~me m6thode, convenablement modifi6e, permettrait de traiter aussi le c a s n = m -- 2, et celui off T e s t de d6terminant 0); (B) Le fait que si, dans le premier membre de (1), on groupe ensemble t o u s l e s termes correspondant ~ des formes Si appartenant ~ un m~me ((genre spinoriel)~, la somme obtenue a la m~me valeur p o u r t o u s l e s ((genres spinoriels)) contenus dans le genre de S. Lorsque S est une forme inddfinie, il r6sulte d ' u n th6or6me de Eichler et Kneser (cf. [2]), qu'il y a identit6 entre les ((genres spinoriels~) et les classes du genre de S; en ce cas, (B) permet donc de conclure que tousles termes du premier membre de (1) ont m~me valeur, ce qui est le r6sultat m~me de Siegel, relatif aux formes ind6finies, affranchi de la condition (2) et soumis seulement aux conditions 6nonc6es ci-dessus dans (A). Pour le c a s n = 1, ces r6sultats se trouvent d6j~ dans un travail r6cent ([3]) de M. Kneser; comme celui-ci a bien voulu me le communiquer, il les avait en fait obtenus p o u r n quelconque (y compris certains cas off l'on a n = m -- 2 ou det (T) -- 0); sa m6thode ne diffSre pas substantiellement de celle qui sera suivie ici. Si je me suis permis n6anmoins de revenir sur la question, c'est principalement afin de mettre en 6vidence le fait que la condition (2) ne tient pas h la nature du probl6me, ce qui ne peut apparaitre clairement tant q u ' o n se b o r n e au cas n = 1. 1. C o m m e corps de base, nous adoptons un corps de nombres alg6briques k, dont les compl&ions distinctes seront d6sign6es par kv, v p a r c o u r a n t l'ensemble des ((places)) de k; on notera p u n id6al premier de k, ainsi que la place correspondante, de sorte que les k~ d6signent toutes les compl&ions p-adiques de k; et on notera A l'anneau des ad61es de k. Une lois p o u r toutes, nous nous plagons dans un espace vectoriel de dimension m sur k, que nous identifions h k m par le choix d ' u n e base; nous nous y donnons une forme quadratique non d6g6n6r6e, d6termin6e, p o u r la base choisie, par la matrice sym6trique S ~t m lignes et m colonnes, h coefficients dans k. On d6signera par G le groupe o r t h o g o n a l O+(S), form6 des matrices X de d6terminant 1 satisfaisant ~ S[X] = S; suivant l'usage, on note Gg, Gk v 11

474

[1962a] ou plus bri6vement Gv, et GA, les groupes des solutions de det X = 1, S[X] ~- S, ~t coefficients dans k, dans kv, et dans A, respectivement; ces groupes sont munis de leurs topologies naturelles. Supposant d6sormais m t> 3, nous d6signerons par G;, le groupe des commutateurs de Gv (c'est-/t-dire, bien entendu, le sous-groupe de Gv engendr6 par les commutateurs d'616ments de Gv) et par G~ celui de GA. Nous noterons r la ((norme spinorielle)) de G; celle-ci d6finit, pour tout v, une repr6sentation continue de Gv dans le groupe fini kv/(kv) 2, et, sur GA, une repr6sentation continue de GA dans Ik/(lx) 2, Ix d6signant le groupe des idbles de k. Si S est d'indice/> 1 dans kv (c'est-/i-dire si la forme quadratique d6finie par S ((repr6sente z6ro>) dans kv), on sait que G~ n'est autre que le noyau de v dans G~; il en est ainsi en toute place h l'exception au plus des places/~ l'infini si m i> 5, et en presque toute place p o u r m =- 3 ou 4. Si S est d'indice 0 (c'est-/t-dire si la forme S ((ne repr6sente pas z6ro)~) dans kv, G~ est encore le noyau de ~ saul si m = 4 et si v e s t un id6al premier de k, ne divisant pas 2; dans ce dernier cas, G~ est un sous-groupe ouvert de Gv, d'indice 2 dans le noyau de v, et Gv/G~ est un groupe de type (2,2,2). On conclut aussit6t de l~t que G.~ est un sous-groupe ferm6 de GA, d'indice fini (6gal h une puissance de 2) dans le noyau G~i de ~; p o u r m ~ 4, on a G,~ = G j ; dans tous les cas, t o u s l e s 616ments de GA/G~ sont d ' o r d r e 2. De plus, au moyen d ' u n th6or~me &((approximation faible)) t o u t / t fait 616mentaire, on v6rifie imm6diatement que, m~me p o u r m = 4, on a G,[ G~c = GJi Gx; comme c'est seulement le groupe G,[ Gx qui nous importe dans ce qui suit, on pourrait donc, sans rien changer, y substituer GJi /i G,i. N o t o n s enfin (bien que cette remarque ne soit pas utile p o u r notre objet) que la norme spinorielle d6terlnine un isomorphisme de GA/G,~ Gx sur le groupe Ix/k* (ID z, dont la structure est ddtermin6e par la th6orie du corps de classes (c'est le groupe de Galois du compos6 de toutes les extensions quadratiques de k). 2. Pour tout sous-groupe ouvert Ga de GA, o n peut consid6rer les classes bilat6res dans GA suivant les sous-groupes Go et Gk; l'ensembte de ces classes s'appellera par d6finition le genre orthogonal d6termin6 par G e t Go. Si les Ui sont des repr6sentants des classes bilat6res en question, on aura d o n c :

(3) 12

GA = U Go U~ Gk.

[1962a]

475 I1 r6sulte d'ailleurs des travaux de Siegel que les classes d'un genre orthogonal sont toujours en nombre fini; mais nous n'aurons pas faire usage de ce r6sultat. Si de plus G,~ est comme ci-dessus le groupe des commutateurs de GA, l'ensemble Ga G.~ Gk est un sous-groupe ouvert de GA; les classes suivant ce sous-groupe s'appelleront les sous-genres du genre orthogonal d6termin6 par G e t Go; chacun d'eux est 6videmmerit r6union de classes du genre en question (qu'on appellera, par abus de langage, les classes de ce sous-genre). Comme Go GA Gk contient le groupe des commutateurs de GA, les sous-genres forment le groupe commutatif GA/GoGAG~; d'aprbs ce qui prdcbde, c'est un groupe discret dont tousles 616ments sont d'ordre 2. De plus, comme GA/Gg est de mesure finie (6gale au nombre de Tamagawa de G si la mesure est normalis6e convenablement), des r6sultats g6n6raux connus sur la mesure de Haar montrent que GA/G~G.~Gk est de mesure totale finie, donc que c'est un groupe fini puisqu'il est discret; bien entendu, cela r6sulte aussi, si l'on veut, du fait que les classes d'un genre sont en nombre fini. On en conclut que les sous-genres forment un groupe de type (2,2, ..., 2). Supposons en particulier que la forme quadratique d6finie par S soit ((ind6finie )), c'est-~-dire qu'il y air au moins une place /~ l'infini v de k telle que la forme S repr6sente z6ro dans kv. I1 s'ensuit alors du th6orbme d'approximation de M. Kneser (cf. [2]) que le noyau de la norme spinorielle, donc ~ plus forte raison le groupe GA, est contenu dans G~ Gk, donc aussi dans U-1 Go UGk quel que soit U ~ GA; cela donne UGA s Go UGg, et par suite

Go UG,~ G~ C G~ UGk. Comme G~ est le groupe des commutateurs de Ga, le premier membre n'est pas autre chose que le sous-genre UG9 GA Gg. On a donc montr6 que, dans le cas d'une forme ind6finie, il n'y a pas de distinction ~t faire entre classes et sous-genres d'un genre orthogonal. 3. Pour retomber sur les notions classiques de la th6orie des formes quadratiques, on supposera que k est le corps Q des nombres rationnels, que S est ~ coefficients entiers rationnels, et qu'on a pris Go = H Gary), off Gut v) ---- Gv quand v est la place ~ l'infini de k, et off Gate) d6signe, pour tout nombre premier p, l'ensemble des 616ments de Gp ~ coefficients entiers p-adiques. Soit alors R 13

476

[1962a]

le r6seau Z m des points fl coordonn6es enti6res dans Qm; pour tout p, soit de m~me Rp = Z~ (c'est l'adh6rence de R dans Q~ ). Pour chaque i, soit R(~) le transform6 de R~ par l'automorphisme de Q~ d6fini par la coordonn6e de Ui -1 relative ~t p (coordonn6e qui, par d6finition de GA, est un 616ment de G~). I1 y a alors, comme on sait, un r6seau R(~) et un seul dans Qm dont l'adh6rence dans Q'~ est R(~) quel que soit p. Soit S, la matrice qui exprime la forme quadratique d6finie par S lorsqu'on prend pour base de Qm, au lieu de la base canonique, une base (arbitrairement choisie) du r6seau R(t). On v6rifie facilement, dans ces conditions, que les S, sont des repr6sentants des classes du genre de la forme quadratique S, au sens classique de ces roots. Quant aux (( sous-genres )), ils ne sont pas autre chose dans ce cas que les ((genres spinoriels)) introduits par Eichler dans la th6orie des formes quadratiques (cf. [11). Un peu plus g6n6ralement, prenons pour k un corps de nombres : alg6briques; soit R u n r6seau de k m, c'est-~t-dire un module de type fini sur l'anneau des entiers de k, contenant m vecteurs lin6airement ind6pendants sur k; et, p o u r tout id6al premier p de k, soit R~ l'adh6rence de R darts k~. Posons G(~) = Gv pour toute place ~t l'infini de k; et, pour tout id6al premier p de k, prenons pour G ~) le groupe des 616ments de G~ qui transforment R~ en lui-m~me. Consid6rons le genre orthogonal d6fini par G e t par Go =/-/Gr ~. Les Ul 6tant comme plus haut des repr6sentants des classes de ce genre, d6signons de nouveau par R~ ) le transforms de R~ par la coordonn6e de U, -~ relative ~t p, et par Rr *) le r6seau dont l'adh6rence dans k'~ est R~ ~ quel que soit p. Les r6seaux R(i), dans l'espace k m muni de la forme quadratique S, constituent ~ nouveau des repr6sentants des classes du genre de S, au sens classique. On notera d'ailleurs qu'on peut, sans changer Go, remplacer les Rp par des r6seaux c~R~ respectivement homothdtiques aux R~o, condition que l'on ait c~ =- 1 pour presque tout p; en proc6dant ainsi, on pourra par exemple faire en sorte que la forme quadratique d6finie par S soit ~t valeurs enti~res sur R~ (ou bien que la forme bilin6aire d6finie par S soit h valeurs enti~res sur R~ x Rp, ces conditions 6tant 6quivalentes s i p ne divise pas 2), et qu'il n'y ait pas de r6seau homoth6tique/t R~, contenant R~ et distinct de R~, qui satisfasse ~t la m~me condition. I1 est clair qu'alors les r6seaux R~ ) ont la m~me proprietY. Ici encore nos ((sous-genres)~ ne sont autres que les ((genres spinoriels)). 14

