Geometric Galois Actions: The inverse Galois problem, moduli spaces, and mapping class groups

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[1

243

Geometric Galois Actions 2. The Inverse Galois Problem. Moduli Spaces and Mapping Class Groups

Edited by

Leila Schneps & Pierre Lochak

CAMBRIDGE UNIVERSITY PRESS

LONDON MATHEMATICAL MATHEMATICAL SOCIETY SOCIETY LECTURE LECTURE NOTE NOTE SERIES SERIES Managing Editor: Professor Professor J.W.S. Cassels, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 ISB, University 1SB, England England difficulty, from Urtiversity Press. Press. The titles below are available from booksellers, or, in case of difficulty, from Cambridge University

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JAMES Representations Diophantine equations over function function fields, R.C. MASON Varieties of constructive mathematics, mathematics, D.S. BRIDGES & F. RICHMAN RICHMAN Localization in Noetherian rings, A.V. JATEGAONKAR Localization differential geometry in algebraic topology, M. KAROUBI & Methods of differential & C. LERUSTE LERUSTE Stopping time techniques for analysts and and probabilists, probabilists, L. EGGHE Elliptic structures on 3-manifolds, 3-manifolds, C.B. THOMAS THOMAS ERDELYI & WANG WANG SHENGWANG A local spectral theory for closed operators, operators, I.I. ERDELYI Compactification of Siegel moduli Compactification moduli schemes, schemes, C.-L. CHAI CHAI Diophantine analysis, J. LOXTON LOXTON & A. VAN DER POORTEN (eds) An introduction to surreal numbers, numbers, H. GONSHOR GONSHOR ideals, D. REES Lectures on the asymptotic theory of of ideals, Lectures on Bochner-Riesz Bochner-Riesz means, means, K.M. K.M.DAVIS DAVIS&&Y.-C. Y.-C.CHANG CHANG Representations of algebras, PJ. WEBB algebras, P.J. WEBB (ed) Skew linear groups, groups, M. Skew linear M.SHIRVANI SHIRVANI&&B. B.WEHRFRITZ WEHRFRITZ Triangulated categories in in the the representation representation theory theory of of finite-dimensional finite-dimensionalalgebras, algebras, D. HAPPEL Proceedings of Groups ROBERTSON & C. CAMPBELL CAMPBELL (eds) (eds) Groups -- St Andrews 1985, 1985, E. ROBERTSON Non-classical continuum mechanics, R.J. KNOPS KNOPS & A.A. LACEY LACEY (eds) mechanics, R.J. Descriptive set theory and the structure structure of of sets sets of of uniqueness, uniqueness, A.S. A.S. KECI-IRIS KECHRIS & A. LOUVEAU P.8. KLEIDMAN The subgroup structure of of the the finite finite classical classical groups, groups, P.B. KLEIDMAN & & M.W. M.W. LIEBECK LIEBECK Model theory and modules, and modules, M. PREST PREST Algebraic, extremal & metric combinatorics, combinatorics, M.-M. M.-M. DEZA. DEZA, P. P. FRANKL FRANKL & & 1G. I.G. ROSENBERG ROSENBERG (eds) Whitehead groups of finite finite groups, groups, ROBERT ROBERTOLIVER OLIVER Linear algebraic monoids, MOHAN algebraic monoids, MOHAN S. S. PUTCHA PUTCHA Number theory theory and and dynamical dynamical systems, systems, M. Number M.DODSON DODSON & & J.J. VICKERS VICKERS (eds) Operator algebras and applications, I,1, D. EVANS & M. TAKESAKI (eds) Analysis at Urbana, I,I, E. BERKSON, T. PECK, PECK, & & J.J. UHL UHL (eds) (eds) Analysis at Urbana, II, II, E. BERKSON, T. PECK, PECK, & & J.J. UHL UHL (eds) (eds) Advances in homotopy theory, S. SALAMON, B. STEER & W. SUTHERLAND (eds) Geometric aspects of Banach spaces, E.M. PEINADOR & A. RODES (eds) Geometric aspects of Banach spaces, E.M. PEINADOR Surveys in combinatorics 1989, 1989, J. SIEMONS (ed) Introduction to uniform spaces, I.M. JAMES Homological questions questions in in local localalgebra, algebra, JAN Homological JAN R. R. STROOKER STROOKER Cohen-Macaulay modules over Cohen-Macaulay rings, rings, Y. YOSHINO Helices and andvector vectorbundles, bundles, A.N. AN. RUDAKOV Helices RUDAKOV eteta! al Solitons. nonlinear nonlinear evolution evolution equations equations and and inverse inverse scattering, scattering, M. ABLOWITZ P. CLARKSON CLARKSON Solitons, ABLOWITZ & P. Geometry of low-dimensional manifolds I,1, S. DONALDSON & & C.B. C.B. THOMAS THOMAS (eds) (eds) Geometry of low-dimensional manifolds 2. 5. DONALDSON & C.B. THOMAS (eds) Geometry 2, S. DONALDSON Oligomorphic permutation groups, groups, P. CAMERON L-functions and arithmetic, J. COATES & M.J. TAYLOR (eds) L-functions (eds) Classification theories theories of of polarized polarizedvarieties, varieties, TAKAO Classification TAKAO FUJITA Twistors in mathematics mathematics and andphysics, physics, TN. Twistors in T.N.BAILEY BAILEY&&R.J. R.J. BASTON BASTON (eds) (eds) Geometry of Banach spaces, P.F.X. MULLER & W. SCHACHERMAYER SCHACHERMAYER (eds) Groups Andrews 1989 volume 1, I, C.M. Groups St Andrews 1989 volume CM. CAMPBELL & E.F. E.F. ROBERTSON ROBERTSON (eds) (eds) (edt) Groups St Andrews 1989 1989 volume 2, C.M. CM. CAMPBELL CAMPBELL & E.F. ROBERTSON ROBERTSON (eds) Lectures on block block theory, theory, BURKHARD BURKHARDKULSHAMMER KULSHAMMER Harmonic analysis and and representation representation theory, theory, A. FIGA-TALAMANCA & C. C NEBBIA NEBBIA Topics in varieties varieties of of group group representations, representations, S.M. VOVSI Quasi-symmetric designs, designs, M.S. SHRIKANDE & S.S. SANE Surveys in combinatorics, combinatorics. 1991, 1991, AD. A.D.KEEDWELL KEEDWELL(ed) (ed) Representations of algebras, H. TACHIKAWA & S. BRENNER (eds) (eds)

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Boolean function Boolean function complexity, M.S. PATERSON PATERSON (ed) (ed) Manifolds with singularities and the Adams-Novikov spectral sequence, B. BOTVINNIK Manifolds Squares. A.R. RAJWADE Squares, Algebraic varieties, varieties, GEORGE Algebraic GEORGE R. R. KEMPF KEMPF Discrete groups groups and and geometry, geometry. W.J. Discrete W.J.HARVEY HARVEY&&C. C.MACLACHLAN MACLACHLAN (eds) (eds) Lectures on on mechanics, mechanics, i.E. J.E. MARSDEN MARSDEN Adams memorial memorial symposium symposium on on algebraic algebraic topology topology 1, I, N. RAY & G. WALKER (eds) Adams Adams memorial symposium on algebraic algebraic topology topology 2, 2, N. RAY & G. WALKER (eds) Applications of of categories categories in in computer computerscience, science, M. JOHNSTONE & & A. A. PITTS PIUS (eds) Applications M. FOURMAN, FOURMAN, P. JOHNSTONE (eds) and L-theory, A. RANICKI Lower KK-and RANICKI Complex projective projectivegeometry, geometry, G. eta! Complex G. ELLINGSRUD ELLINGSRUD et al Lectures on on ergodic ergodic theory theory and and Pesin Pesin theory theoryon oncompact compactmanifolds, manifolds, M. POLLICOTT Lectures POLLICOTT Geometric group group theory theory!, Geometric I, G.A. NIBLO & M.A. ROLLER (eds) Geometric group theory II, II, G.A. NIBLO & M.A. ROLLER (eds) Shintani zeta functions, A. YUKIE Arithmetical functions, W. Arithmetical functions, W.SCHWARZ SCHWARZ & &3. J. SPILKER SPILKER Representations 0. MANZ Representations of solvable groups, O. MANZ & & T.R. T.R. WOLF WOLF Complexity: knots, colourings and and counting, counting, D.J.A. WELSH Surveys in combinatorics, combinatorics, 1993, 1993, K. WALKER (ed) Local analysis analysis for for the the odd odd order order theorem, theorem, H. H.BENDER BENDER & & G. G. GLAUBERMAN GLAUBERMAN Locally presentable and accessible categories, categories, J. ADAMEK & J. ROSICKY ROSICKY Polynomial invariants of finite finite groups, groups, Di. D.J.BENSON BENSON Finite geometry and combinatorics, F. DE CLERCK CLERCK et eta! al Symplectic geometry, D. SALAMON (ed) Computer algebra and differential equations, E. TOURNIER (ed) (ed) Independent random variables and rearrangement invariant spaces, M. VERMAN M.BRA BRAVERMAN Arithmetic blowup algebras, algebras, WOLMER Arithmetic of blowup WOLMER VASCONCELOS V ASCONCELOS Microlocal analysis for for differential differential operators, operators, A. GRIGIS & J. SJOSTRAND SJOSTRAND Two-dimensional homotopy and combinatorial group theory, C. HOG-ANGELONI, HOG-ANGELONI, METZLER & A.J. SIERADSKI (eds) (eds) W. METZLER Al. SIERADSKI The algebraic algebraic characterization characterization of of geometric geometric4-manifolds, 4-manifolds, iA. J.A.HILLMAN HILLMAN n Invariant potential potential theory theory in in the the unit unit ball ball of of C Cit, , MANFRED MANFREDSTOLL STOLL Invariant The Grothendieck theory of dessins denfant, d'enfant, L. SCHNEPS (ed) Singularities, JEAN-PAUL JEAN-PAULBRASSELET BRASSELET (ed) (ed) The technique of pseudodifferential pseudodifferential operators, operators, HO. H.O.CORDES CORDES Hochschild cohomology of von Hochschild von Neumann Neumann algebras, algebras, A. SINCLAIR SINCLAIR & R. SMITH Combinatorial and and geometric geometric group group theory, theory, A.J. DUNCAN, N.D. GILBERT & J. HOWIE (eds) Ergodic theory and its connections with harmonic harmonic analysis, analysis, K. PETERSEN & I. SALAMA (eds) 3. MADORE An introduction to to noncommutative noncommutative differential differential geometry geometry and and its its physical physical applications, applications, J. Groups of Lie type and their geometries, W.M. W.M.KANTOR KANTOR&&L. L.DI DIMARTINO MARTINO(eds) (eds) NEWSTEAD & W.M. OXBURY OXBURY (eds) Vector bundles in algebraic geometry, N.J. HITCHIN, P. NEWSTEAD Arithmetic of diagonal hypersurfaces hypersurfaces over finite fields, fields, F.Q. GOUVEA & N. YUI Hilbert C*-modules, E.C. LANCE LANCE Groups 93 CAMPBELL et eta! 93 Galway Galway // St StAndrews Andrews1,I, C.M. CM. CAMPBELL al eta! Groups 93 Galway /St / StAndrews Andrews11, II, C.M. CM. CAMPBELL CAMPBELL et al Generalised Euler-Jacobi inversion formula and asymptotics beyond all orders, V. KOWALENKO, N.E. FRANKEL, M.L. GLASSER GLASSER & T. TAUCHER TAUCHER Number theory theory1992—93, 1992-93, S. DAVID(ed) DAVID (ed) Stochastic partial differential differential equations, A. ETHERIDGE ETHERIDGE (ed) (ed) Quadratic forms with applications to to algebraic algebraic geometry geometry and and topology, topology, A. PFISTER Surveys in in combinatorics, combinatorics, 1995, 1995, PETER Surveys PETER ROWLINSON ROWLINSON (ed) Algebraic set theory, A. JOYAL & & 1.I. MOERDIJK GARDINER Harmonic approximation, Si. S.J.GARDINER Advances in linear logic, logic, i.-Y. J.-Y. GIRARD, GIRARD, Y. Y.LAFONT LAFONT & &L. L.REGNIER REGNIER (eds) (eds) semigroups and and semilinear semilinear initial initialboundary boundaryvalue valueproblems, problems, KAZUAKI TAIRA Analytic semigroups KAZUAKITAIRA Computability, enumerability, unsolvability, unsolvability, S.B. COOPER. COOPER, T.A. SLAMAN & S.S. WAINER WAINER (eds) (eds) A mathematical introduction to to string string theory, theory, S. ALBEVERIO, J. JOST, S. PAYCHA, S. SCARLATTI SCARLATTI Novikov conjectures, index index theorems theorems and and rigidity rigidityI,I, S. FERRY, A. RANICKI & & J. J. ROSENBERG ROSENBERG (eds) (eds) index theorems theorems and and rigidity rigidityII, II, S. FERRY, A. RANICKI & J. ROSENBERG (eds) Novikov conjectures, index d Ergodic theory of Zd POLLICOTT&&K.K.SCHMIDT SCHMIDT(mis) (eds) Z actions, M. POLLICOU Ergodicity for infinite dimensional systems, G. DA PRATO & I. J. ZABCZYK ZABCZYK Prolegomena to a middlebrow arithmetic of curves of genus 2, J.W.S. CASSELS & & E.V. E.V. FLYNN FLYNN and its its applications, applications, K.H. HOFMANN & M.W. MISLOVE (edt) (eds) Semigroup theory and The descriptive set theory of of Polish Polish group group actions, actions, H. BECKER & AS. A.S.KECHRIS KECHRIS Finite fields and and applications, applications, S. COHEN & H. NIEDERREITER (eds) Introduction to subfactors, V. JONES & V.S. SUNDER Introduction SUNDER Number theory Number theory1993—94, 1993-94, S. DAVID (ed) The James forest, forest, H. FETTER & & B. B. GAMBOA GAMBOA DE DE BUEN BUEN GREAVES. Sieve methods, exponential exponential sums, and their applications applications in in number number theory, theory, G.R.H. GREAVES, Sieve G. HARMAN & & M.N. M.N. HUXLEY HUXLEY (eds) (eds) MARTSINKOVSKY & G. TODOROV (eds) Representation theory and algebraic geometry, geometry, A. MARTSINKOVSKY Clifford Clifford algebras and and spinors, spinors, P. LOUNESTO Stable groups, groups, FRANK Stable FRANK 0. O.WAGNER WAGNER Geometric Galois actions 1, I, L. SCHNEPS SCHNEPS & & P. P. LOCHAK LOCHAK (eds) (eds) Geometric Galois Galois actions actions 11. II, L. SCHNEPS & P. LOCHAK (eds)