[1962a]

477

4. D6sormais, nous conviendrons une fois pour toutes de prendre pour Go un sous-groupe ouvert de GA de la f o r m e / / G ( ~ ), off les G(~) sont soumis aux conditions suivantes : (a) G(~) - - Gv pour toute place ~t l'infini v de k; (b) pour tout id6al premier p de k , G(~) est un sous-groupe ouvert de G p ; (c) pour presque tout p, G(~~est le groupe des 616ments de G~ ~ coefficients entiers dans kp. On sait d'ailleurs que, si 0 est une repr6sentation lin6aire fid61e de G, ayant k pour corps de rationalit6, la condition (c) est 6quivalente h la suivante : (c') pour presque tout p, G(~) est le groupe des 616ments X de G:o tels que 0 (X) soit h coefficients entiers dans k~. On d6signera par Go~ le produit 17 Gv, 6tendu aux places h l'infini de k, et par Ge le groupe compact 1I G(~), off le produit est &endu aux id6aux premiers de k; on a Go = Go~ • Go. Sur le groupe alg6brique G, on choisira, une lois pour toutes, une ((jauge)), c'est-~t-dire une forme diff6rentielle (au sens alg6brique) invariante h droite et h gauche, de degr6 6gal h la dimension de G, ayant k pour corps de rationalit6. Cette forme d6termine, comme on sait (cf. [6]) une mesure de Haar mr sur chacun des groupes Gv; de plus, si on pose, pour tout id6al premier p ae k, tz(p) = mp(G(~)), le produit i n f i n i / / # ( p ) est absolument convergent. On peut donc d6finir sur Go la mesure produit des my; la mesure de Haar m sur GA qui coincide sur Go avec ID I-dlm(O)12Hmv (Off D est le discriminant de k, et dim ( G ) = m ( m - 1)/2) est ind6pendante du choix de Go, et du choix d'une jauge sur G, et s'appelle la mesure de Tamagawa sur GA. On en d6duit, par passage au quotient, une mesure (qui sera encore not6e m) sur le quotient de G.4 par n'importe quel sous-groupe discret de GA, et par exemple sur GA/Gk. Par d6finition, le ((nombre de Tamagawa)) de G est m (GA/Ge); on sait qu'il est 6gal ~t 2. i1 s'ensuit que m(GoGe/Ge) a une valeur finie, n6cessairement > 0 puisque Go est ouvert; cette valeur est 6gale h m(F) si F est un 1 4). Nous d6signerons par H la vari6t6 affine d6finie, dans 16

[1962a]

479 l'espace ~t m n dimensions des matrices X ~t m lignes et n colonnes, par l'6quation S[X] = T; nous supposerons que H a au moins un point rationnel sur k, ou, ce qui revient au m~me en vertu d ' u n th6or6me classique de Hasse, que H a des points rationnels sur chacun des corps kv. On notera He, Hv, HA les ensembles de points de H ~ coordonn6es dans k, dans kv et dons A, respectivement. Choisissons une lois p o u r toutes un 616ment M de H~c, et d6signons par g le sous-groupe de G form6 des 616ments X de G tels que X . M = M . Si ml . . . . . mn sont les n colonnes de la matrice M, et si Vest le sous-espace de k m engendr6 par les vecteurs mi, T e s t la matrice, par rapport '2 la base (ml, ..., mn) de V, de la forme quadratique induite sur V par la forme d6termin6e par S dons kin; comme on a suppos6 T de d&erminant non nul, les m/ sont lin6airement ind6pendants, et le sous-espace W de k m orthogonal ~t V par rapport ~t S est suppl6mentaire de V dans k m, donc de dimension m -- n; de plus, la forme quadratique induite par S sur W est non d6g6n6r6e. Le groupe g est alors le sous-groupe de G qui laisse invariants tous les points de V, et il s'identifie d ' u n e mani~re 6vidente au groupe orthogonal de la forme induite par S sur W. La vari6t6 H n'est pas autre chose, dons ces conditions, que l'espace homog6ne G/g; plus pr&is6ment, le th6orSme de Witt montre, comme on sait, que l'application X - + X . M d6termine, par passage au quotient, des isomorphismes de Gk/g~ sur Hk, de Gv/gv sur Hv et de GA/gA s u r HA : en particulier, X - + X . M est une application ouverte de GA s u r HA, et, p o u r tout v, c'est une application ouverte de Gv sur Hv. Soit Go un sous-groupe ouvert de GA, et soit go = ga c~ Go; go est un sous-groupe ouvert de gA, et il est clair que l'intersection ga ,', Go UG~ de gA avec une classe du genre d6termin6 par G et Go est une r6union (qui peut &re vide) de classes du genre d6termin6 par g et go. Plus pr6cis6ment, consid6rons, dans HA, l'ensemble ouvert U - 1 G o M = (U -1 Go U) 9 U - 1 M , qui est une orbite p o u r le sous-groupe ouvert U -1 Go U de GA; et posons : Ho(U) = Hg ~ U - 1 G o M . L'ensemble Ho(U), qui peut ~tre vide, est 6videmment stable pour le sous-groupe Go(U) ---- Gk ~ U -1 GoU de Gg, et peut donc s'6crire 17

480

[1962a]

comme r6union d'orbites par rapport h ce groupe; si les MQ sont des repr6sentants de ces orbites, on pourra 6crire chacun d'eux sous la forme Me = U - 1 X e M avec Xe E Go, et aussi sous la forme Me = Y o M avec Ye e G~ puisque Gg op~re transitivement sur Hg. Si, dans ces conditions, on pose Vo = X~-~UYe, on v6rifie facilemerit que l'on a (6)

gA ,', G~ UG~---- IA g~ V~gk

et que les ensembles qni figurent au second membre sont disjoints. 6. Si de plus Ga satisfait aux conditions (a), (b), (c) du d6but d u n ~ 4, il est clair que go y satisfera aussi; on pourra alors appliquer la formule (4) d u n ~ 4 aux termes du second membre de (6). Pour cela, on choisira une jauge dans g, au moyen de laquelle on d6finira la mesure de Tamagawa m' dans ga, des mesures de H a a r mb dans les groupes gv, et, comme au n ~ 4, une mesure de Haar m~o dans le groupe go~. On peut alors, d'une mani6re et d'une seule, choisir une jauge dans H (c'est-&-dire une forme diff6rentielle au sens alg6brique, invariante par G, de degr6 6gal h la dimension de H, et ayant k pour corps de rationalit6) de telle sorte que la jauge dans G soit formellement le produit des jauges dans g e t dans H. Au moyen de cette jauge, on d6finira des mesures invariantes m~, m.~ dans les espaces homog6nes Hv, HA, et on aura, au sens de la th6orie des espaces homog~nes, m~ -- mv/m~, m ~ = mA/rn~4. Cela pos6, soit W un 616ment quelconque de GA; dans la formule (6), remplagons U par WU et Go par WGo W - l ; il vient : gA r~ W G t l U G k = [.J (gA ~ W G t l q

W-1) Vogk,

off les V0 sont d6termin6s comme suit: on 6crit l'ensemble H• ,~ U -1 Ga W - 1 M comme r6union d'orbites Go(U)" Mo distinctes par rapport au groupe Go(U) = Gx ,', U -1 Go U,

et, pour chaque ~, on 6crit Me -

18

U-1XoW-1M---- Y o M

[1962a]

481 avec Xq ~ Go, Yo ~ G~, puis Vo = W X -1 UYQ. Appliquons (4) au r6sultat ainsi obtenu; il vient

m'((gA ta WGa UGk)/g~) = ~ m~(go~/(gk ~ y~IGo(U) Ye))

]--I m'~(gp ~ Wp G~~) w~l),

off Wp d6signe la coordonn6e de W relative ~t p. Mais, pour chaque p, W~ G(~) W~1 M est une partie ouverte compacte de Hp, orbite de M par rapport au groupe W:oG (~) W~ 1" comme le stabilisateur de M dans ce groupe est l'intersection de celui-ci avec gp, on a donc :

m'~(gp c~ Wp G(~~) W~1) = mp(W~ G (p) W~I) 9m~(Wp G(o~) W ; 1 i ; 1. Au second membre, le premier facteur est 6gal ~t m~(G(~)), c'est-hdire h #(p), et le second ~t m~(H~P)) -1 si l'on a pos6, pour tout v 9

H(av) = G (v) Wv 1M. De plus, nous ~crirons aussi :

.o = [-[.(:'= Go w- M. C'est l~t une orbite de Go dans HA, et r6ciproquement toute orbite de Go dans HA s'6crit ainsi pour un choix convenable de W. On notera qu'en vertu de r6sultats g6n6raux 616mentaires sur les vari6t6s ad61iques (cf. [6]), H(o~) est, pour presque tout p, l'ensemble des points de H ~t coordonn6es dans l'anneau des entiers de k~; il est facile d'ailleurs de v6rifier cette assertion directement. La formule obtenue plus haut peut s'6crire maintenant : (7)

m'((ga ~ WG~ UGk)/gD 1--[ m~ (H~~))

= ]--[ #(.p) ~, m~(g~/(g~ c~ Y~IGo(U) Yo)), et les YQ qui figurent au second membre s'obtiennent en 6crivant (8)

He ~ U-1Ho = I,.J Go(U)Me,

Q

Me = YQM,

YQ e G~, 19

482

[1962a] off la r6union qui apparait au second membre de la premiere de ces formules est une partition du premier membre en ensembles disjoints. 7. Comme dans la formule (3) d u n ~ 2, d6signons par Ui des repr6sentants des classes du genre d6termin6 par Go dans GA. Si, pour chaque i, on substitue. Ui /1 U dans (8), on pourra 6crire (9)

H~ ~ UT,1Ho = (J Go(U~) "Mio, Mio = Y~oM, Yto c G~; 0

bien entendu, le nombre de termes qui figurent au second membre de la premibre de ces formules d6pend de i (pour certaines valeurs de i, il peut s'annuler). Si maintenant on fait la somme des formules obtenues en substituant Ui h U dans (7), on obtiendra (10)

m'(gA/gg) ]-~ m~ (H(ff )) 20

-= ]--[ tt(p) ~_, m~(go~/(gk ,~ Y ~ Go(U 0 YIQ)). On peut obtenir une formule plus pr6cise en faisant porter la sommation, non pas sur tousles U,, mais seulement sur les repr6sentants Uj des classes d ' u n sous-genre UGo G:4 G~; on 6crira 27' au lieu de X, pour indiquer que la sommation porte seulement sur ces Uj, pour un sous-genre U d o n n 6 . Au lieu du facteur m'(gA/gD du premier membre de (10), il faudra alors 6crire (i 1)

m'((gA ~ WUGo G'A Gk)lgg).

Mais on a GA = gA Go Gd G~; cela r6sulte par exemple du fait que, puisque m -- n >/ 3, g contient un sous-groupe g' isomorphe h u n groupe orthogonal ~ 3 variables, donc isomorphe au quotient d ' u n groupe de quaternions de norme ~ 0 par son centre; pour ce dernier, la (<norme spinorielle>> n'est autre que la norme quaternionique, d'ofi r6sulte aussit6t que tout id61e dont les coordonn6es l'infini sont > 0 est norme spinorielle d ' u n 616ment de gA, donc ~t plus forte raison d ' u n 616ment de gA; cela donne GA =gA Go~G'fi, off G~i est le noyau de la norme spinorielle dans GA, &Off (puisque G~i C GA GD le r6sultat annonc6. I1 s'ensuit qu'il y a un 618ment Z de ga tel que WU ~ Z G o G d G~; on a alors

gA r~ WUGo G~ G~ ~ Z " (gA r~ Go GA Gg), 20

[1962a]

483 ce qui prouve que (11) a une valeur ind6pendante de WU; comme la somme des valeurs de (11), lorsqu'on y substitue ~ U des reprdsentants de t o u s l e s sous-genres, est m'(gA/gk), on voit que (11) a la valeur 2-Vm'(gA/g~), Off 2~ est le nombre des sous-genres dans G.4. Cela donne : (12)

2-:'m'(gA/g~) l---[ mp(H(~)) p t

= [ I .(p) P

,, r # Go(V,)Y.)), J,~

off, comme il a 6t6 expliqu~, la sommation du second membre porte sur les repr6sentants Uj des classes d'un sous-genre UGaGAGk. En particulier, s'il s'agit d'une forme ind6finie S, chaque sous-genre ne contient qu'une classe, et il n'y a pas ~ sommer sur j au second membre. Dans les formules (10) et(12), les premiers membres contiennent les = 0; more precisely, the quadratic f o r m F is positive and n o n - d e g e n e r a t e unless a t least one of t h e simple components of the Lie algebra of G is of dimension 3. In fact, we can w r i t e f = f ' + f " , w h e r e f ' = (f:j) is s y m m e t r i c and f " = (f[S) is altern a t i n g in the indices i, j. If we define the bilinear f o r m F ( f ' , f " ) by F ( f ' , f " ) = F ( f ' + f " ) -- F ( f ' ) -- F ( f " ) ,

we have, in view of (9): F(f',f")

P " ~ ~er ,err . ~f~f~5 + Z.~,j.~, ~R ~k~J~jJhk ,

= ~

as t h e first sum changes sign if we e x c h a n g e i, j , and the second sum does so if we e x c h a n g e i with j and h w i t h k, this is 0. T h e r e f o r e F ( f ) = F ( f ' ) + F ( f " ) . Also we have, for a similar reason

and t h e r e f o r e , using (8), (9) and (10): ~

.

~ h ~ .R i l t k.JrJ,i j, Jrh,. k, -- .

1 X-~

2. Z--'l J ~ ~

R ~ j n krJ ~, ,j 2r~, k,

~

1

-~o~

Ig-" c c ~ i J r,,l~ ~..g~,j J i 3"]

"

This proves t h a t F ( f " ) >= O, and also t h a t F ( f " ) can be 0 only if f " = 0. Now we w a n t to show t h a t F ( f ' ) > O. In fact, if we make on the X~ a n y orthogonal substitution X~-* ~ j ~ i X ~ , this will c h a n g e f~j into ~ ~f~, and t h e r e f o r e the quadratic f o r m Y~,~,J~T~T~ into

we m a y t h e r e f o r e choose t h a t orthogonal substitution in such a way t h a t a t a given point, the quadratic f o r m ~ , ~ f ~ T~ T~ becomes a diagonal f o r m Y~ a~ T:. T h e n we have, a t t h a t point, fl[~ = a ~ j . B y considering t h e effect of the same substitution on the R ~ , one sees immediately t h a t it does not c h a n g e the value of the f o r m F ( f ' ) . T h e r e f o r e , in order to show t h a t F ( f ' ) >= O, it is enough to show t h a t this is so a t a point w h e r e fi'~ = a ~ . Now, a t such a point, we have F(f') = ~

a~ + ~ , ~ r~a~a~ ,

w h e r e we have put r~j :

--R~H : ~-~'~( c ~ ) ~ ,

so t h a t we have r~j = r ~ >= 0 and also, in view of (11):

[1962b]

493 585

DISCRETE SUBGROUPS

(17)

~jr,j----1

L e t ,o be an eigenvalue of the quadratic f o r m ~ , j e i g e n v e c t o r belonging to p, we h a v e

(l. L e t G' be the group of all the right-translations in G ('> which leave t h a t solution invariant. Then r0(F) is contained in the group of the elements of G whose G(nLcoordinate is in G'; as G/ro(F) is compact, and as we have seen in no. 12 t h a t G' is closed in G on),this implies t h a t G(~>iG' is compact, and therefore, by t h e final result in no. 12, t h a t G' is discrete. This proves t h a t , unless the projection o f r0(P) on G (n) is discrete i n G Cn),all the fJ~) m u s t be 0 i n some neighborhood o f t = 0 i n L 14. For convenience, we shall identify F w i t h r0(F) from now on. Collect into a partial product G" of G all the simple factors G (n) of G of dimension 3 such t h a t the projection of F on G (n~ is discrete; let G' be the product of all the other simple factors of G. We know t h a t , if G ~*~is a n y one of the factors in G', all the fJ?,) are 0 for t in a suitable neighborhood of 0, which we m a y assume to be L Then we have IX, Y] = 0 for every X in t h e Lie algebra of G', so t h a t Y is invariant under left-translations by elements of G'; therefore, if ~(t, s) is as in no. 11, (19) is valid for all t e L s e G , s' ~G'.

[1962b]

499 DISCRETE SUBGROUPS

591

Call F', F " the projections of F on G', G"; and call A', A" its intersections w i t h G', G" considered as subgroups of G. The definition of G" implies t h a t F " is discrete in G"; t h e r e f o r e we can apply corollary 3 of A p p e n d i x II. This shows in p a r t i c u l a r t h a t F' is discrete in G', t h a t A' has finite index in F', and t h a t G'/A' is compact. Now, if we apply (18) and (19) to a n y ~' e A', we g e t (21)

~(t, s6') = ~(t, s)rt(6') ,

~(t, $'s) = F~(t, s ) .

F o r s = e, t h e s e relations show t h a t , if we modify r, by t h e inner autom o r p h i s m of G d e t e r m i n e d b y ~(t, e), i.e., if we replace it b y ~(t, e). f t . ~(t, e) -1, r, induces t h e i d e n t i t y on A'. Then, if 7 = (7', 7") e F, t h e projection of rt(7) on G" depends only upon t and 7" and m a y be w r i t t e n as r['(7"), w h e r e r 7 is a o n e - p a r a m e t e r f a m i l y of r e p r e s e n t a t i o n s of F " into G" such t h a t r~' is t h e identity. Now, applying (21) to a n y s = s" e G", we see, in view of t h e f a c t t h a t rt(F) = 6' and #'6' = 6's", t h a t ~(t, s") c o m m u t e s w i t h e v e r y 6' e A'; t h e r e f o r e , for e v e r y s" e G", t h e projection of ~(t, s") on G' is in the normalizer N(A') of A' in G'; as t h a t projection is e for t = 0, and N(A') is discrete b y corollary 1 of A p p e n d i x II, this shows t h a t ~(t, s") is in G" for all t e I and s" e G". T h a t being so, if we t a k e s = e and 7 e A" in (18), we see t h a t r~(7) e G" f o r all t e I and 7 e A"; t h e r e fore, if 7 = (7', 7") e F, t h e projection of r,(7) on G' depends only upon t and 7' and m a y be w r i t t e n as r;(7'), w h e r e r; is a o n e - p a r a m e t e r f a m i l y of r e p r e s e n t a t i o n s of F' into G' such t h a t r ' is t h e identity. As r; is t h e i d e n t i t y on A', corollary 2 of A p p e n d i x II shows t h a t it is the i d e n t i t y on F'. This gives, for all 7 = (7', 7") e F: =

If now we apply t h e results of D ~j u s t as in no. 11, and if we use t h e noration ~I(F, G) in t h e m a n n e r explained t h e r e , we see t h a t we h a v e proved t h e following t h e o r e m : THEOREM 2. Let G be a product of connected non-compact simple Lie groups. Let F be a discrete subgroup of G such that G/F is compact. Let G" be the product of those simple factors G (") of G which are o f dimension 3 and such that the projection o f F on G (p) is discrete in G (p), and let G' be the product o f all the other simple factors of G. Then the projections F', F" o f f on G', G" are discrete in G' and in G", respectively; G'/F' and G"/F" are compact; and F is o f finite index in F' • F". Moreover, i f j is the injection m a p p i n g of F into F' x F", and i f ~ ' is the set o f the representations of F' into G' induced on F' by the inner automorphisms o f G', then (r', r") --) (r' • r") o j is a homeomorphism o f

500

[I962b]