London Mathematical Society Society Lecture Lecture Note Series. Series. 243 243

Galois Actions Geometric Galois 2. The TheInverse Inverse Galois Galois Problem, Problem, Moduli Moduli Spaces and Mapping Class Groups Groups Edited by

Leila Schneps Schneps CNRS CNRS and Pierre Lochak CNRS

CAMBRIDGE UNIVERSITY UNIVERSITY PRESS PRESS

CAMBRIDGE UNIVERSITY PRESS PRESS Cambridge, New York, Melbourne, Madrid, Cape Cape Town, Town, Singapore, Singapore, São Sao Paulo, Paulo, Delhi Delhi

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Table of of Contents Table Introduction

Abstracts

vii vii

11

Part I.I. Dessins Dessins d'enfants d'enfants Unicellular cartography and Galois orbits of plane trees Unicellular cartography N. Adrianov Adrianov and G. Shabat Galois groups, monodromy groups and cartographic groups monodromy groups groups G. Jones and M. M. Streit Streit Permutation techniques techniques for for coset coset representations representations of of modular subgroups subgroups T.Hsu T. Hsu Dessins Dessins d'enfants en en genre genre 11 L. Zapponi Zapponi

13 25 25 67 67

79

Part II. II.Inverse Inverse Galois Galois problem problem The regular inverse inverse Galois Galois problem problem over over large large fields fields P. Debes and B. B. Deschamps Deschamps The symplectic symplectic braid group and Galois Galois realizations realizations K. Strambach and and H. H. Völklein Volklein Applying modular towers to the Applying the inverse inverse Galois Galois problem problem M. Fried and Y. Y. Kopeliovich Kopeliovich

119 139 151 151

Part III. III. Galois Galois actions, actions, braids braids and and mapping mapping class class groups groups Galois Galois group GQ, GQ, singularity E7 E7 and and moduli moduli M3 M3 M. Matsumoto Monodromy of of iterated iterated integrals and non-abelian unipotent periods Monodromy periods Z.Wojtkowiak Z. Wojtkowiak

179 179

219 219

Part IV. IV. Universal Universal Teichmüller Teichmiiller theory

The universal Ptolemy group and its its completions completions 293 R. Penner 293 Sur du groupe Sur l'isomorphisme l'isomorphisme du groupe de Richard Richard Thompson Thompson avec avec le le groupe groupe de de Ptolémée Ptolemee 313 M. Imbert 313 The universal universal Ptolemy-Teichmuller Ptolemy-Teichmuller groupoid groupoid 325 P. Lochak and L. L. Schneps Schneps 325

Intro duct ion Introduction This grew out out of the conference which was was held held at at Luminy This volume volume grew conference which Luminy in August on the the theme "Geometry August 1995 1995 on "Geometry and Arithmetic Arithmetic of of Moduli Moduli Spaces". Spaces". In some sense, sense, it was was conceived conceived as a sequel sequel to to the the1993 1993Luminy Luminyconference conference on "The d'Enfants", which on "The Grothendieck Grothendieck Theory Theory of Dessins Dessins d'Enfants", which gave gave rise rise to proceedingsbearing bearingthe the same same title title (this series, proceedings series, number number 200). 200). The The second second conference revolved revolved mostly mostly around around some "multidimensional" versions conference versions of of the the themes consideredinin the the first first one. one. All themes considered All these these themes themes are are developments developments of ideas expressed Grothendieck's Esquisse Esquisse dd'un seminal ideas expressed in Grothendieck's 'un programme. This seminal text has now now been published published in companion volume (this text has in the companion volume to to this this one (this series, series, number 242). section of of the the Esquisse, Grothendieck Grothendieck sketches sketchesout out what what he In the second section theory". This This theory theory is an an approach approach to the study terms "Galois-Teichmüller "Galois-Teichmuller theory". of the the absolute absolute Galois Galois group group Gal(Q/Q), Gal(Q/Q), via the action of this group of group on the the —profinite— Teichmüller modulargroups groups(alias (aliasmapping mappingclass class groups), groups), which -profiniteTeichmuller modular are the fundamental fundamental groups groups of of the moduli moduli spaces spaces of of Riemann Riemann surfaces surfaces with with marked Grothendieck's theory marked points. points. By Grothendieck's theory of of the the fundamental fundamental group, group, ifif XX is a scheme defined over over Q, Q, then Gal(Q/Q) acts acts on on the the algebraic algebraic fundamental fundamental group *1 TTI(X Esquisse Grothendieck singles out the case of (X 0(8>Q). In the Esquisse Grothendieck singles out the case of the Q). moduli spaces as being of particular interest. Note that the "fine" moduli moduli spaces as being of particular interest. Note that "fine" moduli spaces schemes but however aa similar similar theory applies (one spaces are are not not schemes but stacks, however theory applies can Oda in in the the companion companion volume volume to this one). one). can consult consult the the article by T. Oda Grothendieck points points out out that "even" In the third section section of the Esquisse, Esquisse, Grothendieck "even" case is already interesting. This the simplest simplest case already extremely extremely interesting. This case case concerns concerns the of spheres with four four ordered ordered marked smallest non-trivial moduli moduli space, space, that of which is removed. The The points, which is isomorphic isomorphicto to the the sphere sphere with with three points removed. study of the the Galois Galois action action on on the the fundamental fundamental group group of of this this space space (the (the study profinite completion generators) is equivalent equivalent to profinite completion of of the the free free group group on two generators) study of of the the Galois Galois action action on on finite finite covers covers of the projective line ramified ramified the study ByBelyi's Belyi's famous famous theorem, theorem, every every algebraic algebraic curve over at most three points. By defined over defined over aa number number field fieldcan can be be realized realized as assuch suchaacover. cover. The The study study of this subject action is known known as as the the theory of of "dessins "dessins d'enfants", d'enfants", and formed the subject of the 1993 1993 conference. conference. The 1995 1995 conference conference dealt with the generalization dealt generalization of this theme to all all moduli moduli spaces, spaces, as discussed discussed in the second second section of the Esquisse. Esquisse.

During the conference, we learned learned that that Jean Malgoire conference, we Malgoire had obtained obtained from from Grothendieck Grothendieck permission permission to publish some of his mathematical manuscripts, manuscripts, among which was the the Esquisse. We We are very grateful to to him him for for suggesting suggesting to to

viii

Introduction

us that itit appear appearas aspart partofofaavolume volume of of proceedings. proceedings. It took took some some thought to to decide on the best best way way of of including including the the Esquisse Esquissein in such such aavolume; volume; moreover moreover we were were agreeably agreeablysurprised surprisedby bythe the quantity quantity of of papers papers submitted submitted both by by participants participants at at the theconference conference and and also also by certain non-participants non-participants who who were were unable to come come but would would have liked to. The The result result isisthe thetwo-volume two-volume series series Geometric Galois Actions, Actions, of of which which the the first first volume contains the Esquisse volume contains Esquisse itself, a letter letter from from Grothendieck Grothendieck to Fait ings on subject of of anabelian anabelian itself, to Faltings on the subject geometry, and various contributions all all of which play a role geometry, and various contributions which play role in in clarifying clarifying specific themes themes introduced introduced by byGrothendieck Grothendieck—- or some some of the some of the specific themes raised raised by bythese theseclarifications! clarifications!—- whereas the second, present, volume themes original research papers submitted at at the theconference. conference. contains the original

Let us To begin us briefly briefly survey survey the the contents contents of of this this volume. volume. To begin with we we include the the abstracts include abstracts of the the talks talks which which were were actually actually given given at at Luminy; Luminy; they were divided into into five five short short courses of two two lectures lectures each, each, a series they were divided courses of series of individual talks, and an evening seminar seminar given given by by graduate graduate students, aimed individual aimed at understanding the basics basics of Teichmüller Teichmuller and and moduli modulispaces. spaces. The The rest rest of of the volume volume isis divided dividedinto intofour fourseparate separatebut but related related sections. sections. The The first first part, previous Dessins dd'enfants, 'enfants, contains contains five five articles articles forming forming aa bridge bridge with with the previous conference. considers a special conference. The first one, by N. Adrianov Adrianov and G. Shabat, considers subclass aim of of determining determining finer finer Galois Galois subclass of of genus genus zero zero dessins, dessins, with with the aim invariants than the well-known valency lists. The paper by G. Jones and invariants than well-known valency lists. The paper by M. Streit contains some introductory sections sections to to dessins dessins which which are are basically basically self-contained, and also gives self-contained, gives aa flavour flavour of ofwhat what has has been been happening happening recently in the field. field. The next paper by by T. T. Hsu Hsu illustrates illustrates how how group group theoretic theoretic in the The next and combinatorial combinatorial techniques on arithmetic arithmetic problems, problems, techniques can can be be used used to bear on thanks to the very very visual nature of of dessins. dessins. Lastly, Lastly, the paper by L.Zapponi L.Zapponi concentrates on on the the genus genus one one case, exploiting some of the specific specific techniques techniques pertaining to to the the theory theoryofofelliptic elliptic curves. curves. The projective line with three points points removed removed can be viewed, as Grothendieck saw saw it, it, as as the simplest moduli space, space, but but of of course course itit can can also dieck simplest moduli also be generalised Then generalised to to the projective projective line line minus minus nn points points for for arbitrary arbitrary n, n > 3. Then on to study families of unramified unramified coverings coveringsof ofthese theseobjects, objects, and and one can go on families of this leads to the theory theory of of Hurwitz Hurwitz spaces, spaces, which plays role in in plays an important role of the the Inverse Galois the investigation of Galois problem. problem. A A recent aspect of of this story can be found Debes and B. Deschamps, which surveys the found in the article by P. Dèbes state of of affairs affairs of of the the regular regular inverse inverse Galois Galois problem problem over over so-called so-called "large" "large" fields, aa notion notion introduced introduced by by Florian Florian Pop. Pop. The fields, The paper paper by by K. K. Strambach Strambach deals with with the regular realization of and H. Völklein Volklein deals of finite groups as Galois Galois groups via via the existence of rational rational points on Hurwitz spaces; groups existence of spaces; it introduces introduces considering certain certain subvarieties subvarieties of ofthese thesespaces, spaces,rather rather than the new idea of considering just its its irreducible irreducible components. paper in in this this section, section, by M. Fried components. The third paper

Introduction

ix

Kopeliovich, uses usesHurwitz Hurwitz spaces spacesto tostudy study the the ramification ramification properties and Y. Kopeliovich, of realizations realizations as as Galois Galois groups groups of ofthe the characteristic characteristic finite finite quotients quotients of of the the of universal p-Frattini cover universal cover of a finite group. The third part, part, Galois Galoisactions actionsand andmapping mappingclass classgroups, groups, contains contains two two which exemplify exemplifythe the kind kind of oftechniques techniquesthat that are are currently being used used papers which Galois action action on on fundamental fundamental groups. In his contribution, in the study of the Galois contribution, M. Galois action mapping class class groups groups M. Matsumoto Matsumoto determines determines the the Galois action on the mapping of connection between between the genus genus 33 moduli moduli space space of genus genus 3, 3, via via a remarkable connection and the deformation deformation space of E7-singularity. ^-singularity. The paper paper by by Z. Z.Wojtkowiak Wojtkowiak studies monodromy of studies the the monodromy of iterated iterated integrals, integrals, making making ample ampleuse useofofHodge lodge theory for for fundamental fundamental groups, which which makes possible to use Lie algebras makes it possible and differential differential techniques. techniques. In the fourth fourth section, section, Universal UniversalTeichmüller Teichmuller Theory, Theory, we we have have gathered gathered three closely related papers. papers. R. Penner three closely related Penner first first gives gives a gentle gentle survey survey (in (in his his own words) words) of of aa recent recent and and promising promising theory of his which grew out out from own which grew from his previous studies of the "classical" "classical" decorated Teichmüller Teichmuller and and moduli moduli spaces. spaces. His theory, but it also also contains contains new His paper paper serves serves as as an an introduction introduction to to this theory, features. It investigates features. investigates the the applications applications and and completions completions of of his his "universal "universal Ptolemy which was Ptolemy group" group" G, a group which was already already known known to to group group theorists theorists as Richard isomorphic to Richard Thompson's Thompson's group, group, and and which which isis in in fact fact isomorphic to the group PPSL (Z) of piecewise PSL2(Z) transformations. The identification PPSL22 (Z) of piecewise PSL2 (7L) transformations. The identification of of G with Thompson group sight; first noted by with the the Thompson groupisis not not apparent apparent at at first first sight; first noted M. subject of his M. Kontsevitch, Kontsevitch, itit was was proved proved by by M. M. Imbert Imbert and and forms forms the the subject contribution. the last last paper, paper, by byP.P.Lochak Lochak and and L. L.Schneps, Schneps, itit isisshown shown contribution. In the that by by suitably enlarging enlarging and completing the Ptolemy group into into aa profinite profinite Ptolemy-Teichmuller this new new Ptolemy-Teichmüller groupoid, groupoid, one one can can let let the Galois group act on this object. object. This action actually occurs naturally as the restriction to Gal(Q/Q) of Grothendieck-Teichmuller group of an an action of the Grothendieck-Teichmüller group (cf. (cf. the the survey survey on this group volume to this one). one). group in the companion volume