592

ANDR]~ WEIL

~ ' • ~I(F", G") onto ~RI(F, G). Here we have denoted by r ' • r " the representation of F' • F" into G' • G" which maps (7', 7") onto (r'(7'), r"(7")). 15. In order to obtain a complete result for a product of simple noncompact groups, we still have to deal w i t h the case G = G". The result is as follows: THEOREM 3. Let G be a produ.ct of connected non-compact simple Lie groups Gp of dimension 3. Let F be a discrete subgroup of G such that G/F is compact and that, for every p, the projection Fp of F on Gp is discrete in G~. Then, for each p, G~/Fp is compact, and F is of finite index in I L F~. Moreover, i f j is the injection mapping of F into I L Fp, (rp) --~ ( I L r~) o j is a homeomorphism of I L ~l(Fp, Gp) onto ~l(F, G). Here we have denoted by II~ r~ the representation of rip F~ into G which maps (7,) onto (r,(7~)). The first assertion in our theorem follows at once from corollary 3 of Appendix II by induction on the n u m b e r of the factors in G. The second p a r t will be proved in our usual m a n n e r by dealing first w i t h a one-param e t e r family r, of representations of F into G. Notations being the same as before, we shall write, on I • G: Y:

0 +~ 0t

~/~/v~l. ~ ~ '

t h e n the ~pk'~satisfy the equations (6) w i t h f , f ' replaced by 0, f ; this can be w r i t t e n X ~k~'1~1 = 0 "Y/z

(p' :~ p)

9

The l a t t e r equations show t h a t ~ ~'~ , , as a function on I • H , G~, depends only upon t h e / - c o o r d i n a t e and the G ; c o o r d i n a t e , so t h a t we can write

y_

a + ~ ot

.

'~

(t,x~)

. X ~

at the point (t, (x,)) of I • 1-in G,. This implies t h a t the solutions (t, ~(t, s)) of the differential system determined by Y on I • G are of the form (22)

}(t, (sp)) = ((},(t, s~))

where, for each p, (t, }p(t. s~)) is the solution t h r o u g h (0, s~) of the differential system determined in I • G~ be the vector-field ~ ~p~tt Y~ = - ~ y + 2_.,~,'e~ , , x0). X~p)

[1962b]

501 DISCRETE SUBGROUPS

593

I f we a g a i n i d e n t i f y F w i t h r0(F), we see now a t once, by combining (18) w i t h (22), t h a t t h e Gp-coordinate of r~(7), for 7 = (%) c F, depends only upon t and % a n d m a y be w r i t t e n as r~PS(%), w h e r e r~') is a o n e - p a r a m e t e r f a m i l y of r e p r e s e n t a t i o n s of F~ into G,. The conclusion of T h e o r e m 3 follows n o w b y m a k i n g use of the r e s u l t s of D I in t h e s a m e w a y as in no. 11. 16. N o w t a k e f o r G a n y connected semisimple g r o u p w i t h o u t c o m p a c t c o m p o n e n t s , and let Z be its c e n t e r . N o t a t i o n s b e i n g as before, it has b e e n s h o w n in no. 5 t h a t Y is i n v a r i a n t u n d e r Z, so t h a t (19) gives ~(t, zs) = z~(t, s) f o r z ~ Z; if now z ---- r0(7) f o r some 7 e F, a comparison w i t h (18) s h o w s t h a t z -- rt(7) for all t e L L e t n o w F be a discrete s u b g r o u p of G such t h a t G/F is c o m p a c t ; b y m a k i n g use of t h e r e s u l t s of D ~ in t h e s a m e w a y as in no. 11, we see now t h a t , w h e n e v e r r, r ' a r e in ~I(F, G) and an e l e m e n t 7 of F is such t h a t r(7) = z e Z, t h e n r'(7) = z. T h e r e f o r e e v e r y r e ~ ( F , G) c a n be e x t e n d e d to an injective r e p r e s e n t a t i o n r* of t h e g r o u p F* = Z . F into G b y p u t t i n g r*(zT) -- z . r(7) for z e Z, 7 e F. Moreover, as Z - F is c o n t a i n e d in the n o r m a l i z e r N ( F ) of F in G, corollary 1 of A p p e n d i x I I shows t h a t it is discrete in G; as G/F is c o m p a c t , G/F* is so; for similar reasons, t h e g r o u p r*(F*) ---- Z - r(F) is discrete in G, and G/r*(F*) is c o m p a c t , f o r e v e r y r e ~I(F, G). T h u s we h a v e proved t h a t e v e r y e l e m e n t of ~ ( F , G) induces t h e i d e n t i t y on F R Z, and t h a t r -* r* is a h o m e o m o r p h i s m of ~ ( F , G) onto ~ ( F * , G). F r o m now on, a s s u m e t h a t F contains Z. L e t Z1 be a s u b g r o u p of Z; p u t G' = G/Z1; call p t h e canonical h o m o m o r p h i s m of G onto G'; and p u t F' ----2(F), so t h a t F = p-~(F'). As e v e r y r e ~ ( F , G) induces the i d e n t i t y on Z, t h e relation p o r = r ' o p d e t e r m i n e s a m a p p i n g r --* r ' of ~ ( F , G) into ~li(V', G'), which is obviously continuous. Choose n o w (as in D ~, no. 11) a finite set (%) of g e n e r a t o r s of F, such t h a t P is defined b y a finite set of relations Rx(%) = e b e t w e e n t h e %; choose a finite set (~j) of g e n e r a t o r s f o r Z1; and, f o r e a c h ~'j, choose an e x p r e s s i o n ~'j = Fj(%) of ~'j in t e r m s of t h e %; t h e n , if we p u t 7'~ = p(7~), F' h a s t h e g e n e r a t o r s 7~ and is defined b y t h e r e l a t i o n s Ra(73 = e', F j ( 7 ~ ) = e' b e t w e e n t h e m . Take any r'0 e N~(F', G'); p u t s~ = r~(7'~); and, f o r each i, choose s~ e G such t h a t p(s~) = s~. L e t V be a neighborhood of e in G, such t h a t V R Z1 = {e}; choose an open neighborhood U of e in G such t h a t R,~(u~s~) ~ V . Rx(s~) and Fj(u~s~) e V . F~(s~) for all ?~, j w h e n e v e r all t h e u~ a r e in U, and also such t h a t p induces on U a h o m e o m o r p h i s m of U onto its i m a g e U ' in G'; call q> t h e inverse of t h a t h o m e o m o r p h i s m . L e t 1I' be t h e open neighborhood of r~ in ~tt~(F', G'), consisting of t h e r e p r e s e n t a t i o n s r ' such t h a t r'(7~) e U's~ for all i. T h e n it is e a s y to see t h a t if, for a n y r ' e t t ' , t h e r e is an r ~ N~(F, G) such t h a t p o r = r ' o p, this m u s t be g i v e n by f o r m u l a s

502

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r(7~) = z; ~. q~(rt ( Tt i ) t--I s i ) . s~, where the zi are elements of Z1 satisfying the relations Rx(z~) = R~(si) ,

F~(z~) = ~ j . Fi(s~) .

As these relations are independent of the choice of r' in W, one concludes immediately from this, and from obvious continuity considerations, t h a t the inverse image of 11' under the mapping r -+ r ' is the union of neighborhoods of the points in the inverse image of r~, and t h a t this mapping determines, on each one of these neighborhoods, a homeomorphism onto 11'. In other words, for this " na t ur al " mapping, !~(P, G) is a covering space of ~ ( F ' , G')% In particular, we can apply this to the case Z~ = Z; G' is t hen t h e adjoint group of G and is a product of simple groups, so t h a t the structure: of ~I(F', G') is fully determined by Theorems 2 and 3. In view of t h e known facts on fuchsian groups, this shows for instance t h a t ~ ( F , G) is a manifold. Alternatively, we may write G - G J Z , where G~ is the simply connected group with the same infinitesimal st ruct ure as G, and Z~ is a subgroup of the center of G1; then, if F~ is the inverse image of F in G , we see t h a t ~l~(F~, G~) is a covering space of ~R~(F, G); here again G1 is a product of simple groups, so t h a t the s t r u c t u r e of ~(P~, G~) is again giver~ by Theorems 2 and 3.

APPENDIX I Through an oversight, the main theorem of D e, as formulated there in no. i, is not quite strong enough for the application which is made of it in D r, no. 12; this is to be corrected now. We wish to show that the theorem in question is valid with the following addition: L e t G, F, r0, ~, 11 be as i n that theorem; then, J b r e v e r y g ~ G, there is a neighborhood W o f g i n G, a n d a neighborhood 11' o f ro i n 1I, s u c h that the u n i o n o f the sets r - l ( W ) , f o r all r e 1I', is a f i n i t e set. Assume first t h a t G is simply connected. L et all assumptions and notations be as in nos. 7-10 of De; put W --- soIU'-~sog. A s , m a p s U' x S • F onto G, we can write sog = us7, with u e U', s c S, 7 e F. If the representation 7 -* ~ is close enough to the identity, the point ~ determined by us7 = ~s~(s)-I~(s)~ 4 If G h a s no c o m p o n e n t of d i m e n s i o n 3, t h e s e two s p a c e s a r e a c t u a l l y isomorphic. T h i s follows f r o m T h e o r e m I and f r o m a r e s u l t of Borel [2, Corollary 4.4]. If the s a m e could be proved for g r o u p s of d i m e n s i o n 3, t h e n T h e o r e m s 2 a n d 3 would s h o w t h a t it r e m a i n s t r u e for all s e m i s i m p l e g r o u p s .

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will be in U'. Now assume t h a t , for such a r e p r e s e n t a t i o n , and for some 7' e F, U is in W; this means t h a t we have ~'so~' -= sog = ~s3(s)-1~(s)7 w i t h some ~' e U'. This can also be w r i t t e n as

so,

-1)

s, e ) ,

w h i c h a m o u n t s to saying t h a t (~', so, 7'7 -1) and (~, s, e) are equivalent for /~; this implies t h a t {So, s} e N ( S ) and 7'7 -1 = 7(s0, s). So 7' m u s t have one of t h e finitely m a n y values 7(s0, s)7. This proves our assertion w h e n G is simply connected. Reasoning as in D ~, no. 11, one e x t e n d s it a t once to the g e n e r a l case. Now let assumptions and notations be as in D ~, no. 12. F r o m w h a t we h a v e proved above, it follows t h a t , w h e n P is made to act on X • G by

(x,

= (x,

it operates p r o p e r l y on X x G. This m e a n s t h a t , given a n y two compact subsets K, K ' of X • G, t h e r e are at most finitely m a n y elements 7 of F such t h a t K 7 m e e t s K ' . T h e r e f o r e t h e space S -- ( X x G)/F is s e p a r a t e d (i.e., " H a u s d o r f f " ) . While this f a c t had not expressly been s t a t e d in no. 12 of D ~, some of t h e assertions made t h e r e would not m a k e sense unless this w e r e so. Once t h e above addition to the main t h e o r e m of D ~ has been obtained, it is a trivial m a t t e r to s t r e n g t h e n it as follows: one can choose W a n d p, so that the sets r-~( W ) , f o r r e 1I', are all e m p t y i f g r ro(r) a n d all equal to {g} i f g e roW).

APPENDIX I I

L e t G be a connected Lie group of dimension n; in D ~, no. 6, we defined a G - s t r u c t u r e on a manifold V of dimension n as being given by n everyw h e r e linearly i n d e p e n d e n t differential f o r m s w ~ on V s a t i s f y i n g t h e M a u r e r - C a r t a n equations for G. I f t h e X~, a t e v e r y point of V, are t h e vectors defined by i(Xx)w ~ = 3~, t h e Xx m a k e up n e v e r y w h e r e linearly i n d e p e n d e n t vector-fields on V, s a t i s f y i n g the s t r u c t u r a l equations (1) for G; a G - s t r u c t u r e m a y be considered as given b y n such vector-fields, just as well as by s t r u c t u r a l f o r m s cox. The g r o u p G itself is always to be considered as c a r r y i n g t h e G - s t r u c t u r e d e t e r m i n e d by t h e r i g h t - i n v a r i a n t vector-fields Xx (cf. no. 1); t h e n , if F is a discrete subgroup of G, G/F (the space of r i g h t cosets sF in G) carries a G - s t r u c t u r e , d e t e r m i n e d in an obvious m a n n e r b y t h a t of G.

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[1962b] ANDR~ WEIL

As observed in D ~, no. 6, the automorphisms of the G-structure of G are the right-translations; if F is a discrete subgroup of G, a right-translation x ~ xs determines an automorphism of G/F if and only if sPs -1 = F, i.e., if and only if s belongs to the normalizer N(F) of F; and it is easily seen (using the elementary facts noted in D ~, no. 6, i.e., essentially nothing more than Frobenius's theorem) t h a t all automorphisms of G/F can be obtained in this manner. The group of automorphisms of G/F may t herefore be identified with N(F)/F. L et N0(F) be the component of e in the closed subgroup N(F) of G; it is a connected Lie group. For each 7 e F, the image of N0(F) by x ~ XTX-1 must be a connected subset of F, containing 7, and is t h e r e f o r e {7}. Thus N0(F) is also the component of e in the centralizer Z(F) of F (consisting of the elements of G which commute with e ve ry 7 c F). In particular, the Lie algebra ~(F) of N0(F), which may be identified with those of Z(F) and of N(F)/F, consists of the vectors X i n the Lie algebra ~ of G which are invariant under Ad(7) for every 7 c F (as usual, we denote by Ad(s) the automorphism of g induced by the inner automorphism x ~ sxs -~ of G). Now assume t h a t G/F is compact, i.e., t h a t t here is a compact subset K of G such t h a t G = K F . Then, if p is any representation of G in a finitedimensional vector-space A over R, any vector a c A which is invariant under p(7) for eve r y 7 e F has a compact orbit under p(G). In this situation, we can apply the following lemma: LEMMA. Let p be a r e p r e s e n t a t i o n o f a topological group G i n a finited i m e n s i o n a l vector-space A over R. Let A' be the set o f all the vectors i n A whose orbit u n d e r p(G) is r e l a t i v e l y compact. Then A' is a subspace o f A , i n v a r i a n t u n d e r p(G); a n d p induces on A ' a r e p r e s e n t a t i o n p' o f G such that p'(G) is contained i n a compact group o f a u t o m o r p h i s m s o f A r,

The first assertion is obvious. Now let al, . . . , a,, be a basis for A'; as the vectors p(x)al belong to a bounded subset of A' for all x e G, G induces on A' a relatively compact set of endomorphisms of A', hence also a relatively compact subset of the group of automorphisms of A'. Now let again G be a Lie group with the Lie algebra g; call c the set of the vectors in g whose orbits under the adjoint group are relatively compact. It is clear t h a t this is not only a vector-subspace of .q but a Lie subalgebra of g, invariant under Ad(G) and even under all automorphisms of g. The adjoint representation x -~ Ad(x) of G induces on c a representation whose kernel is the centralizer C of c in G; in view of the lemma, this implies t h a t G/C has an injective representation into a compact group. In the case with which we are mainly concerned in this paper, G is con-

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n e c t e d and semisimple w i t h o u t compact components, so t h a t it has no non-trivial r e p r e s e n t a t i o n into a compact group; t h e r e f o r e , in t h a t case, we have C = G, hence c = {0}. I f now F is a discrete subgroup of G such t h a t G/F is compact, it follows f r o m w h a t has been said above t h a t e v e r y v e c t o r in n(F) has a compact orbit u n d e r the adjoint group, so t h a t n(P) c c; t h e r e f o r e : THEOREM. Let G be a connected semi-simple Lie group without compact components; let F be a discrete subgroup of G such that G/F is compact; let ~(F) be the set of all the vectors in the Lie algebra of G which are i n v a r i a n t under Ad(7) for every 7 e r. Then ~(F) = {0}. This is of course contained in a d e e p e r r e s u l t proved by Borel for t h e case w h e n G/F is m e r e l y assumed to have finite m e a s u r e [2, Corollary 4.4]. COROLLARY 1. Let G and F be as in the theorem; then the normalizer N ( F ) of F in G is discrete, G/N(P) is compact, and F is of finite index in N ( F ) . In fact, the first assertion a m o u n t s to saying t h a t N0(F) = {e}, and this is e q u i v a l e n t to our t h e o r e m . The o t h e r assertions follow f r o m this a t once.

COROLLARY 2. Let G and F be as in the theorem, and let A be a subgroup of finite index of F. Then the space of the representations of F into G which induce the identity on A is discrete. The subgroup A' of F which induces the i d e n t i t y m a p p i n g on the homogeneous space F/A is of finite index in F (at most equal to d! if d is the index of A in F) and is a normal subgroup of F; replacing A by A', we see t h a t it is e n o u g h to consider t h e case w h e n A itself is normal in F. As A is of finite index in F, G/A is compact, so t h a t N(A) is discrete by corollary 1. Now, if a r e p r e s e n t a t i o n r of F into G induces t h e i d e n t i t y on F, we have r(F) c N ( A ) . T h e r e f o r e , if r, r ' are t w o such r e p r e s e n t a tions, and if (%) is a finite set of g e n e r a t o r s for F, we m u s t have r(7~) = r'(%) for all i, and t h e r e f o r e r = r', as soon as r ' is close enough to r. COROLLARY 3. Let G', G" be two connected semisimple Lie groups without compact components; let F be a discrete subgroup of G = G' • G" w i t h compact quotient. Let A', A" be the intersections of F with G' and w i t h G" considered as subgroups of G, and let P', F" be its projections on G' and on G". Assume that F" is discrete in G". Then P' is discrete in G'; G'/P' and G"/F" are compact; F has a finite index in F' • P", and A' • A" has a finite index in F. L e t K be a compact subset of G such t h a t G = K F ; call K ' , K " its projections on G' and on G". F o r a n y x' in G', we can w r i t e (x', e") = k7

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w i t h k e K and 7 = (7', 7") e F; t h e n 7" belongs to K ''-~ f3 F", which is a finite set since F " is discrete and K " is compact. Choose a finite n u m b e r of elements % = (7~, 77) of F so t h a t K ''-1 g / F " = {7;', " " , 7"}. Then, if x', k, 7 are as above, t h e r e is an i such t h a t 7 = %$' with 5' e A'. This gives

x' e (U~ K'73. A', which shows t h a t G'/A' is compact. T h e r e f o r e , by corollary 1, N(A') is discrete in G'. On t h e o t h e r hand, one sees a t once t h a t F' is contained in N(A'), so t h a t it is discrete. E x c h a n g i n g now G' and G" in the above proof, we see t h a t G"/A" is compact. Our o t h e r assertions are now obvious.

APPENDIX III

(This appendix reproduces, w i t h minor verbal changes, a communication o f L. Hgrmander to the author, and is published here with his permission). L e t notations be as in nos. 2 and 3; t a k e a n y vector-field Y0 e ~ on S, and consider, as in no. 11, t h e solutions of t h e differential s y s t e m d e t e r m i n e d on S by t h a t vector-field. F o r e v e r y point M o f S, call O ( M ) the point w h e r e t h e solution of t h a t s y s t e m which goes t h r o u g h M cuts t h e fibre 7c-1(0). Then, for each t, 9 induces on t h e fibre ~-l(t) a h o m e o m o r p h i s m of class C ~ of t h a t fibre onto 7~-~(0), and the m a p p i n g M - ~ (re(M), ~ ( M ) ) is a h o m e o m o r p h i s m of class C ~ of S onto I • 7~-~(0). P u t V = ~-~(0); t h e n t h e m a p p i n g of 7c-~(t) onto V induced by (P will map t h e vector-fields Xx onto n vector-fields Xx(t) on V; similarly, if Y and the f ~ are as in no. 3, it will map t h e functions induced by t h e f ~ on z-l(t) onto functions f~(t) on V; it maps t h e volume e l e m e n t d 3 on ~-l(t) onto a volume e l e m e n t d~2, = ~(t)~d~2 on V, w i t h a density ~(t) ~which is n o w h e r e 0; and t h e X~(t), t h e f~(t) and ~(t) are all of class C ~ on I • V. A f t e r replacing t h e Xx(t) by $(t)Xx(t), w r i t i n g f,~(t) instead of ~(t)f~(t), c,~p(t) instead of $(t)cL and q~ instead of 9 ~, we can now s t a t e t h e variational problem of no. 3 as follows: f o r each value of t, the functions q~ are to be chosen so as to

m i n i m i z e the integral (23)

fv~

~ (f~x(t) § Xx(t)q% § ~

c~xp(t)q%)2d~2 .