Abstracts of the talks Abstracts of talks Short Courses

Fields of Covers; Embedding Problems Problems over Large Fields of of Definition Definition of Covers; Embedding Large Fields. Fields. Pierre PierreDèbes. Debes. I. The joint work workwith withJean-Claude Jean-ClaudeDouai. Douai.Let Letff : X X —> -÷ The first talk deals with joint defined over over the the separable separable closure closure KK8 of aa field fieldif, K, with cover defined B be a finite cover s of B algebraic variety isomorphic to each each B an algebraic variety defined definedover overK. K. Assume Assume that that f/ isis isomorphic /K). The field K is called the field of moduli of its conjugates conjugates under G(K under G(K8/K). The field K is called the field of moduli S cover. Does Does itit follow follow that the the given given cover cover can be be defined defined over over K? K? of the cover. is an obstruction obstruction to tothe thefield fieldof of moduli moduli The answer is "No" in general: general: there is measured? We being a field of of definition. definition. Still, Still, how how can can the the obstruction obstruction be measured? present general approach The obstruction obstruction is is entirely entirely present aa general approach for for this this problem. problem. The of cohomological nature. This was was known known only only in in the thecase caseofofG-covers, G-covers, of a cohomological nature. This i.e., Galois covers special i.e., Calois covers given giventogether togetherwith with their their automorphisms. automorphisms. This special case happens to be the simplest one. In the situation of mere covers, the case happens to simplest one. In the situation of mere covers, the problem problem is shown shown to to be be controlled controlled not not by one, as for for G-covers, G-covers, but but by several 2 (K, Z(G), G{K8/K) characteristic characteristic classes classesininHH2(K, Z(G),L) L) (for (foraacertain certainaction actionLLofof G(K8/K) on center Z(G) Z{G) the the group group of of the the cover). cover). Furthermore Furthermore our approach approach on the center reveals one, called called reveals aa more niore hidden hidden obstruction obstruction coming coming on on top top of of the main one, the first obstruction and and which which does does not not exist exist for for G-covers. G-covers. Our Main Theorem yields quite concrete criteria for for the the field of moduli to be a field of of definition. definition. Such Such criteria criteria were were not not available available in in the the general general situation of mere mere covers. covers. Furthermore Furthermore the the base base space space BB can be here an algebraic variety of characteristic. of any any dimension dimension and and the ground field K aa field of any characteristic. All classical results, for which the base space was the projective line P1 P 1 over over classical results, for which the was Q, are contained as as special special cases. cases. also leads leads to some local-globaltype type results. results. For exOur Main Theorem Theorem also some local-global ample we prove this local-to-global principle: a G-cover f : X —* is defined local-to-global principle: a G-cover / : X —B >• B is defined over Q Q ifif and only if it is for all primes p. This This was was conjecconjecover is defined defined over Qp for of tured by E. Dew and proved by the author author in in the thespecial special case case of of G-covers G-covers of 1 result and other related PP'. . We Wewill willdevelop develop this this local-to-global local-to-global result related questions. questions. We will will prove prove in in particular particular this global-to-local (or GWe global-to-local principle principle for for covers covers (or covers): if a cover covers): cover (or (or aa G-cover) G-cover) f/ : X : X—+ -» B has a number number field K as as field of moduli, moduli, then then it may not be of be defined defined over over K, but but ititisisnecessarily necessarily defined defined over all all but finitely many completions K,, of K. over completions K of K. v II. The each finite group group a quotient of II. The Inverse Inverse Galois Galois Problem — is each of G(Q)? — and Safarevic's conjecture — G(Qlb) is a free free profinite group — are two questions generally generally asked asked about about the absolute Galois main questions Galois group G(Q) of of Q.

22

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A fruitful fruitful approach consists in studying the the action action of of G(Q) G(Q) on oncovers covers of of P1 P1 (T). defined over Q, or equivalently, equivalently, on extensions of Q(JT). defined The goal of this talk is twofold. First we present The goal of this is twofold. First we present aa "Main "Main Conjecture" Conjecture" that contains contains all all basic basic conjectures conjectures of of the the area, area,including including the theInverse Inverse Galois Galois Problem and Safarevic's Safarevic's conjecture conjecture but also also the theFried-Völklein Fried-Volklein conjecture conjecture G(K) projective — G(K) projective ++ K hilbertian hilbertian =>G(K) G(K)pro-free pro-free — .AAspecial specialcase caseofof the Main Main Conjecture is is that, that, ififKK isisany anygiven given field, field, then then all all split split embedding embedding problems for G(K(T)) G(K(T)) have problems for have a strong solution. The The general general form form of the Main Conjecture same is is true for for split embedding problems with "some "some Conjecture is that that the same extra constraint on K". K". In the second In second part, we we present present a "Main "Main Theorem" Theorem" that unifies unifies a whole whole absolute Galois Galois group series of of results results that that have recently recently appeared appeared about the absolute G(K(T)) when whenKKisisaaalgebraically algebraicallyclosed, closed, or orPseudo PseudoAlgebraically Algebraically Closed, Closed, G(K(T)) local field, or or is a local or is Pseudo Pseudo S Closed Closed {e.g. (e.g. the field of of totally totally real (or padic) Main Conjecture Conjecture adic) algebraic algebraic numbers). numbers). The The Main Main Theorem Theorem isis that that the Main is true if the field K is large. By definition, a field K is large each is true if the field K is large. By definition, a field K is large ifif each geometrically irreducible infinitely many smooth geometrically irreducible curve curve defined defined over overKK has infinitely if-rational provided that is at at least least one. one. The Theexamples examples above above K-rational points provided that there is algebraically closed, closed, etc.) are areexamples examplesofoflarge largefields. fields. (K algebraically contributions to the Main Main Theorem Theorem are are due due to to D. D. Harbater who The main contributions who introduced some some very efficient efficient "patching and glueing glueing techniques" for for covers covers of PP',1 ,F.F.Pop Popfor forthe thearithmetic arithmeticingredients ingredients of of the the proof, proof, in in particular, particular, his his work on on the the property property "K large" work large" and and M. M. Fried, Fried, for for the the idea idea of of working working on families covers. Q. Liu, H. Völklein, Volklein, D. were other families of of covers. D. Haran Haran and and the author were contributors to this result. result. .

Coordinates on on Teichmüller Teichmiillerand andmoduli modulispace. space.Adrien AdrienDouady. Douady. I. In the the first first lecture, lecture, pants pantsdecompositions decompositions (also (also known known as as maximal maximal muticmuticurves) of a surface surface are described, described, together with some urves) some elementary topological topological and enumerative properties. We We then then show show how, how, picking picking aa fixed fixed decompodecomposition, one can define define the corresponding corresponding Fenchel-Nielsen Fenchel-Nielsen coordinates which which provide real analytic analytic coordinatization coordinatization of ofTeichmüller Teichmiiller space. space. These These provide a global real coordinates can can also also be be used used in in the the study study of the moduli and their coordinates moduli spaces spaces and compactifications. II. The The second second lecture lecture is is devoted to an introduction introduction to to the thecomplex complexanalytic analytic theory. Starting Starting from from the definition definition and existence existence statement statement of of Strebel Strebel we show show how how to to introduce introduce deformation deformation parameters quadratic differentials, differentials, we which complex analytic coordinatization of Teimüller Teimiiller space. which afford afford aa local local complex analytic coordinatization More differential are associated horizontal and and More specifically, specifically,toto aa quadratic quadratic differential are associated vertical matter of of convention), convention), vertical foliations; foliations;picking pickingone oneofofthem them (this (this isis a matter

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one associates associates to aa Strebel Strebel differential differential aa decomposition decomposition of of the the underlying underlying Riemann surface cylinders. The deformation parameters surface into cylinders. parameters which which are are used used in coordinatization may may be intuitively intuitively described described as the mutual mutual sliding sliding in the coordinatization and cylinders, or and twisting of these these cylinders, or more more visually visually "stove "stove pipes". pipes". Once these holomorphic descend holomorphic coordinates coordinates on on Teichmiiller Teichmüller space space are are obtained, obtained, they descend to the moduli moduli space, space, thus thus providing providing one one of of the several several ways ways to endow endow this last space with a complex complex structure. structure.

Topologicalfield fieldtheory theoryand and connections connectionswith with CFT, CFT, and and the topoTopological logy of configuration logy configurationspaces. spaces.Ruth RuthLawrence. Lawrence. I. This This short short course course aims aims to to show show some of the structures naturally naturally arising arising from topological, geometric and and combinatorial from topological, geometric combinatorial approaches approaches to topological topological quantum field field theory theory as as well well as as relations relations between betweenthem. them. The first quantum first lecture lecture concentrates on the definition concentrates definition of of aa topological topological field theory and and shows shows how how simple decompositions of manifolds manifolds into 'elementary 'elementary simple topology topology relating relating to decompositions pieces' in two two pieces' forces forces the the existence existenceof ofaa tight tight algebraic algebraic structure structure for for TQFTs in and three three dimensions. dimensions. II. In In the thesecond second lecture, lecture, we we concentrate concentrate on on the theconnections connections with with conformal conformal theory and and the geometry of configuration spaces of of points. points. In particufield theory configuration spaces particusee that the in the representation theory of lar, we see the combinatorics combinatorics involved involved in quantum groups forms forms a bridge between the two two approaches. approaches.

On universal universal monodromy monodromy representations representations of ofGalois-Teichmüller Galois-Teichmiiller modular groups. modular groups.Hiroaki HiroakiNakaniura. Nakamura. I. In In this this course, course, II explain explain the the basic basic setting setting and and recent recent advanced advanced results results in the study of of pro-i pro-£ exterior exterior Galois Galois representations representations arising arising from from algebraic algebraic curves. When the the algebraic algebraic curve curve varies, varies, the Galois Galois representation representation is decurves. When formed, but but itit turns out that formed, that there thereisisan aninvariant invariant portion portion common common to to all all algebraic curves. These towers are defined defined by the universal universal monodromy monodromy reprepresentations of Galois-Teichmiiller weight Galois-Teichmüller modular modular groups groups together together with the weight filtrations The problem problemof ofshowshowfiltrat ions in in the the so-called so-called pro-^ pro-i mapping class groups. The ing stability properties of these fields of definition with respect to genera ing stability properties definition with respect genera and marking points was posed fairly estabposed by Takayuki Takayuki Oda, Oda, and recently fairly lished Takao, R. Ueno Ueno and lished by by cooperation cooperation of of Y. Y. Ihara, Ihara, M. Matsumoto, N. Takao, the author. author. II. In In the the second second part, part, we weexplain explain graphs graphs of of profinite profinite groups associated with combinatorial data on maximally combinatorial maximally degenerate curves.

44

Abstracts Abstracts

survey of of Grothendieck-Teichmüller Grothendieck-Teichmiillertheory. theory.Leila LeilaSchneps. Schneps. A survey I. The The first first part partof ofthis thisshort shortcourse coursesurveys surveys recent recent work work on on the the following following topics: ics: new new properties properties of of the theGrothendieck-Teichmüller Grothendieck-Teichmiiller group GT GT analogous analogous to known to be contained known properties properties of of the absolute absolute Galois Galois group group known known to contained in itit and and conjecturally conjecturally equal equal to to it, it,such suchasastorsion, torsion,behavior behaviorofofcomplex complex conjugation, action on on combinatorial combinatorial structures structures such such as as dessins dessins and and braid conjugation, action groups, and others, particularly particularly the theGalois-compatible Galois-compatible action action of of GT GT on on varivarifundamental groupoids of geometric moduli spaces spaces ous fundamental geometric objects, objects, particularly moduli of Riemann surfaces surfaces with marked marked points. points.

II. II. In Inthe thesecond second part, part,we weconsider consider universal universal Teichmüller Teichmuller theory. We We introduce the isomorphism by M. M. Imbert) between isomorphism (proved (proved by between Richard u ThompThompson's group, the group PPSL2(Z) piecewisePSL2(Z)-transformations, PSL2(Z)-transformations, and PPSL^Z) of piecewise Penner's universal Ptolemy group group G. G. Penner's Penner's universal Ptolemy Penner's presentation presentation of this group group emphasizes emphasizes a remarkable remarkable pentagon equation which which indicates indicates aa connection connection with the Grothendieck-Teichmiiller Grothendieck-Teichmiiller group. explicit group. This connection is made explicit result that thatGT GTacts actsonona suitable a suitablegroupoid-profinite groupoid-profinitecompletion completionofof via the result the universal Ptolemy group. group.