H e r e all t h e d a t a are assumed to be of class C ~ on I • V; for each t, t h e Xx(t) are e v e r y w h e r e linearly i n d e p e n d e n t vector-fields; and one wishes to show t h a t t h e problem has a unique solution, of class C ~ on I • V, u n d e r t h e assumption t h a t , for each value of t, the s y s t e m

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SUBGROUPS

X~(t)9~ + ~ p c,xp(t)9, = 0

(24)

has no solution of class C ~ on V. o t h e r t h a n O. On t h e space L s of functions of i n t e g r a b l e square on V, we use the / r

~

\V2

n o r m II g II = (W d ~ )

; similarly, for a s y s t e m g = (g~x) of such funetions,

N~

we use t h e n o r m given by

II g IIs =

~ x . , II g,x ]is = f ~'-x,~ (g~x) sdt2 ;

w i t h this norm, t h e space of such s y s t e m s will also be denoted by L ~ (this will cause no confusion). On t h e o t h e r hand, for q~ = (q%), we introduce t h e norm given by Ill e Ills = ~ , ~

II X~(O)e~ 11s + ~

1l 9~ IIs ,

and we call H t h e space of the q) = (q)~) for which this is finite, i.e., for which all the q~. are in L s and t h e i r first derivatives, in t h e distribution sense, are also in LL W e shall now prove t h a t t h e inequality (25)

Ill 9 llts G C s ~ x ~ . 1IXx(0)q)~ + ~;~'~pGxp(0)~Pp Iis

holds, for a suitable choice of t h e c o n s t a n t C, for all q~ ~ H. In f a c t , if this w e r e not so, t h e r e would be a sequence q~"} w i t h Ill 9 c*~Ill = 1 such t h a t t h e r i g h t - h a n d side of (25) tends to 0. By a well-known principle based on Poinear6's i n e q u a l i t y (el. e.g., Courant-Hilbert, Vol. 2, pp. 488490), e v e r y sequence for which III q~(*)Ill is bounded has a subsequence which c o n v e r g e s in LS; t h e r e f o r e we m a y assume t h a t q~(~) c o n v e r g e s in L s tow a r d s a limit ~p. As t h e r i g h t - h a n d side of (25) t e n d s to 0 for this sequence, this implies t h a t t h e Xx(0)q~ converge in LL T h e r e f o r e ~p is in H, we h a v e ]l] q~ ]]I = 1, and ~p is a solution of t h e s y s t e m (24) w i t h t = 0. As (24) for t ---- 0 implies ~

X~(0)(X~(0)~ + ~

c~(0)~) = 0,

and as this is an elliptic system, t h e t h e o r y of elliptic systems (cf. e.g., [5] or [6]) shows t h a t q~ can be modified on a null-set so as to become C ~. We have t h u s obtained a non-zero solution of (24) for t = 0, of class C% a g a i n s t our assumption. This proves (25). F r o m the c o n t i n u i t y of t h e d a t a in t, it follows now at once t h a t we have the inequality (26)

Ill ~ Ill~ < 4 c s ~ x ~ I1xx(t)~p, + ~ , G~,(t)q~p lIs

for all q~ e H and all t in some neighborhood I ' of 0 in L F r o m now on. we assume t h a t t is in I'.

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F r o m (26), it follows t h a t the mapping G, of H into L 2 given by

9 ~ o~(9) = (x~(t)9~ + Ep c~(t)~p) has a closed range. L e t P(t) be the projection on this range in LL Then the problem of minimizing (23) has the unique solution

~p -_ _G(t)-ip(t)f in H, and it follows from (26) t h a t Ill q~JII =< 2C l[f]r. The solution q~ of our variational problem m u s t also s a t i s f y the Euler equations of t h a t problem

E . c ~ f ~)d~'~

E.

for all ~ = ( ~ ) of class C~ on V. This is a w e a k form of a system (27)

L~q) = F ~ ,

where the leading t e r m in L~ is - - ~ x X:9~, so t h a t this is an elliptic system; F~ is the effect of an operator of the first order acting on f , so t h a t F~ is of class C ~ on I ' • V. Therefore, for each t e I', ~ can be modified on a null-set so t h a t it will be of class C ~ on V (cf. [5], [6]). The s y s t e m (27) has no other solution t h a n 9 in H. In fact, assume now t h a t ~p is a n y solution of (27) in H. In view of the definition of L~, we have

f r~.~ (X~qg~ + ~_,p c~p~)~dFt = I r ~ ( L ~ 9 ) ~ d F t = f ~

F~q>~dFt .

Combining this w i t h (26), we get

Ill 9 Ill 2 < 4 c ' II F l l . I1 ~ iI < 4C ~ II E l l . Ill ~ Ill, and t h e r e f o r e (28)

lll~l]l =< 4C"I[F]I 9

In particular, F = 0 implies q~ -- 0, as we asserted. If a = ( a , . . . , c~.), where the c~ are integers ~ 0 , we shall write, as usual, I a ] = ~ a~; and we write X~ for the differential operator of order [a I given by x g = x~(o) o ~ . . . x ~ ( o y ~ .

W i t h this notation, the proofs for the r e g u l a r i t y of solutions of elliptic systems (cf. again [5], [6]) give estimates supv I Xg~P~ I - for the bilinear form on V • V' which defines the duality between V a n d V'; as usual, denote by ~p(s), for s e F, the transpose of p(s), i.e., the automorphism of V' defined by < p ( s ) x , x ' > = <x, ' p ( s ) x ' > .

By definition, i' is the dimension of the space of vectors x' e V' such t h a t 1, il n'y a pas plus de groupes simples que sur le corps complexe. Cependant ils n'avaient pas tir6 au clair la classification de ces groupes sur un corps de base quelconque. La vraie difficult6 de ce probl6me tient/l la prdsence des groupes exceptionnels et au d6sir bien naturel de ne pas traiter ceux-ci comme des monstres mais de les englober darts une th6orie g6n6rale off ils n'apparaissent plus comme tels. Pour cela il faut, comme on dit, "soulever des poids et arracher des racines"; c'est un exercice qui a toujours ddpass6 mes forces et que j'ai dfi,/~ regret, abandonner/t des col16gues mieux douds pour cela. Du moins, comme me le fit observer un amateur de cocycles tr6s connu, mon travail de 1956 sur la descente du corps de base, en donnant le moyen de "tordre" une structure par un groupe d'automorphismes, permettait de retrouver 616gamment l'id6e de Siegel, suivant laquelle tous les groupes classiques peuvent atre ddcrits comme groupes d'automorphismes d'alg6bres fi involution. Lorsque le corps de base est le corps des r6els, on retrouve 6galement ainsi, sous forme g6om6trique, la construction donn6e par Siegel pour les espaces riemanniens symdtriques correspondants. En r6digeant ce travail en 1958-59, mon secret espoir 6tait de pouvoir inclure dans mon expos6 au moins quelques-uns des groupes exceptionnels; pour cela je voulais substituer, aux alg6bres associatives classiques, l'alg6bre de Cayley ou "alg6bre des octaves", ~ laquelle justement Borel avait consacr6 l'un de ses premiers travaux. Pour bien faire j'aurais souhait6 aussi pouvoir traiter les formes exceptionnelles que peut prendre le groupe orthogonal/t 8 variables sur les corps admettant des extensions de degr6 3. Apr6s quelques essais infructueux, je dus y renoncer. J'avais destin6 ce travail au volume "jubilaire" du Journal of the Indian Mathematical Society; il rut insdr6 darts la seconde partie de ce volume, qui parut seulement en 1961 ; c'est 15, l'explication de la remarque finale de la note, p. 589, que les 6diteurs voulurent bien ne pas prendre en mauvaise part. Quant aux initiales de cette note, elles se rdf6rent 5, l'Office of Useless Research of the Poldavian Air Force (cf. [1954b]).

[1960 c] On discrete subgroups of Lie groups A Princeton mon coll6gue Atle Selberg s'6tait pos6 depuis quelque temps la question de l'arithm6ticit6 des sous-groupes discrets, /t covolume fini, des groupes semisimples; il avait formul6/l ce propos diverses conjectures, dont certaines portaient plus particuli6rement sur les groupes /t quotient compact, et avait obtenu des rdsultats partiels mais encourageants (cf. sa communication au Bombay Colloquium on Function Theory, T.I.F.R. 1960, pp. 147 164). I1 y a lfi une s6rie de probl6mes qui n'ont pas encore 6t6 tous rdsolus, mais sur lesquels de grands progr6s ont 6t6 faits dans les derni6res ann6es, principalement gr/tce fi G. A. Margulis (cf. J. Tits, SOm. Bourbaki, n ~ 482, f6v. 1976, qui contient une abondante bibliographie). Lorsqu'on se borne au cas des groupes "co-compacts" c'est-5~-dire/~ quotient compact, il s'agit, gdomdtriquement parlant, du probl6me des formes de CliffordKlein d'une structure gdom6trique homog6ne donn6e; si la structure en question

550

Commentaire

est celle d'espace riemannien /t courbure constante n6gative, on retrouve bien le probl6me classique des formes de Clifford-Klein de la g6om6trie hyperbolique. S'agissant d'un groupe semisimple quelconque, on peut, comme le faisait Selberg, appliquer cette id6e/l l'espace riemannien sym6trique associ6, pourvu du moins que le groupe discret op6re sans point fixe; d'apr6s un th6or6me de Selberg (loc. cit.), on peut toujours se ramener/t ce cas en rempla~ant au besoin le groupe donn6 par un sous-groupe d'indice fini. Pour moi, en abordant cette question, je trouvai plus avantageux d'op6rer dans le groupe de Lie lui-m~me, consid6r6 comme muni de la structure de G-vari6t6 d6finie dans [1960c], p. 374. J'y 6tais d'autant plus enclin que, dans l'avant-projet destin6/l Bourbaki dont il a 6t6 fait mention dans [1953]*, j'avais 6tudi6 ces structures avec quelque d6tail, sous le nom d'"espaces de Lie"; ce sont en somme les espaces qui sont "localement homog6nes principaux". Le passage au quotient par un groupe discret F donne alors sans autre hypoth6se une G-vari6t6, compacte si F est co-compact. Dans ses recherches, Selberg avait fait usage du pavage d'un espace riemannien sym6trique par des domaines fondamentaux poly6draux. I16tait plus naturel pour moi d'avoir recours ~ la m6thode des recouvrements ouverts, qui en mainte circonstance, combin6e ou non avec un minimum de cohomologie, m'avait rendu de si utiles services (v. [1941]*, [1946a], [-1949d], [1952a], [1952c], [-1955a]; cf. [1957c]*). Encore une fois elle r6pondit bien ~ ce que j'en attendais. Les ouverts dont j'avais besoin 6taient fi peu pr6s les "group-chunks" de [1955a], et je n'eus qu'~ imiter la proc6dure suivie dans [1952a], w pour v6rifier en toute g6n6ralit6 l'une des conjectures de Selberg; c'est le th6or6me 6nonc6 dans [1960c], p. 370.

[1961a] Ad61es and algebraic groups J'ai dit plus haut (v. [1959a]*) l'impression que m'avait faite en 1958 la d6couverte de Tamagawa; j'y consacrai mon cours fi l'Institute en 1959-60; c'est ce cours, excellemment r6dig~ par Demazure et Ono, et augment~ d'un appendice sur le groupe G2 par Demazure, qui forme le contenu du volume [1961a]. Les deux premiers chapitres traitent de ce qu'on pourrait appeler la g6om6trie et l'analyse ad61iques; les deux autres traitent des groupes classiques, et principalement du calcul du nombre de Tamagawa. Siegel, lorsqu'il avait d~couvert ses th6or6mes sur les formes quadratiques, avait donn~ des d6monstrations compl6tes pour les formes fi coefficients rationnels; pour les formes fi coefficients dans un corps de nombres alg~briques, il avait trait6 des cas particuliers typiques, puis avait laiss6 au lecteur le soin de suivj:e ses indications dans le cas g6n~ral. La m6thode ad61ique, elle, permet de traiter d'un seul coup tous ces cas et m6me celui off le corps de base est un corps de fonctions de caract6ristique p > 1. En ce sens, la premi6re d6monstration compl6te des th6or6mes de Siegel qui ait 6t6 publi6e est celle qui est contenue dans [1961 a], avec le compl6ment que devait lui apporter [1962a]; cela suffirait, s'il en 6tait besoin, &justifier l'emploi de la m6thode ad61ique dans ces questions. Quant fi l'intervention de fonctions zfita au Chap. IV de [1961a] (sugg6r6e elle aussi par des travaux de Siegel; v. ses Ges. Abh., n ~ 30-31, vol. II, pp. 41-96), elle

Commentaire

551

prend la place des 6valuations d'int6grales qui forment chez Siegel la "partie analytique" des d6monstrations. C'est ainsi que, dans la d6termination du nombre de classes par Dirichlet, les fonctions L avaient pris la place des 6valuations de volumes dont s'6tait servi Gauss dans sa solution (rest6e in6dite de son vivant) du mfime probl6me.

[1961b] Organisation et d6sorganisation en math6matique Au cours d'un nouveau s~jour au Japon, on me demanda de faire fi la Maison Franco-Japonaise de Tokyo (celle m~me dont Delsarte devint le directeur un peu plus tard) une causerie en fran~ais pour un public plus litt6raire que scientifique. J'en profitai pour m'attaquer au monstre de l'organisation et de la planification scientifiques, qui de nos jours pr6tend tout r6genter. "Le veau se cogne au chine", comme dit Soljenitsyn . . . .

[1962a] Sur la th6orie des formes quadratiques J'ai dit plus haut que la d6termination du nombre de Tamagawa pour lc groupe orthogonal 6quivaut au th6or~me principal de Siegel sur les formes quadratiques; mais ce n'est pas tout fi fait exact. Siegel avait donn6 une formule pour le hombre de repr6sentations d'une forme cp fiv variables par une f o r m e f ~ n variables, sur un corps de hombres alg6briques, pour tout v 0; plus prdcis6ment, il est "en g6ndral" 6gal/l la dimension du centre de G, ce qui entraine une correction dans la formule finale, p. 157.

Appendix I: Correspondence, by XXX A correspondent, who wishes to remain anonymous, writes as follows: 9 . . U n a notissima congettura di F. Severi (Rend. P~,l. 28 (1909), p. 45) asserisee ehe "ogni variet~ dotata di p~nti multipli si pu() considerate come

lira ire di una senza singolarit~, apparlenente allo stes.~o .~'pazio." Secondo una notizia orora diffusa da Naneago dall'agenzia " U n i t e d Press," l'ipotesi d e l F i l h s t r e autore sarebbe staia eonfutata dall'egregio geom e t r a franecse l~enato Thorn, basandosi sull'esempio dei toni del 3 ~ or(line nello spazio S~ e mediante assai delicate eonsiderazioni topolog'iehe. Forse non dispiaccr'~ ai lettori del Suo pregiato periodieo trovare qui una trattazione geometriea elementare dell'esempio di Thorn. Di fatti, si d e t e r m i n e r a n n o tutte le variet~ del 3 ~ ordine in uno spazio S,~ q u a h n q u e . Questo trarr~ con s~, come conseguenza immediata, la falsit~ dell'ipotesi suddetta. Sia Vr una variet/l del 3~ ordine nello spazio S~, di dimensione r < ~ - - 1, non eontenuta in un iperpiano. Sia C il eono, proiettante la Vr da un punto sempliee qualunque M di V,.; sia M' un altro punto di V~, sempliee sul eono C. I1 cono C ~ del seeondo ordine; quindi la sua proiezione da M ' una variet~ lineare di dimensione r - I - 1 , sieehg la V~ ~ contenuta in una varieth lineare di dimensione r - l - 2 . J~ dunque r ~ n - - 2 , e C h~ ]a dimensione n - - 1 . Lo stesso vale per il eono C', proiettante V~ da 3iq I eoni C, 6" sono distinti, giaeehg M" ~ sempliee sul eono C ; la loro intersezione dunque una varieth ridueibile del 4=.~ ordine, spezzata nella V~ e una varieth lineare. Siano A ~ 0, B ~ 0 le equazioni di quest'ultima. Allora si possono serivere le equazioni dei eoni C, C' nella f o r m a :

A P = BQ,

(1)

A P ' = B(2',

denotando con P, Q, P', (2' quattro forme lineari helle coordinate omogenee hello spazio. Sia s il numero di forme indipendenti fra ]e sei forme A, B, P, (2, P', (2'; questo ~ almeno 4, giaceM, se fosse 2, i eoni C, C' non sarebbero irriducibili, e, se fosse 3, la loro intersezione si spezzerebbe in quattro variet~ lineari distinte o coincidenti. I valori possibili per s sono quindi 4, 5 e 6. Received June 3, 1957. 951 Reprinted from the Am. Y. of Math. 79, 1957, pp. 951-952, by permission of the editors, 9 1957 The Johns Hopkins University Press.

555

556

Appendix I 952

x.x.x.

Sia L la varietlt lineare, di dimensione . n - - s , definita dalle equazioni A~B=0, P=Q=P'=Q'~o. Ogni varlet5 lineare, proiettante da L un punto dell'intersezione di (7 e C', 6 eontenuta in questa, come si vede subito sulle equazioni (1). P r o i e t t a n d o V,- da L, si ottiene quindi una variet5 W del 3 ~ ordine di dimensione s - - 3 in uno spazio S,-1; e V,. non ~ altro the il cono proiettante g da L. Nello spazio S, 1, si pu6 prendere per coordinate omogenee s forme indipendenti fra le forme A, B, P, Q, P', Q'; allora le equazioni (1) vi definiscono una varietg del 4 ~ ordine, spezzata nella W e una variet~ lineare. Adesso distinguiamo ire casi: (a) s = 6 : IF ~ la varietg di Segre Wa immersa in S~, immagine birazionale senza eceezione del prodotto di un piano e una t e t r a ; come 6 nolo, priva di punti multipli. (b) s ~ 5: IF ~ sezione iperpiana della sopradetta IF;. 1~3 faerie vedere the tutte le sezioni iperpiane irridueibili della IFa sono proiettivamente equivalenti fra di loro; denotando (:on IF., una di esse, 5 anch'essa una variet~ razionale senza punti multipli. (c)

s = 4: IF ~'~ la ben nota c~;biea razionale IF~ immersa in Sa.

Cosl ~ dimostrato ehe ogni variet~'~ del 3 ~ ordine appartiene a uno dei seguenti tipi : 1~ le variet~k di dimen,~ione r, eontenute in una varieta lineare di dimensione r q- 1 ; 2~

le tre variet'5 razionali W;~,, W.,, IV1 enumerate disopra;

3 ~ i eoni proiettanti una di queste tre da una variet~ lineare di dimensione qualunque. Evidentemente, una variet& del secondo o terzo tipo non pu6 essere limite di una varietg del primo tipo, e perci6 una variet~ del terzo tipo, di dimensione maggiore di 3, non pub essere limite di varietfi prive di punti multipli. X. X. X.