Individual talks talks Characterizing curves Jean-Marc curves by bytheir theirdessins. dessins. Jean-MarcCouveignes. Couveignes. We We recall that defined over over Q Q that aaBelyi Belyi function function 0 is a function from some curve C defined outside {O, 1, oo}. oo}. Then (C, is called a Belyi pair. A morphism unramified outside {0,1, (C, (j>) is called a Belyi pair. A morphism such that that fol = (j>i> between from C1 C\ to C2 such between two two such such pairs pairs isis aa map I/ from An An isomorphism Belyi pairs is is called called aa dessin dessin following following Grothendieck. Grothendieck. isomorphism class class of Belyi We first first give give aa sketch sketch of ofproof prooffor forsome someslight slightimprovement improvement of ofBelyi's Belyi'stheorem theorem stating existence of a Belyi Belyi function function with no no automorphisms automorphisms on any any stating the existence curve each curve curve C defined defined over we curve denned defined over over aa number number field field K. K. To each over K we dessin (C, associate aa dessin (C, (j>) with with nono automorphisms automorphisms(or(or thethe setset of of allall of of them). them). characterize this this curve curve up up to toisomorphisms isomorphisms defined defined over over This is enough to characterize K. We in order order to test We then then look look for for some some explicit explicit examples examples in test which which kind kind of arithmetic arithmetic information information on read on on the the topological topological structure structure of on CC can can be be read of its characteristic characteristic dessins. We insist naive use use of of Belyi's Belyi's theorem theorem of dessins. We insist that aa naive is not likely likely to provide provide any such non trivial example. Instead, we obtain any such non trivial example. Instead, families of dessins dessins by by an indirect families of indirect way, way, using using coverings coverings ramified ramified over over four four points and the the corresponding corresponding moduli moduli spaces spaces (called (called Hurwitz Hurwitz spaces spaces after after points For example, example, we we present present aa family family of of dessins dessins of of genus genus zero zero with no no Fried). For automorphims, indexed n, p, p, q, and associated associated to to automorphims, indexed by by four four parameters parameters m, in, ii, curves Cm,n,p,q Cm^^^q ininP3 the curves P3given givenby bythe the equations equations ma ma + + nb + pc + qd == 00 2 ma2 ++ nb2 nb2 ++pc qd2 = 0. We algebraic and topological topological and ma2 pc2 + qd2 = 0. We give give both both the algebraic

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5 5

description of these these dessins dessins and and prove prove that that any conic can be set in description of in the the form form of some sonie C Cm,n,p,q for suitables values of the parameters. We finish by a brief , ,g for suitables values of the parameters. brief m>n p — 1, 1, n = 2, p = 3, where we we prove prove that that the conic conic study of the case m m= 3, q = 7 where has bad reduction at at 77while while 77 is prime to the the degree degree of of and any ramification ramification in our associated associated dessin. dessin. The Sylvain Presentation of of the theMapping MappingClass ClassGroup. Group. SylvainGervais. Gervais. The Mapping Class Group M(g, M(g, n) of a genus g surface surface 5S with n boundary comcomponents is the group group of of diffeomorphisms diffeomorphisms of S which leave fixed its boundary, boundary, modulo whose whosewhich whichare areisotopic isotopictotothe the identity. identity. M(g, modulo M(g,n)n) is is generated generated by by Dehn twists for give aa presentation presentation of M(g, M(g, n) considerDehn for all g and n; we we will give considering all all twists twists as as generators. generators. When When gg isis greater greater than than or or equal equal to to 2, M(g, n) ing M(g,n) is generated generated by by twists twists along along non-separating non-separating curves. curves. A A presentation presentation is also is given with with these these generators. generators. In both cases, given cases, all all the the relations relations live live in in surfaces surfaces of genus genus 00 with with 44 boundary boundary components components or of genus genus 1 with with 1 or 2 boundary of components.

Arithmetic aspects aspects of ofTeichmüller Teichmullertheory. theory.Bill BillHarvey. Harvey. We Wefocus focus attention on two two ways ways in which which the theory theory of of uniformisation uniformisation for for Riemann surfaces which are hyperbolic hyperbolic (covered (covered by real hyperbolic hyperbolic disc disc D2) D2) surfaces XX which by the real for X X as has points of contact with with fields fields of definition definition for as an an algebraic algebraic curve. curve. 1) Following FollowingBelyi's Belyi'stheorem, theorem, we weknow knowthat that X X can be defined defined over Q if and only if a Fuchsian uniformising group for X is contained in a triangle group. only if a Fuchsian uniformising group for X is contained in

For many For many (perhaps (perhaps all) such such surfaces, surfaces, there there exist exist holomorphic holomorphic quadratic quadratic differential forms forms CJ w on X such differential such that thatthe theassociated associatedTeichmüller Teichmullerdeformation deformation disc D([X], w) (here disc D([X],a;)ST(X), (here T(X) T(X) denotes denotes the the Teichmiiller Teichmiiller space space of the the closed surface surfaceX), X), has has the the property property that the closed the subgroup subgroup of of the the mapping mapping class class group group Mod(X) Mod(X) which which preserves preservesDD isis aa Fuchsian Fuchsian group group (of (of the the first kind) kind) intermediate between between aa triangle group q, oo} oo} and and aasubgroup subgroup intermediate group of of type type {p, {p,q, representing lri(X*), surface X X with representing TTI(X*), the surface with the the zero zero set set of ofwLJremoved. removed. 2) The classical Schottky uniformisation classical Schottky uniformisation of a (complex) (complex) Riemann surface surface has analogue for for non-Archimedean non-Archimedean fields has an analogue fieldsdue duetoto Mumford. Mumford. This This motivates search for for aa choice choice of Schottky uniformisation uniformisation of of curves curves definable definable vates the search over over aa given given number number field. field. If If we we choose choose aa class class of of stable stable degeneration degeneration for for a genus g curve (via its dual graph), the work of Bers and Maskit provides genus g curve (via its dual the work of Bers and Maskit provides complex uniformisation of of such such curves. curves. I have have complex coordinates coordinates for for aa natural uniformisation shown

Theorem. of Maskit's coordinates Theorem. A modification modification of coordinates determines determines Schottky Schottky cocoMumford curves curves at a specific field ordinates which define define Mumford specific set of places of the field generated by by the matrix entries generated entries of of the the (classical) (classical) Schottky Schottky group. group.

6

Abstracts

A complex interpretation A interpretationofofFricke Frickecoordinates. coordinates.John JohnHubbard. Hubbard.No No abstract submitted. submitted. ofcurves curves over over arithmetic arithmetic ground On 7r1 TTI of groundfields. fields.Yasutaka YasutakaIhara. Ihara. Let Let X be a smooth X smooth irreducible irreducible curve over over a field field kk (assumed (assumed to be finite over a prime field). field). The first topic is prime is aa simple simple but but useful useful observation observation of of A. A. TamaTamagawa. When When k is finite and and g(X) g(X) >>0,0, he gawa. he characterized characterized group-theoretically group-theoretically which section section s : Gal(k/k) —+ iri(X) of the which -» ni(X) the projection projection iri(X) ni(X) —÷ —>•Gal(k/k) Gal(fc/A;) ~ fromaa fc-rational k-rationalpoint pointPP eG X(k). X (k). The second Z comes comes from second subject the subject is the quotient by the the normal normal subgroup subgroup generated generated by all all conjugates conjugates of quotient of of TTI(X) iri(X) by of s(Gal(A;/fc)), where 5 is a section corresponding to some P. This quotient, s (Gal(k/k)), where s section corresponding to some P. This quotient, denoted by 7r1 TTI(X)(P), (X) (P),isisthe theGalois Galoisgroup groupof ofthe the tower tower of of finite etale covers of splits completely. completely. We We first first discuss discuss what can can be be said said in in gengenof X X in which P splits eral about this group group iri(X) TTI(X)^P) (various "derived" quotient groups must be (F) (various "derived" groups be "small", using "abelian" "abelian" mathematics, mathematics, and restrict to Shimura Shimura "small", etc.) etc.) using and then then restrict curves phenomena. curves to discuss deeper phenomena.

A interpretation of A cohomological cohomological interpretation of the the Grothendieck-Teichmüller Grothendieck-Teichmiiller group. Pierre group. PierreLochak. Lochak. The Theresult resultgiven given in in this thistalk, talk,following following from from joint work with with L. of work L. Schneps, Schneps, isis the thefollowing. following. The The three three defining defining relations relations of the Grothendieck-Teichmüller group are cocycle relations for certain nonGrothendieck-Teichmiiller group cocycle relations nonsets of of cyclic cyclic groups groups with with values values in braid groups. commutative cohomology cohomology sets groups. These sets are are very very simple, simple, and and one can actually compute These cohomology cohomology sets compute exexplicit coboundaries representing the elements of of GT. GT. The The methods methods for for calcucalculating the the cohomology cohomology sets are due to Serre, Brown and Scheiderer; Scheiderer; the the same same gives the result result that thatcomplex complexconjugation conjugation isisself-normalizing self-normalizing methods also gives in GT. GT.

Etale covers of aa generic generic curve curve in in characteristic characteristic pp>> 00 and ordinarcovers of ordinarity. Michel Michel Matignon. Contrary Contrary to tothe theaffine affine case case (Abhyankar's (Abhyankar's conjecture) conjecture) there is no conjecture finite quotients quotients of of the the fundamental fundamental group conjecture concerning concerning finite of a smooth projective projective curve. curve. The fundamental fundamental group codes the Hasse-Witt invariants of etale covers; this gives an infinite infinite set of invariants whose study study was begun by by H. H. Katsurada Katsurada and S. Nakajima. was begun Nakajima. In this this talk talk we we concentrate concentrate on ir1 TTI of a generic generic curve of genus g over over an an algebraically algebraically closed closed field k of characteristic pp>>0;0;such suchaacurve curvedegenerates degenerates into into proper proper stable stablecurves curves over over g; then aa theorem k of arithmetic genus #; theorem due due to to M. M. Saldi Sai'di gives gives the the profinite profinite fundamental group of the graph of groups, built on the intersection intersection graph graph of of stable curve curve and with group data the the tame tame fundamental fundamental group group of of irreirrethe stable ducible components along ducible components minus minus the the intersection intersection points points and amalgamation along

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77

tame inertia at at these these points, points,asasaaquotient quotientofof7r1. 7Ti. The Thecase caseof ofmaximal maximal dede1 1, oo}). generationsisisrelated relatedto to —{O, {0,1, oo}).An Anoptimistic optimisticconjecture conjectureisisthat that generations i4 TT^P (IF'1 — every etale cover of generic curve is known known for for abelian abelian every etale cover of aa generic curve is is ordinary; ordinary; this is covers. A first first attempt attempt towards towards this this conjecture conjecture is is the the study study of of covers covers of covers. (such covers covers appear Legendre elliptic curve which which are étale etale outside outside oo (such the Legendre degenerating into generic elliptic by degenerating into a chain of generic elliptic curves); curves); one one expects expects that which don't factorise factorise through an an isogeny isogeny are still ordinary; the the case case of those which hyperelliptic hyperelliptic and and (in (in particular) particular) genus genus two two curves curvesisisexamined. examined. In In the the last we look look at the the geometry geometry of of the the "Deligne-Mumford" "Deligne-Mumford" boundary part we boundary of the Teichmuller give a group group theoretic theoretic Teichmüllertower towerofofM M9; in this this way way one one hopes hopes to give g; in of ir1 TTI which etale covers. covers. description of quotients of which give give rise rise to to ordinary etale

Galois actions on Galois on braid-like braid-likegroups. groups.Makoto Makoto Matsumoto. Matsumoto. Let Let V V be a geometrically geometrically connected by connected variety variety defined defined over overaa number number field field k.k. Then, by have an outer action of Grothendieck's theory, we have of the absolute absolute Galois Galois group group Gk the profinite profinite completion completion of of the the topological topological fundamental fundamental group group Gk of of k on the of We denote denote this representation representation by By Belyi's Belyi's theorem, theorem, this of V(C). We by py. Pv. By action projective line minus minus three and each each action is is faithful faithful ifif V V is the projective three points, and a element )i fa) in 7L Z xx [F , F2]. The [F2, F2]. The element ao of of Gk Gk has has its its unique unique "coordinate" "coordinate" (x( (x(°), 2 basic we use coordinate in order order to describe describe pv py for for basic question question is: is: can can we use this coordinate V other other than than projective projective line line minus minus three points? points? This This occured occured in in Ihara's Ihara's V geometric geometric proof proof of of the the embedding embedding of of GQ GQinto into GT GT (done (done by Drinfeld's) Drinfeld's) using two-dimensional tangential base point. two-dimensional tangential We show show two two examples examples where where same same kinds kinds of of arguments arguments can be applied: We applied: (1) V configuration space of of some some points points on on an an open curve, and (2) configuration space (2) moduli moduli curves of of genus genus g, g, with with #g = 2,3. space of curves 2,3. The former example can be applied of Belyi's Belyi's Injectivity Injectivity theorem for nonabelian affine to show show a generalization generalization of afrine curves. The latter example curves. example for for g = —33uses usesaaclose closerelationship relationshipbetween betweenthe the deformation space of .EV-singularity E7-singularity and the moduli deformation moduli stack stack of of genus genus 33 curves.

On Grothendieck's Grothendieck's anabelian anabelian conjecture. conjecture.Florian FlorianPop. Pop. The Theidea idea of of Grothendieck's anabeliangeometry geometryisisthat that under under certain certain "anabelian" Grothendieck's anabelian "anabelian" hypotheses the isomorphy type, hence the geometry and the potheses the isomorphy type, hence the geometry and the arithmetic arithmetic of schemes, isis encoded encodedin intheir their etale étale fundamental fundamental groups. Relatively schemes, Relatively speaking, if SS is some some base base scheme, scheme,and and AA isis an an anabelian anabelian category of 5-schemes, S-schemes, then for every every object object X X and for and YY of of AA one oneshould should have havethe thefollowing following

Isom(X,Y) • A 2

*3 v V > . . . Am_i

Jmv

V »• A

m

]p>l ((H\ — Jr {^).

Maximality of of the the chain Maximality chain implies implies that all all the themonodromy monodromy groups groups Mon(f1) Mon(/i) 1,., . . . ,, m) are primitive. —1 primitive. (i = algebraic curves. Then Let f/ :: X Let X —* —> Y and and gg::YY —* —>ZZ be be morphisms morphisms ofofalgebraic curves. Then the set of of composition composition factors of the monodromy monodromy group group Mon(g Mon(# oo /f)) is concomposition factors factors of ofMon(/) Mon(f) and Mon(g) tained in the union of the sets of composition Mon(g) (see [GTh], Prop. Prop. 2.1). Hence to prove prove that the (see [GTh], Hence it suffices suffices to the composition composition facfactors of monodromy monodromy groups groups of any morphism fi are contained contained in the the above above list. Since /f has Since has oniy only one one pole, for any morphism morphism f2 fi there exists a point point P1 Pi E G equals the the total degree Xi such such that the the ramification ramification degree degree of of fi over Pi equals degree of fi. The monodromy monodromy group of fi is a factor factor of of the the fundamental fundamental group group n(Xi\ramification cyclic permutation ir(X1 \ramification points) and hence it contains a cyclic permutation which corresponds loop around around P1. Pi. corresponds to a loop Therefore we we have have reduced reduced the the problem problem to to the case Therefore case of of primitive permupermucontains aa cyclic cyclicpermutation. permutation. At this point we tation group which which contains we use two classical results. Burnside's classical Burnside's theorem theorem (1911) (1911) says that that every every non-solvable non-solvable transitive group of 11.7). of prime prime degree degree is is doubly doubly transitive transitive(see (see[Wie], [Wie],Th. Th.11.7). Schur's theorem (1933) guarantees every primitive transitive group group of of Schur's theorem (1933) guarantees that that every composite degree degree which which contains contains aa cyclic cyclic permutation permutation is doubly transitive composite transitive [Wie, Th.25.3J). Th.25.3]). (see [Wie, A more more recent recent result was was proved proved by Feit Feit (1980) (1980) using using the the classification classification of finite simple groups groups (see (see [F], Th.1.49): Let of finite simple [F], Th.1.49): Let G G be benon-solvable non-solvable doubly doubly whichcontains containsaa cyclic cyclicpermutation: permutation: then one transitive group of degTee degree nn which one of the following statements holds following statements holds (i) G ~ A n or S n ; (i) = 11 (F11) oorr G (ii) ii n= 11 and G ~PSL 2 (Fn) G - Mi1; Mn; (iii) nn == 23 (iii) 23 and G ~ M23; M23; (iv) nn = (qm (iv) (q171—- 1)/(q \)/{q— 1)1)and andGGisisisomorphic isomorphictotoa asubgroup subgroupofofPFLm(l?q) PFL m (F q ) containing PSLm (lFq). containing PSL m (F g ). Here PFL P['Lm(Fq) sernilinear group, group, i.e. i.e. the extension so-called semilinear extension of of the the m (F q ) is the so-called group PGL PGLm(Fq) by the Frobenius automorphism. For m 3 it can be also group (F<j) by the Frobenius automorphism. For m > 3 it can be also m l considered as the group considered group of of collineations collineations of of the the projective projectivespace spacepm— FTn~'(Fq). (¥q). is solvable solvablethen then its its composition composition factors factors are are cyclic. cyclic. Otherwise Otherwise applying If G is the Burnside, Burnside, Schur and Feit Feit theorems theorems completes completes the the proof. proof. .

.

Since the edge edge rotation rotation group of a dessin is isomorphic isomorphic to the monodromy Since the monodromy group of the corresponding Belyi function we get the following corollary. corresponding Belyi function we get the following corollary.

16 16

Nikolai Adrianov and George Shabat Nikolai Adrianov

Corollary 2.2. factors of of the the edge rotation groups of of Corollary 2.2. The The composition composition factors unicellular dessins are are contained contained in the the list list of oftheorem theorem2.1. 2.1. Remark. We X there exists only a finite finite Remark. Wesuppose supposethat thatfor forany anyfixed fixed genus genus of of X number (n,q) thegroup groupPSLTh(Fq) PSL n (F q ) can number of possible possible pairs pairs (n, q) such such that the can occur occur as as factor of of Mon(/) Mon(f) for a meromorphic function /f with a composition composition factor meromorphic function with aa single single pole on on X. pole X. This is a particular case of the Guralnick-Thompson conjecture (see (see [GTh]) according to which which the analogous statement holds according holds for for all all simple simple non-abelian non-abelian and an an arbitrary function groups except A n and function f. /.

2.2. 2.2. Edge Edge rotation groups groups of of irreducible plane trees trees We call call a plane We plane tree T T irreducible irreducible if itit has has no nonon-trivial non-trivial morphisms morphisms T —* —> T'. X"'. The The irreducibility irreducibility of of a plane tree T is is equivalent to the primitivity primitivity of the the group group ER(T). of Let TT be Theorem 2.3. 2.3. Let be an an irreducible irreducible plane plane tree treewith with nn edges edgesand andG G== ER(T). Then one of the following statements holds: ororSn; (1) G n (l)G~A (2) G ~ C n or D n and n is is prime; prime; PSL2(1F11) and nn = 11; 11; (.9) (3) G G ~ PSL 2(Fu) and M11 M23 11 or 23; (4) G ~ M n or M 23; 23 and n = 11 m — 1)/(q — and (5) G ~ iTL (F ), n = (q \)/{q - 1) 1) and PFLm(Fq), n (qm m 9 (a)m=2 (a) m = 2 andq=5,7,8,9; and q = 5, 7,8,9; (b)m=3 (b) m = 3 andq=2,3,4 and q = 2,3,4 or or 2. (c) m = 4,5 4, 5 and q = 2. See the proof in [Adr]; an an equivalent equivalent result for for primitive monodromy groups of polynomials wasobtained obtained by by Miiller Muller [Miil]. [Mull. The complete of polynomials was complete list of of irreirreducible plane trees with ER(T) isomorphic ducible plane isomorphic to one one of of the groups groups (3)-(5) (3)-(5) will will be published in [AKS]. Below we wepresent present all all such such plane plane trees trees with 9, 10 [AKS]. Below 10 and also[AKSS}. [AKSS]. 11 edges; edges; for for the the trees trees with ER(T) c^ M M23 2 3 see also

§3. Combinatorial structures structures help to see the edge §3. Combinatorial edge rotation groups groups The common way of of calculating calculating the the edge edge rotation rotation groups groups is to switch on common way the computer computer and and use use some some software software based based on on Sims' Sims' algorithm. algorithm. (This (This is how most of of ER's ER's of the present how most present paper paper were were calculated.) However, However, in this this hand. section we give give two two examples examples of of calculations by hand.

Galois Orbits of of Plane Trees Trees

17

4

b)

a)

7

Figure Figure 1. A plane tree with with 77 edges edges and the Fano Fano plane. plane.

3.1. Fano plane 3.1. Consider the the 7-edged 7-edgedplane planetree tree in in fig. fig.la. la. In the specified numeration of of Consider specified numeration of the the tree the generating the edges edges of generating substitutions substitutionshave havethe thefollowing following form: form: = (2567) (34), a := = a:=r. (2567)(34),

b := ro = (12)(35) (12)(35)

(the edges are numbered numbered in in such such aa way way that that ab == (1234567)). edges are (1234567)). We assign assign the the edges edgesof ofthe the tree tree to to the points of the Fano We Fano plane plane as as shown shown on fig. fig. lb. Under on Under such such aacorrespondence correspondence the the group group ER(T) ER(T) preserves preserves the the collinearity subgroup collinearityofofthe the points points of of the the Fano Fano plane plane and and therefore therefore itit is a subgroup of PSL3(F2) ~PSL 2 (F 7 ). We of the collineation group PSL3(F2) We show showthat that the order of of the group ER(T) isis 168. 168. Indeed, the group ER(T) is Indeed, is transitive transitive on on the the seven seven lines lines of the Fano plane (e.g., hence the stabilizer of the (e.g., the operator ab ab permutes them cyclically), cyclically), hence line (4,6, 7) 7) isis aa subgroup subgroup in in ER(T) ER(T) of of index index 7. 7. In In particular, this stabilizer line stabilizer 3 3 2 contains the the substitutions substitutions (ab)(ab)3a(ab)3 contains a(ab) = (1235)(67) and and (ab)a2(ab)1 (ab)a (ab)~1 == (15)(46). The last substitutions, substitutions, acting acting on the set set {l, {1,2,3,5}, generate the (15)(46). 2,3, 5}, generate full full symmetric symmetric group group S4. S4. Therefore, Therefore,the the order order of of the the edge edge rotation rotation group group is is #ER(T) > 7 • 4! = 168, and hence ER(T) ~ PSL (F ). 4! = 168, and hence ER(T) #ER(T) 7 . 3(IF2). 2

3.2. Biplane 3.2. Biplane example is is related related to the The second second example the exceptional exceptional action action (known (known already already on 11 points. to Evariste Galois, Galois, see see [Con]) [Con]) of PSL2(F11) PSL 2 (Fn) on a:=r•=(1 a:=r.=(l 1O1I)(279)(346) 10 11 )(2 7 9)(3 4 6) b:=r0o=(2 =(2 1O)(3 b:=r 10)(3 7)(5 7)(56)(8 6)(89)9)

4—o I

10

22

77

33

44

Figure 2. A plane tree with Figure with 11 11 edges. edges.

Considerthe the plane plane tree tree in in fig. fig. 2. 2. We Consider We are are going going to show show that its its edge edge rotation group group PSL2 (IF1 1). is known that groupisisisomorphic isomorphictotothethe group PSL (Fn). It is known 2

Nikolai Adrianov and George Nikolai Adrianov George Shabat Shabat

18 18

the group group PSL2 P S L ^ F n ) can be represented represented as the automorphism automorphism group group of of the the "biplane" (see [ATL]), i.e. the system system of 11 "points", which label by the "biplane" [ATL]), i.e. which we label natural numbers numbers from from 11 to 11, 11, and and of of 11 11 "lines" "lines" of the form form Li = {i + 1, i + 3, i + 4, i + 5, i + 9}

(modil), (mod 11),

i=l,...1l. i = 1 , . . . 11.

through any two different points and exactly two lines lines passing passing through different points There are exactly any two different different lines lines meet meet in in exactly exactly two points. any We identify identify the points of We of biplane with with the the edges edges of of 11-edged 11-edged tree (fig. (fig. 2) 2) in such such aa way waythat that the edge with the the automorphism in edge rotation group coincides coincides with group of of the biplane. group biplane. Similarly Similarly to the the previous previous example example we we label the the edges edges such a way way that product ab ab has has the the form form (1,2,. ( 1 , 2 , ... .. ,, 111). 1). of the tree in such of that the product It is is easy easy to see see that the thepermutations permutations aaand andbbpreserve preserve the thelines; lines; hence hence Since 11 11 is is prime prime our tree isis the group ER(T) E R ( T )isisaasubgroup subgroupofofPSL2(1F11). P S L 2 ( F n ) . Since irreducible, hence by irreducible, by theorem theorem 2.3 2.3we wehave have ER(T) ER(T)=PSL2(1F11). =PSL2(Fn).

Generalized Chebyshev polynomials of of cartographically cartographically spe§4. Generalized Chebyshev polynomials cial trees cial We call call an irreducible plane tree TT with cartographicallyspecial special We irreducible plane with nnedges edgescartographically if ER(T) is not isomorphic if isomorphic to one of the groups groups C n , D n , A n and S n . We present all the cartographically special trees trees with with 9, 10 present cartographically special 10 and 11 11 edges edges and and their generalized generalized Chebyshev Chebyshev polynomials polynomials (see (see e.g. e.g. [ShZv]). [ShZv]). Some of these polynomials Some of polynomials defined defined over over Q; Q; they were were computed computed earearlier by Matzat Computation of the lier Matzat (see (see [Mull). [Miil]). Computation the generalized generalized Chebyshev Chebyshev polynomial for for the the plane plane tree tree with with ER(T) ~ M M1i polynomial n was was also also performed performed by Yu.Matiyasevich with with the help of of the the Couveignes-Granboulan Couveignes-Granboulan algorithm (cf. Yu.Matiyasevich (cf. [CoGr]). [CoGr]).