Appendix II: Correspondence, by R. Lipschitz We have received the following letter, purporting to come from an ultramundane correspondent :

SIR, It iS sometimes a matter of wonder, to us in Hades, that what we had believed to be our best work remains buried under thick layers of dust in your libraries, while the very talented young men in the mathematical world of the present day strive manfully against problems which are by no means as novel as they think. For instance, it is not so long ago that the very remarkable algebraic systems discovered by my friend Professor Clifford shortly before leaving your world have again attracted the attention of your algebraists after many years of oblivion. When, during my lifetime, I first became interested in them, I, too, fancied that they were new; I soon found out my mistake, and hastened to acknowledge Professor Clifford's prior discovery. It is now a matter of great satisfaction to me to hear that his name has been given to them, as a fitting tribute to his memory among the living. On the other hand, as Professor Clifford has told me himself, it had not occurred to him to apply these algebraic systems to the study of the substitutions which transform a sum of squares into a sum of squares (or, as my young friend and colleague Hermann Weyl would say, of the orthogonal group); he kindly insists that this idea was wholly mine. As you may well believe, we have often discussed this topic since I had the honour of joining the distinguished company of the mathematicians in the Elysian Fields; incidentally, without the many delightful conversations which I have had with him, I should hardly be able now to write to you in English (a feat which ! could have accomplished only with g r e a t difficulty d u r i n g m y lifetime). I t is not, h o w e v e r , in o r d e r to a s s e r t m y claims to f a m e in this m a t t e r t h a t I a m n o w a s k i n g f o r the h o s p i t a l i t y of y o u r j o u r n a l . In w h a t y o u a r e pleased to call t h e n e t h e r world, we a r e h a p p i l y f r e e f r o m v a i n g l o r i ous feelings. B u t it m a y be u s e f u l to a f e w of y o u r c o n t e m p o r a r i e s to h a v e t h e i r a t t e n t i o n d r a w n upon s o m e f o r m u l a s c o n t a i n e d in m y m e m o i r U n t e r s u c h u n g e n iiber die S u m m e n von Q u a d r a t e n (a b r i e f a c c o u n t of which m a y be f o u n d in t h e B u l l e t i n des Sciences M a t h ~ m a t i q u e s f o r 1886), since t h e y would be s o u g h t in vain, unless I a m m u c h m i s t a k e n , in v a r i o u s 247 Reprinted fromAnn. of Math. 69, 1959, by permissionof Princeton UniversityPress. 557

558

Appendix H

248

CORRESPONDENCE

learned volumes r e c e n t l y published on this v e r y subject. U n f o r t u n a t e l y , it appears t h a t t h e r e is now in y o u r world a race of vampires, called r e f e r e e s , who clamp down mercilessly upon m a t h e maticians unless t h e y k n o w the r i g h t passwords. I shall do m y best to modernize m y l a n g u a g e and notations, b u t I am well a w a r e of m y shortcomings in t h a t r e s p e c t ; I can assure you, a t a n y r a t e , t h a t m y intentions are honourable and m y results i n v a r i a n t , probably canonical, p e r h a p s even functorial. But please allow me to a s s u m e t h a t the characteristic is not 2. Call e , - 9 ",G the g e n e r a t o r s of P r o f e s s o r Clifford's algebraic system; this m e a n s t h a t e l - - 1 for all i, and eje~= --v,ej for i < j. For each set [ = {i~, - - - , i,,,} of indices, w r i t t e n in t h e i r n a t u r a l o r d e r < i~< . . - < i , ~ = n ,

1~i, put e(f)

=

% % . . - %,~,

with e(I) = 1 if m = 0; the set I and the unit e(I) will be called even if m is even, odd if m is odd. L i n e a r combinations of e v e n (resp. odd) units will be called e v e n (resp. odd) quantities. Now t a k e an a l t e r n a t i n g m a t r i x X = ( x ~ ) , and assume at first t h a t the d e t e r m i n a n t of E + X ( w h e r e E is the unit m a t r i x ) is not 0. My learned and illustrious colleague P r o f e s s o r C a y l e y was, I believe, the first one to o b s e r v e t h a t , if X is such a m a t r i x , the f o r m u l a (1)

U=

(E--X)-(E+X)-'

defines an o r t h o g o n a l m a t r i x U, and t h a t c o n v e r s e l y X can be expressed in t e r m s of U by the f o r m u l a

(2)

X = (E--

U).(E

+ U ) -~ .

For each e v e n set J----{j~, - . . , j~,} of indices ( w r i t t e n , as always, in t h e i r n a t u r a l order), p u t

x(J) =

1 ~ 2Pp!

s(J, H)xI,~1, x~:/, ""x1%~_,,~

w h e r e the s u m m a t i o n is e x t e n d e d to all p e r m u t a t i o n s H of d, and ~(J, H) is q 1 or - 1 according as the p e r m u t a t i o n is e v e n or odd; p u t x ( J ) = l for p -- O. Consider the e v e n q u a n t i t y

(3)

~ = ~S(J)e(J)

w h e r e the s u m m a t i o n is e x t e n d e d to all e v e n sets of indices J. On the o t h e r hand, t a k e two vectors ~ = ( ~ , , - . - , G ) , ~-(r~,, . . . , ~5,) such t h a t 5 = U~; by the definition of U, this can be w r i t t e n as

Appendix H

559 CORRESPONDENCE

249

or a g a i n as

S~e~ § ~ j xLje~ej.$ jej = ~e~ § ~

7ijej.x~je~ej .

Multiply this to t h e l e f t w i t h 1

2p ~ - Xl~lT~2XhS~z4

"

9

9

Xlb21)_l&21ehi~lt2

*

9

9

el~2~~

a n d t a k e t h e s u m o v e r all s e q u e n c e s ( h , . . . , h,~, i) of distinct indices in odd n u m b e r . I t is easily verified t h a t t h e r e s u l t can be w r i t t e n as

c o n v e r s e l y , if (5) holds, a c o m p a r i s o n of the coefficients of the units et on b o t h sides will s h o w t h a t (4) is satisfied, so t h a t ~ = US. Thus, to e v e r y o r t h o g o n a l m a t r i x U which can be e x p r e s s e d as in (1), we h a v e associated an e v e n q u a n t i t y ~ such t h a t (5) is e q u i v a l e n t to rj = US; as (2) shows, this will be so w h e n e v e r d e t ( E + U ) is not 0. N o w , for e v e r y set I of distinct indices i , . . . , i~,, d e n o t e b y D(I) t h e d i a g o n a l m a t r i x w h o s e diagonal coefficients 3~ a r e such t h a t a~ is - I or + 1 a c c o r d i n g as i is in I or not. I f A is an a r b i t r a r y m a t r i x , one can easily v e r i f y t h e i d e n t i t y ~flet(E

+ D(J)A) = 2~-~(1 + d e t ( A ) ) ,

w h e r e t h e s u m is t a k e n o v e r all e~zen sets J ; in p a r t i c u l a r , if det(A) is not -- 1, a t least one t e r m on t h e l e f t - h a n d side m u s t be o t h e r t h a n 0. T h e r e f o r e , if U is an o r t h o g o n a l m a t r i x of d e t e r m i n a n t + 1, t h e r e is a t l e a s t one of the o r t h o g o n a l m a t r i c e s U' = D ( J ) U which can be e x p r e s s e d b y (1), so t h a t w e can associate w i t h it an e v e n q u a n t i t y t2' such t h a t t h e relation is e q u i v a l e n t to w i t h ~ e q u a l to -- l or + l a c c o r d i n g a s i i s in J or not. B u t t h e n (5) will be e q u i v a l e n t to ~ = US p r o v i d e d we p u t t 2 = e ( J ) t 2 ' . I f U had b e e n a n o r t h o g o n a l m a t r i x of d e t e r m i n a n t --1, one could h a v e r e a c h e d a similar conclusion, p r o v i d e d n is e v e n , b y considering odd instead of e v e n sets. N o t only h a v e we t h u s p r o v e d t h a t , to e v e r y o r t h o g o n a l s u b s t i t u t i o n U of d e t e r m i n a n t + 1 , t h e r e is an e v e n q u a n t i t y ~q such t h a t (5) is e q u i v a l e n t to ~.= US, b u t we h a v e also g i v e n an explicit m e t h o d f o r the c o n s t r u c t i o n of t2 (which, as m a y readily be seen, is uniquely d e t e r m i n e d

560

Appendix H

250

CORRESPONDENCE

by this condition up to a scalar factor). Therein, I believe, lies t h e a d v a n t a g e of the m e t h o d followed in m y m e m o i r . To e v e r y q u a n t i t y ~ , let us associate a n o t h e r one ~ * b y p u t t i n g (ehei ~ - ' - ei~)* = e~ 9

e~2e~I ,

and e x t e n d i n g this to all q u a n t i t i e s by l i n e a r i t y . If, in f o r m u l a (1), we s u b s t i t u t e - X for X, this c h a n g e s U into U -~ and ~2 into t2*. F r o m this, one concludes at once t h a t , if St is t h e q u a n t i t y associated w i t h a n y o r t h o g o n a l s u b s t i t u t i o n , ~st* m u s t be a scalar; in m y memoir, this scalar was called the n o r m of t~ and d e n o t e d by N(t2); now, as I a m told, it is k n o w n as the spinor norm. In particular, if t2 is g i v e n by (3), its n o r m is g i v e n b y N(a)

= aSt*

jx(J) .

=

Since x(J) is the pfaffian of the m a t r i x consisting of t h e x,j for i e J , j e J , its s q u a r e x(J) ~ is the d e t e r m i n a n t of t h a t m a t r i x . F r o m this it follows at once t h a t N ( ~ ) is n o t h i n g else t h a n t h e d e t e r m i n a n t of E + X; e x p r e s s i n g X in t e r m s of U by (2), we g e t N(t~) = 2 ~ . d e t ( E + U) -1 Similarly, if J is an e v e n set such t h a t t h e d e t e r m i n a n t of E + D ( J ) U is not 0, p u t U' = D(J)U, and call X ' and ~2' t h e m a t r i x and t h e q u a n t i t y derived f r o m U' as X and ~2 w e r e d e r i v e d f r o m U. T h e n we h a v e N(t2') = 2 ~. d e t ( E + D(J)U) -1 . On the o t h e r hand, we see as above t h a t ~ and e(J)~' can differ only b y a scalar factor; c o m p a r i n g t h e coefficients of e(J) in these two quantities, we g e t

t2 = x( J)e( J)t2' , so t h a t we h a v e d e t ( E + D(J)U) = 2~N(t2)-lx(J)'~. As I m e n t i o n e d in m y memoir, t h e s e f o r m u l a s can be combined into a single one as follows. L e t t = ( t l , . . . , t~) be a vector; w r i t e diag(t) for t h e diagonal m a t r i x with the diagonal coefficients $i, . . . , t~; and put, for each e v e n set of indices J : A(J) = d e t ( E + D(J)diag($)). T h e n we h a v e N(t2) d e t ( E + diag(t)U) -- ~ j

A(J)x(J)~;

m o r e o v e r , as this f o r m u l a is h o m o g e n e o u s in t h e coefficients of t h e e v e n

Appendix H

561

251

CORRESPONDENCE

quantity 12, it remains valid if t2 is multiplied by a scalar factor. Therefore it holds whenever U is an orthogonal matrix of determinant + 1, provided is an even quantity, given by (3), such t h a t (5) is equivalent to ~]= US. I can still vividly recall my pleasure when I first came across this result in bygone days. But I fear t h a t I am becoming garrulous, and t h a t your patience with me may be exhausted by now. I have the honour to be, etc. R. LIPSCHITZ