4.1. Results 4.1. Results

a)

b) b)

c) c)

Figure 3. 3. ee == 9 and ER(T) Figure E R ( T ) ~ PFL2 P r L 2(IF'8). (F8). (1) The The generalized generalized Chebyshev Chebyshev polynomial polynomial for the 9-edged 9-edged tree in in fig. fig. 3a 3a is is given given by

Galois Orbits of Plane Trees Galois Trees

19 19

P=(x2_x+ (x3+3x2+3.7_1

x - 22(2 -

^

values are are 00 and and c == —25(1 —- y/—3)(2 - \ / - 3 ) 7 , and we Its critical critical values we have have (x— F— 4)(x4 +2x3 P - cc= = (x - 4) j x4 + 2x3 + 3(-4 + S v ^ ) ^ 2 + 2(2+ 2(2 +

^3)?

(2) The The generalized generalized Chebyshev Chebyshev polynomial for for the 9-edged 9-edged tree in fig. 3b is given by 2 3 P =x9 =x9 + 2223372x7 3 7V ++ 233373x6 233373x6 + 2 • 3774x5 3774x5 ++ 233576x4+ 233576*4+

223476307x3 223476307x3 + 23367737x2 23367737x2 + 3678372x 3678372x ++ 233679139. 233679139.

Its critical values are ±c, where c = 2 13 3 4 7 9 \/ r:: 3. values are (3) The generalized Chebyshev polynomial polynomial for for the 9-edged generalized Chebyshev 9-edged tree in in fig. fig. 3c 3c is given by

P =x9 =x9 -— 2 • 327(2 327(2 +

+ 32781

6 ( 1 _— y Z 3.. 776(1

3

= 2278(1 are ±c, where Its critical values values are where c = 2278(1 + \fI3)(2 + x/ 1 ^). o

P+e= c - I * + '* . p

(x3

I rp£ _l_ 'Jrp

72

' _3•71 + 2 7x2: - 3 - '

V

j

2

J +

x

+ 2272(1 +

+

3

P—c= P-c= [xz-7x-72 71 +1 +

2 (x3 + 3. 3 •7x2 7x— - 3. 3• 7

'

v

"

x ^ / ^ ( 2 ++ y=3) 2 x—- 2272(1 2272(1++ ^/=3)(2+

Nikolai Adrianov and George Shabat Nikolai Adrianov

20

PFL2(1F9). Figure 4. e = 10 and ER(T) ~ PrL Figure 2 (F 9 ).

(4) The The generalized generalized Chebyshev Chebyshev polynomial polynomial for for the the 10-edged 10-edged tree tree in in fig. fig. 44 is is given by 2 4 95)4. (x2 — P = (x - 20x + 180)(x2 180)(x2 ++5x 5x— - 95) . 4 12 5 Its critical values are 0 and c = 22431255. values are 3 5.

P c = (x4 + + 30x3 30x3 ++ 75x2 75x2— - 4850x -— 39375)(x3 39375) (x3 —- 15x2 15x2 + 75x 75x ++ 550)2. 550)2. P-— C=(x*

(5) The The generalized generalized Chebyshev Chebyshev polynomial for the 11-edged 11-edged tree in in fig. fig. 22 is given by

I 2

2

i-hV/ziT

X

3

)

2

x (x2_3x+(5+vCii)).

Its critical values are 0 and c = values are — —1728. —1728. 3 (x3+4x2+ P—c= P - c = [x + 4x

x

5V-11

(x3+

x ^

x(

—

+(—2+

-

3(1

Figure 5. e = 11 and ER(T) ~ M11. 11 and Mu.

(6) The The generalized generalized Chebyshev Chebyshev polynomial polynomial for the 11-edged 11-edged tree in in fig. fig. 55 is is given by P =(x2 9(19 + Sv^ll))(x — = (x2++2(4 2(4—- \f-U.)x — - 9(19 - 18)x

/I

92

5 + v/^TI

221-h 19v/::IT>

2

2

)

A

Galois Orbits of Plane Trees Galois Trees

21 21

are 0 and Its critical values values are c = _2636(4 -2 6 3 6 (4 — - VZ

+

+

—

We have P - c = I x3 + 3(5 + V^TT)*2 + 3 71 +29V 11^ + (_ 2 1 1 9 + ^ l T 3 591 4x 22 2 6057 + 4743 v /z TT\22 )/ " + 22

2

3

;

4267+1775Vc1i 4267 + 1775v c H 2

of computation 4.2. Techniques Techniques of show by by an an example example that aa priori In this section we show priori knowledge knowledge of the field field definition of of aa dessin dessin can can essentially essentially simplify simplify calculations of corresponding of definition corresponding Belyi function. function. Consider plane tree in fig. fig. 5. 5. This This tree and its mirror reflection are the Consider the plane reflection are 11-edgedplane planetrees trees with with edge edgerotation rotation group M M11. only 11-edged n . Hence they constitute the the Galois Galois orbit orbit over over some some imaginary quadratic quadratic field. field. Using Using techniques techniques similar to those in [Mat], one one determines determines that this similar this field field is is Q(\/—11). can write the desired Chebyshev polynomial polynomial in in the the form We can desired generalized generalized Chebyshev

P P== (x3 - 4x2

b)(x2

c) 4 .

in the the field field of of definition definition of ofthe the trees. trees. Then with the coefficients coefficients in

P' = (x2 + x + c)3Q, degree 4. 4. The roots of where Q is is a polynomial of degree of Q Q are the coordinates of of the tree T. the white vertices vertices of T. polynomial PP assumes the same values values at at the roots of the polynomial The polynomial Q. by Q, Q, write Q. Dividing P by P = QX QX + c3x3 c3x3 + c2x2 c2x2 + c1x cix + c0. c0.

Nikolai Adrianov Adrianov and and George George Shabat Shabat

22

The above condition implies implies that that c1 The above condition c\ = c2 == c3 c3 = 0, 0, or: or: 2 2 144239ac2b — 100362a 100362a2cb 144239ac26 c6 ++ 2373872acb 2373872ac6 4 + 5253120c 5253120c2 — - 13152000a 13152000a

— 23961600b — 11126ac4 239616006 ++ 21043200c 21043200c ++ 523008c3 523008c3 ++ 15648c4 15648c4 11126ac4 — 81920bc3 819206c3 —- 2049056ac2 2049056ac2 —- 12453120ac —- 123440ac3 123440ac3 —- 8239040bc 82390406c 2 2 2 — 1022648bc2 10226486c ++ 10831160ba 108311606a +4 2108528a2c 2108528a c 425097406 +4 5275200a2 5275200a2 + 2509740b2 — 87614a3c — 371048b2a - 1183439ba2 11834396a2 +4 179456a2c2 179456a2c2 87614a3c +4 499152b2c 49915262c 37104862a — 563130a3 563130a3 +4 24183ba3 241836a3 ++ 1814a2c3 1814a2c3 —- 850a3c2 850a3c2 +4 162a4c 162a4c 2 2 4 — 16352acb2 + 42486 4248b2a2 13050a4 = 0 16352ac62 —- 26048b3 2604863 +4 21768b2c2 2176862c2 4a 4+ 13050a 0 2 2 2 — 23779a 23779a2cb 34585ac2b 184c2 — 52642560a 34585ac 6c6 ++ 899443acb 899443ac6 +4 1085 1085184c 3 4 — 13375040b — 5472c3 133750406 ++ 13704960c 13704960c 5472c +4-24576c4 24576c —- 22281bc3 222816c3 — 783020ac2 783020ac2 —- 10463168ac 10463168ac —- 32258ac3 32258ac3 —- 2670632bc 26706326c —- 209351bc2 2093516c2 2 + + 669724b2 13434684a2 — 574642ba2 + 6056367ba 60563676a ++ 2040186a2c 2040186a2c + 66972462 4+ 13434684a 5746426a2 + 103128a2c2 4103128a2c2 -— 86746a3c 86746a3c + 9196062c -— 77176b2a 7717662a 1169544a3 + 91960b2c — 1169544a3 4 + 4 6075ba3 60756a3 —- 2112b3 211263 +4 63129600 63129600 -f + 24300a 24300a4 = = 0 0

3

2

120864c3 + 240306acb + 25557760c 120864c 4240306ac6 4 25557760c +4 12144b2c 121446 c

— 12696640b 126966406 2 — 2465672bc 24656726c —- 14854ac3 14854ac —- 270334a3 270334a —- 608732ac2 608732ac —- 177419bc2 1774196c2 + 14854a2c2 — 7362a3c 4+ 2681547ba 26815476a +4 740302a2c 740302a2c +4 243584b2 24358462 4 14854a2c2 7362a3c — 6160b2a 616062a —- 92147ba2 921476a2 —- 9057728ac 9057728ac —- 48328960a 48328960a +4- 1458a4 1458a4 2 — 4096c4 4096c4 +4 104313600 104313600 +4- 2448384c2 2448384c2 4 0 + 6669644a 6669644a2 = = 0 3

3

The system leads The direct direct solving solving of of this system leads to to an an equation equationofoflarge large degree degree(there (there

10 10 trees with the the same same valency valency sets). sets). Straightforward computations Straightforward computations Grobner basis techniques applied to the the system system demand demand considerable considerable using Gröbner computer resources. resources. However, weare arenot not looking lookingfor forall all the the solutions solutions of of the the system system (*) (*) but However, we only for those from from Q( Q(\/—11). We W e eliminate variables variables using using resultants resultants and obtain the the polynomial polynomial are are

c2), Resb(c2, c3)) R(a) R(a) = Resc(Resb(ci,c 2),Res b(c2,c 3)) of degree 156. 156. We We find of degree find aa prime prime (say, (say, pp == 29) 29) such such that that 1) does not 1) pp does not divide divide the the leading leadingcoefficient coefficient of of R(a); R(a); 2 irreducible modulo modulo p; 2) aa2 4+ 11 11 is is irreducible p; 3) there there exists one quadratic irreducible factor of R(a) modulo 3) exists only only one irreducible factor modulo p. p.

Galois Orbits of of Plane Plane Trees Trees

23 23

Using standard techniques (see e.g. e.g. [Akr]) [Akr])we welift lift this this quadratic quadratic factor factor to Using standard techniques (see a divisor d(a) of R(a) divisor d(a) R(a) in in Z[a]. Z[a\. Adding Adding d(a) d(a) to tothe thesystem system (*) (*)we weeasily easily compute the Gröbner Grobner basis: basis: {49b+256a— {496+ 256a- 18 1 8=- 0 49c — 73 = 0 49c - 18a — - 73 2 3a2+155a+4203= 0} 3a + 155a + 4203 = o} desired polynomial polynomial P is found. The The corresponding and the the desired is found. corresponding polynomial polynomial prepresented in 4.1 (which coefficients)isisobtained obtained from from PP sented (which has integer integer algebraic algebraic coefficients) by a suitable linear substitution.

References N.M.Adrianov, Classificationofofprimitive primitive edge edge rotation rotation groups groups of of N.M.Adrianov, Classification plane trees (in Russian), plane trees Russian), to appear appear in in Fundamentalnaya Fundamentalnaya ii prikiadprikladnaya matematika. matematika. N.M.Adrianov, Yu.Yu.Kochetkov, A.D.Suvorov, A.D.Suvorov,Plane Plane trees with [AKS] N.M.Adrianov, Yu.Yu.Kochetkov, [AKS] with special primitive edge edgerotation rotation groups groups (in (in Russian), Russian), to to appear appear in special primitive Fundamentalnaya i prikladnaya matematika. matematika. algebrawith withapplications, applications,John John [Akr] A.G.Akritas, Elements Elements of of computer computer algebra Wiley & & Sons, Sons, 1989. 1989. [AKSS]N.M.Adrianov, N.M.Adrianov, Yu.Yu.Kochetkov, Yu.Yu.Kochetkov, A.D.Suvorov, A.D.Suvorov, G.B.Shabat, G.B .Shabat, Math[AKSS] ieu groups prikgroups and plane trees trees (in (in Russian), Russian), Fundamentalnaya Fundamentalnaya ii prikladnaya matematika, v.1, ladnaya matematika, v.l, no. 22 (1995), (1995), 377-384. 377-384. [ATL] J.H.Conway, R.T.Curtis, S.P.Norton, [ATL] J.H.Conway, R.T.Curtis, S.P.Norton, R.A.Parker, R.A.Wilson, R.A.Wilson, An An ATLAS of finite finite groups. Oxford Oxford University Press, Press, London, London, 1985. 1985. ATLAS [Be] G.V.Belyi, [Be] G.V.Belyi, On Galois extensions extensions of a maximal maximal cyclotomic cyclotomic field field (in Russian), Izv. Akad. Nauk NaukSSSR SSSR 43, 43,(1979), (1979),269-276. 269-276. Izv. Akad. Three lectures lectures on on exceptional exceptional groups, groups, in in Finite simple simple [Con] J.Conway, J.Conway, Three groups (eds. G.Higman), AC Press, New groups (eds. M.P.Pawel, M.P.Pawel, G.Higman), New York, York, 1971, 1971, 215-247. 215-247. [CoGr] J.-M.Couveignes, L.Granboulan, Dessins from a geometric point of [CoGr] J.-M.Couveignes, L.Granboulan, Dessins from view, in [GTDEJ, 79-114. [GTDE], 79-114. [F] W.Feit, Some [F] Some consequences consequences of classification classification of finite simple groups. groups. Santa Cruz Cruz Conference, Conference, Proc. Proc. Sympos. Sympos. Pure Pure Math. Math. 37, AMS, AMS, Providence, R.I. R.I. (1980), (1980), 175-183. 175-183. [GrHa] Ph.GrifRths, Ph.Griffiths, J.Harris, Principles [GrHa] Principles of of algebraic algebraicgeometry, geometry,Pure Pure Appi. Appl. Math., Math., John Wiley Wiley & & Sons, Sons, N.Y., N.Y., 1978. 1978.

[Adr]

24 24 [Gr] [Gr]

Nikolai Adrianov Adrianov and and George Nikolai George Shabat A.Grothendieck, Esquisse d'un programme, programme, in in Geometric GeometricGalois GaloisAcActions, volume volume I.I.

R.M.Guralnick, J.G.Thompson, J.G.Thompson, Finite Finite groups groups of of genus genus zero. zero. 3. J. Al[GTh] R.M.Guralnick, Al[GThI gebra 131 no.l no.1 (1990), 303-341. gebra [GTDE] The Grothendieck Grothendieck Theory Theory of of Dessins Dessins d'Enfants, [GTBE]The d'Enfants, ed. ed. L. Schneps, Schneps, London Math. Math. Soc. London Soc. Lecture Lecture Notes Notes Series Series 200, 200, Cambridge Cambridge UniverUniversity Press, 1994. 1994. G.A.Jones, M.Streit, M.Streit, Galois groups, monodromy monodromy groups groups and and carto[JS] G.A.Jones, Galois groups, [.JS] graphic groups, this volume. graphic volume. B.H.Matzat, Konstruktive [Mat] B.H.Matzat, Konstruktive Galoistheorie, Galoistheorie, Springer Springer Lecture Notes Notes [Mat] 1284, Berlin-Heidelberg-New York, 1987. 1987. [Mull [Miil]

P. Muller, Miiller, Primitive monodromy monodromy groups of polynomials, in Recent Recent developments Galois problem problem (M. (M. Fried, Fried, ed.), Condevelopments in the the inverse Galois Contemporary Mathematics, vol. 186, 1995, 1995, 385-401. 385-401.

L.Schneps, Introduction Introduction toto\[GTDE], [Schi] L.Schneps, [Schl] ^[GTDE], 1-15. 1-15.

L.Sclineps, Dessins Dessinsd'enfants d'enfantson on the the Riemann Riemann sphere, sphere, in [GTDE], [Sch2] [GTDE], [Sch2] L.Schneps, 47-78. [ShVo] G.B.Shabat, [ShVo] G.B.Shabat, V.A.Voevodsky, V.A.Voevodsky, Drawing curves over number fields, fields, in in The Grothendieck Festschrift III, III, Progress Progress in in Math. Math. 88, Grothendieck Festschrift 88, Birkhãuser, Birkhauser, Basel, 1990, 1990, 199-227. 199-227.

[ShZv] G.Shabat, A.Zvonkin, and algebraic algebraic numbers, numbers, in in JerusaJerusa[ShZv] A.Zvonkin, Plane trees and lem Combinatorics (H. Barcelo, Barcelo, G. G. Kalai, eds.), Contemporary Combinatorics 93 (H. Contemporary Mathematics, vol. Mathematics, vol.178, 178,1994, 1994,233-275. 233-275. [Wie] [Wie]

H.Wielandt, Finite Finite permutation permutation groups. groups.Academic AcademicPress, Press,New New YorkYorkLondon, 1964. 1964.

** Moscow Moscow State State University University adrianov©nw . math.msu.su adrianov@nw. mat h. msu. su ** ** Institute of New Technologies Technologies shabat©int.glas.apc.org [email protected]

Galois Groups, Monodromy Groups Galois Groups Groups and Cartographic Groups Gareth A. Jones and Manfred Manfred Streit

Abstract. The Abstract. TheRiemann Riemannsurfaces surfacesdefined defined over over the the algealgebraic are those those admitting admittingBelyY Belyi functions; functions; such such braic numbers are functions can be represented represented combinatorially combinatorially by maps functions can be by maps called faithfully called dessins d'enfants, d 'enfants,and and these these are are permuted faithfully by Galois group group of algebraic numbers. We dedeby the Galois of the algebraic numbers. We fine the the monodromy monodromy group group and and the cartographic group of a dessin, and show that these permutation permutation groups groups (and (and hence hence many properties of dessins) are many other other properties of dessins) are invariant invariant under under this action. We We give give examples to show how these groups can be computed how they distinguish Galois Galois computed and and how they can be used to distinguish on dessins. dessins. orbits on §0. Introduction

One of of the the most but intractable One most interesting interesting but intractable groups groups in mathmathematics ematics is the absolute absolute Galois Galois group, group, the theautomorphism automorphism group group G field Q Q of algebraic algebraic numbers. Since Since Q is is G = Gal(Q/Q) of the field union of all the algebraic algebraic number number fields C C, C, ititfollows follows the union fields K KC G isis the theinverse inverse limit limit of the finite Galois Galois groups that G groups Gal Gal (A7Q), (K/Q), where K ranges over over the Galois Galois extensions extensions of Q. Q. Thus G is is aa where K ranges profinite group, which one as embodying embodying the the whole whole profinite group, which one can can regard as of classical classical Galois Galois theory theory over Q. recent years years there has developed In recent developed a geometric geometric approach approach to this important group, 'enfants, which group, through through dessins dessinsdd'enfants, which are essentially maps compact Riemann surfaces. surfaces. ItIt isiswellwellsentially maps drawn drawn on compact known Riemann surface compact ifif and only if known that that a Riemann surface XX is compact and only if it corresponds equivalence class of algebraic corresponds to an equivalence algebraic curves. We may choose = 00 choose some somepolynomial polynomialf f e C[x,y]so C[x, y] so that that the curve curve /(#, f(x, y) = is a plane model (possibly singular) representing this class. Belyi model (possibly singular) representing Belyl [Bel] defined over over Q (in the sense that we we [Bel]showed showedthat that XX is defined the sense

26 26

Gareth Jones and Manfred Gareth Manfred Streit

y]) if and only can take take /f E£ Q [#>!/]) can only if if there there is is aa Belyl Belyi function function 1 /3 : X —+ —> E == P'(C) P (C) == CCU {oo}, {oo}, that thatis,is,a ameromorphic meromorphicfuncfunction which 1, oo}. {0,1, oo}. One One can can form form a which isisunbranched unbranchedoutside outside{O, pair (X, /?) /3) by by taking taking a simple dessin 'picture' of of such such aa BelyY BelyT pair dessin T>\ on onthe theRiemann Riemann sphere sphere E, S, such such as asthe thebipartite bipartitemap mapwith with two vertices 1], and using vertices at 00 and and 11joined joined by by aasingle single edge edge [0, [0,1], using l /3 to lift V1 to a dessin V = on X. The maps on Riej3 to lift T>\ to a dessin V = f3~ {T>i) on X. The maps on Riemann surfaces 'erzfants surfaces obtained in in this thisway wayare arecalled calleddessins dessinsd d'enfants view of childish appearance of of some of the simplest in view of the the rather childish examples. Grothendieck [Gro] [Gro]observed observedthat thatthe the natural natural action action of of G on Grothendieck on Belyl pairs .induces induces an an action action on on these these dessins. BelyT pairs dessins. What is is remarkremarkable is that that this action action of of G is is faithful, in the sense sense that each each nonnonidentity element u to a non-isomorphic dessin T>° VU;r;indeed, Schneps [Schi] shows indeed,a aresult resultofofLenstra Lenstraand and Schneps [Schl] shows that G G acts acts faithfully faithfully on on plane plane trees trees (maps (maps on E S with with aa single single face). face). These These apparently apparently simple simple and and explicit explicit combinatorial combinatorial obobtherefore provide Many jects therefore provide aa direct direct insight insight into into the the group group G. G. Many examples of this this connection between Galois Galois theory theory and and plane examples of connection between plane trees have been studied by Shabat and by Shabat and and Voevodsky Voevodsky [SV] [SV] and Bétréma, Péré and Zvonkin [BPZ], and a general theory is emerga general theory is emergBetrema, Pere and Zvonkin [BPZ], ing in the papers of Schneps [Schl] [Schi] and and Shabat Shabat of Couveignes Couveignes [Cou], [Cou], Schneps Zvonkin [SZ]; [SZ]; Wolfart and Zvonkin Wolfart [Woll] gives gives examples examples of of the the action of on torus torus dessins, dessins, and and further further examples examples of of positive positive genus genus are are G on considered in [Jon]. [Jon]. For on different aspects of of For general surveys surveys on different aspects considered in dessins dd'enfants 'enfants see see [CIW, [CIW, Jon, Jon, JS3], JS3], and and for for recent recent research research in this and and related related areas, areas, see see the the proceedings proceedings of of the the 1993 1993 Luminy Luminy conference [Sch2]. [Sch2]. In its action on dessins, dessins, G preserves preserves all the obvious obvious numerical parameters such such as as the the genus, genus, the the numbers numbers of of vertices, vertices, edges edges and and faces, ofthe the vertices, vertices, and and so so on. on. It also preserves the faces, the valencies valencies of group of orientation-preserving orientation-preserving automorphisms, though not not necnecessarily those which which reverse reverseorientation. orientation. (These (These facts facts seem seem to to be essarily those widely-known, but explicit proofs are not so easy to find.) widely-known, but explicit proofs are not so easy to find.) This makes makes it a non-trivial task to to determine determine whether whether two two de.ssins dessins are conjugate conjugate under under G: G: itit isis necessary necessary but not not sufficient sufficient that these these

Monodrorny Monodromy and Galois Galois groups groups

27

invariants should be be equal. equal. A invariants should A rather rather finer finer invariant invariant isis provided provided by the monodromy monodromy group which we define group of a dessin, which define to to be be the monodromy group of the branched branched covering covering /3 ::XX —+ —• E. E. This Thisisis monodromy a 2-generator 2-generator transitive subgroup subgroup G = —(go, (#o,gi) where whereNNisisthe thedegree degreeofof/3;j3\the thepermutations permutationsg0,g1 #o,#i group -1 describethe thebranching-pattern branching-pattern of of /3 (3 above and #00 = (go V and their monodrorny monodromy generators gj which conjugate Theorem.Two Twodessins dessinsVT> whichare are conjugateunder under a groups G C and G have have monodromy monodromy groups and GU G which which are conjugate conjugate in SN, that is, permutation groups. SN, is, which which are are isomorphic as permutation

Since the the orientation-preserving orientation-preserving automorphism group of a desSince sin is the centraliser centraliser of of its its monodromy monodromy group group in in SN, SN, ititfollows follows that this this isis also also invariant invariant under G. G. We Weshall shall show show that that for for each each 1,00, i= = 0, 0,1, oo, the generators generators gj gi and gf of C G and and CU G° are are conjugate conjugate in SN; SN; they therefore therefore have have the the same same cycle-structure, cycle-structure, which which imima plies that V V and and VU V share the same numerical numerical parameters listed plies above. (Note that these these generators generators need need not not be besimultaneously simultaneously above. conjugate, so V and and VU T>acan canbebenon-isomorphic.) non-isomorphic.)UnfortuUnfortuconjugate, so that V nately, although they nately, although these these parameters parameters are are invariant invariant under under G, they always sufficient sufficient to distinguish its different different orbits: we are not always to distinguish orbits: we shall give examples where two dessins share the same parameshall give examples where two dessins share same parameters, but the the monodromy monodromy groups (and hence the dessins) dessins) are not conjugate. Similarly, the the monodromy monodromy group group does not always Similarly, always distinguish distinguish can have have conjugate conjugate monorbits of G; thus thus non-conjugate non-conjugate des.sin.s dessins can ociromygroups, groups,so sothe the converse converseofofour our theorem theorem isis false. false. Under odromy invariant is needed, needed, and and for for this one these circumstances, circumstances, a finer invariant can use group CC of of a dessin; can use the cartographic cartographic group dessin; this transitive transitive subgroup of S N is the monodromy group of the Belyi function subgroup of 52N is the monodromy group of Belyl function 2 4(3(1—[3) X —> and S, and main theorem shows conju—i3) : N: —* ourour main theorem alsoalso shows thatthat conjugate dessins have have conjugate cartographic cartographic groups. groups. Now Nowconjugacy conjugacy of implies conjugacy of monodromy groups, groups, of cartographic groups implies but the converse converse isis false: false: we shall give give examples examples of dessins dessins but the we shall

28 28

Gareth Jones and Manfred Manfred Streit

which have conjugate conjugate monodromy groups groups but non-conjugate carcar-

tographic groups. groups. This that C isis more than G tographic This shows shows that more effective effective than in distinguishing orbitsofofG, G, but but there distinguishing orbits there is a price price to be be payed payed for for this: doubling doubling the degree degree of a permutation group group can can have have a disproportionate effect on on its its size, size, so so itit can can be much harder to disproportionate effect to compute C C than CGfor compute for aagiven givendessin. dessin.Moreover, Moreover, we we will will show show by means of examples that in some cases even C can fail by means of examples that some cases even C fail to to distinguish distinguish orbits of G. G. This paper isis organised organised as as follows. follows. After After briefly describing describing G G we state Belyl's in §1, we Belyi's Theorem with aa few few simple simple examples examples in §2, and and then in §3 we we show showhow howthis thistheorem theorem isis related related to various §2, types of dessins, such as triangulations, hypermaps, dessins, such hypermaps, and and biparbipartite maps. rtiaps. These These are are represented represented algebraically algebraically in §4 §4 in terms of groups, namely namely the monodromy monodromy group and certain permutation groups, cartographic group, group, and and geometrically geometrically in §5 §5 as quotients quotients of of the cartographic We describe certain universal structures on the hyperbolic plane. We how G how G acts on these dessins, and in the Appendix we prove prove that that various parameters §6 we concentrate various 1)arametersare areinvariant invariantunder under G. G. In §6 our attention on on the the simplest simplest dessins dessins which which admit a faithful faithful acnamely the the plane plane trees, trees,showing showing how how these these are are derived derived tion of G, namely from a class of from of polynomials which which generalise generalise the Chebyshev polynomials. In In §7 §7 we we give give a number number of examples examples of nomials. of this this action of G, showing showing how how the monodromy monodromy and cartographic cartographic groups groups can can be used used to distinguish distinguish the orbits orbits of of G G on on plane plane trees trees of of aagiven given he type. combinatorial type.

We are are very very grateful grateful to to the European We European Union Union Human Capital Capital Mobility Program on Computational Computational Conformal Conformal Geometry Geometry and Mobility for for enabling enabling this this collaboration collaboration to take place, to David Singerman for for many many valuable valuable comments comments about about Riemann Riemann surfaces surfaces and and related related and to to Sasha Sasha Zvonkin Zvonkin for for his his constructive constructive criticisms criticisms of matters, and draft of this paper. paper. an earlier draft §1. The absolute Galois Galois group

The algebraic numbers are are the the elements elements a £E C which generate algebraic numbers a finite Q(a) D finite extension extension Q(a) DQ Q of the rational rational field field Q; Q; they form form a field field Q, the algebraic closure of Q in C. For For each each a E G Q, the the

Monodromy Monodromy and Galois groups

29

minimal polynomial polynomial of of aa over C Q Q minimal over Q has a splitting field Ka Ka C which is a Galois Galois (finite (finite normal) extension of Q, and conversely the primitive element element theorem theorem implies implies that every every Galois Galois extenextension D Q arises arises in this way; way; thus sion K D

where AlT JC isisthe the set set of of all all Galois extensions extensions KKofofQQininC. C.For Foreach each (K/Q) of Galois group of K over over Q is a group G(K) G(K) = Gal (K/Q) K EG K, the Galois If K finite group of of order order equal equal to to the the degree degree \K 1K:: Q Q|.I. D L where K, A",LL G /C, then then every every automorphism automorphism of of K K leaves leaves LLinvariant, invariant, G(K) —+ -» so the restriction restriction mapping mapping isisaahomomorphism homomorphismPK,L pxyL • G(K) anepimorphism, epimorphism, since every automorphism G(L); in fact, PK,L PK,L isisan of L can he extended (in K::LI of L can be extended (in \K L\ ways) to an an automorphism automorphism of K. A'. :

The groups groups G(K) G(K) (K (K E£ fC) and the epimorphisms The epimorphisms PK,L px,L form (or projective) projective) system, system, and and G can be an inverse inverse (or be identified identified with its inverse limit: G=limG(K). G = limG(A r ). con This is the subgroup IIA'GX: ^G(K) ( ^ 0 con" subgroup of of the cartesian cartesian product product HKEK sisting sisting of all all elements elements (gx) such that that9KPK,L gxPK}L = 9h whenever K D L, and one identifies each g G K D L, and one identifies each G with with the the element element (ga-) (gx) where gx is the restriction of g to K. (In simple language, E GGisisaaconsistent each g G consistentchoice choiceof ofautomorphisms automorphisms of ofthe thefields fields AlT.) The discrete on the the finite K G /C.) The discrete topologies topologies on finite groups groups G(K) G(K) impose G(K), which which is compact compact by TyTyimpose a topology topology on on Y\Ke>cG(K), chonoff's Theorem. The subgroup chonoff's Theorem. subgroup G inherits inherits aatopology, topology, called called the Krull which isis compact compact since since G G is closed. Krull topology, topology, which closed. Under the Galois Galois correspondence, correspondence, the subfields subfields of Q correspond to the closed subgroups of of G. (See closed subgroups (See [Jac, [Jac, Ch.8] Ch.8] for further details.) details.)

fromthe the above aboveidentification identificationthat that G G has cardinality It follows follows from 2 °, so particuso one one should shouldnot not expect expectthe the structure structure of of G G to be particularly straightforward; for for example, example, being being uncountable uncountable G cannot larly straightforward; be finitely generated. generated. Neukirch has shown shownthat that G is Neukirch [Neu] [Neu] has is comcomplete (that is, is, its centre centre Z(G) and its outer automorphism group group K

30

Gareth Jones and and Manfred Manfred Streit

Out(G) are areboth both trivial), trivial),so soG Gisisnaturally naturally isomorphic isomorphic to to its itsauautomorphism group onecannot cannot imbed imbed tomorphism group Aut(G); Aut(G); this means that one as aa normal normal subgroup subgroup in any larger diG as larger group, group, other other than as aa direct factor, so one one cannot cannot obtain obtain more more information information about about G from Ideally, one would like to have aa fairly explicit explicit such imbeddings. Ideally, faithful some easily-studied faithful representation representation of G G on some easily-studied structure structure<S, S, so that G ESwhose whose Riemann surface function f3 ::XX—* are contained contained in in{0, {0,1, topologically, this is is aa critical values are 1, oo}; topologically, finite covering which points. finite covering which isis unbranched unbranched outside outside these these three points. In these these circumstances circumstanceswe wewill willcall call(X, (X,j3) j3)aaBelyT Belyi pair. pair.

Example 1. X = ES and let 1. Let Let X let /3(x) (3(x) = xn for for some fixed integer 77. > 1. 1. This an n-sheeted n-sheeted covering, covering, branched branched over over 0 and and oo This is an (where cycles of (where the the n sheets come together together in cycles of length length n), n), so it is a BelyT function on Belyi function on S. ii.

Example Example 2. 2.Let LetXXbe bethe then-tb n-thdegree degreeFermat Fermat curve curve Xn, given n n yfl by the algebraic function function x + y = = 1; 1; this is is aa compact compact Riemann Riemann surface 2)/2, surface of ofgenus genus(ri(n— —1)(n l)(n—— 2)/2,defined definedover overQ.Q.The Theprojection projection n : Xn —+ —> S,(x,(x,y) i x^ isor not is not a Belyi function, since critical a Belyl function, since its its critical y) '—+— values are n-th roots roots of of unity, unity, but butififwe we compose compose ir n with the values are the n-th the n function x xi—+ function *-} x (which (which maps maps all allthese these points points to to1)1)we weobtain obtain

Monodromy and Galois Monodromy Galois groups

31 31

a BelyY function ft : Xn -¥ E, Belyi function S, (x, (x,y)y)'—+ H-> xn of degree n2. n 2 . Above Above 2 critical values values 0,1 n sheets sheets come come together the critical 0, 1 and and oo, oo, the n2 together in n cycles of length n. cycles of n. :

Example 3. elliptic curve curve is is a Riernann 3. An elliptic Riemann surface surface X X of of genus genus 1; algebraically, algebraically,itit can can be be put into Legendre 1; Legendre normal form form y2 = x(x - l)(x - X)

(A ( A €EC\{O,1}). C\{0,l}).

Such aa surface defined over Q if and only only ifif AA € Q. Q. Such surface XX = X\ X), is defined In these these cases cases itit is possible possible to obtain obtain aa Belyl Belyi function function ft on on X the projection y) i-÷ has critical by composing composing the projection it n : (x, (#,y) H-> xX (which (which has critical values rational functions functions ES—+ —> E.S. values 0,1, 0, 1,ob ob and and A) A) with with suitable rational good example example is case A A = 11 + y/2 considered considered by Wolfart Wolfart A good is the case [Woll]: an algebraic algebraic number, number, with with minimal minimal polynomial polynomial [Woll]: this this is an p(x) p(x) = — x2 — 2x 2x ——1 1over overQ,Q,and andthe thefunction functionx x \-tww==—p(x) —p(%) sends critical values values of of itn to 1,2, sends the four critical 1, 2,oo oo and and 0 respectively; :

2 the function w w h-> w /4(w —1)1) now now sends sends these these points points to to oo, oo,1,00 1, oo w2/4(w— and 0, and one can verify verify that the composition composition that the 2 2 (x2 2x — 1)2 * (x— -2x-l) 4(w -—1) ft:(x,y)*-> A{w 4x(2 1) = Ax(2 — - x) x)

o /

v

w

of these functions functions with itn is is a Belyl Belyi function function of degree degree 8 on X. X. In 1979, proved the following 1979, Belyl Belyi [Bel] [Bel] proved following theorem: theorem:

Theorem. AAcompact Theorem. compactIliemann Riemann surface surface X X isisdefined defined over over Q if if and only only ifif there thereisisa aBelyY Belyi function function /3: fi :XX—+ -> E. S. The fact The fact that that this thiscondition conditionisissufficient sufficient follows follows from from Weil's Weil's rigidity theorem for the the details. Belyi proved theorem [Wei]; [Wei]; see see [Wo12] [Wol2] for that this this condition condition isis necessary necessary by by composing composing a projection projection itn : X —+ S, E, with X -» suitably with critical critical values values only only in in Q U U {oo}, {oo}, with with suitably chosen chosen rational functions S — theseeventually eventuallyforce force the thecritcrit—+> S, these ical values values into {0, {0,1, giving rise to to aaBelyT Belyi function function ical 1, oo}, oo}, thus thus giving /3: ft : X X -> S as in Examples Examples 2 and and 3. 3.

There is a natural action There is action of of the the absolute absolute Galois Galois group group G on on Belyl /3): both X Belyi pairs (X, (X,ft): X and and /3 ft are defined so one defined over over Q, so

can let each automorphism ao 6E G G act naturally can let each automorphism naturally on on the the defining defining

32 32


a coefficients to give a BelyT pair (X (Xe, coefficients Belyi pair ,/? a ). Let us define define two twoBelyT Belyi 1 pairs (X, /3) and and (X', /3') to be equivalent pairs (X,f3) {X ',/?') equivalent if there is is an an isomorisomor1 such that /?' o i = /?; X -÷ X' such /9; then there is phism i : X -> X is an induced induced action of G pairs. In Examples G on on equivalence equivalence classes classes of of BelyT Belyi pairs. Examples 11 and 2, for instance, everything is defined over Q and is therefore everything is defined fixed by by G, but in fixed in Example Example 33 we we see see a non-trivial Galois Galois action. action. Wolfart's Wolfart's curve X\ = X^^ is defined defined over Q(\/2), which has a Galois group of order 2 generated by \/2 '—+ K> —y/2. This automorphism (which extends extends to to an an element element aa E phism (which € G) G) transforms transforms X\ to the conjugate elliptic = Xl_^ and elliptic curve X\* — and leaves leaves the theformula formula for the Belyi for Belyi function function /9 f3 unchanged unchanged since since this this isisdefined defined over over Q. Q. Now in'general, in general, two two elliptic elliptic curves curves XX = X\ are isomorphic isomorphic (as (as Riernann surfaces) ifif and and only only ifif they they have have the the same same J-invariant Riemann surfaces) 2 3 2 2 J(X) = 4(1 — A + A2)3/27A2(1 — A)2; in our case, J(X) = 4(1 - A + A ) /27A (1 - A) ; in our case, X^^ and and Xx_^ have (19 -— 3\/2)/27 and (19 ++ 3\/l)/27 rehave J-invariants J-invariants (19 and (19 spectively, so non-isomorphic and hence hence the thetwo twoBelyT Belyi spectively, so they they are non-isomorphic pairs are inequivalent. inequivalent.

This action pairs is faithful. action of of G G on onequivalence equivalenceclasses classesof ofBelyT Belyi pairs faithful. In fact, fact, it is is easy easy to see see that itit remains remains faithful faithful when when restricted restricted functions on on elliptic if oa isis any to BelyT Belyi functions elliptic curves: curves: if any non-identity non-identity element of of GG then then some some aa E by cr, a, and element £ Q is is moved moved by and by by solving solving 2 3 2 2 A++A2)3/27A2(1 A ) /27A (1—- A)2 A) == aawe weobtain obtain the equation equation J(X) J(X) = 4(1 4(1 -— A an elliptic curve X — X\ which is not isomorphic to its conjugate curve X = which is not isomorphic to its conjugate since A £E Q, there is a Belyi Belyi function function /9 (3 on on X, X, and and curve X°r; since Belyi pair to an an inequivalent inequivalent pair. pair. the BelyT pair (A", (X, /3) /3) is transformed transformed by a to we shall more useful useful (but (but less less obviously obviously As we shall see see later, later, there is aa more faithful) action on onBelyT Belyi pairs of of genus 0. 0. faithful) action

BelyTfunctions functions and and dessins §3. Belyt pair (X, /3) (and hence A good good way way of visualising visualising a BelyT Belyi pair hence of seeing how G G acts) acts) is to take seeing how take some some simple simple combinatorial combinatorial structure on E, such such as a map, map, aa triangulation triangulation or aa hypermap, hypermap, and and to use (3 /3 to lift lift itit to to X, X,thus thusobtaining obtaining aasimilar similar structure structure on on XX which covers the original structure on S. The action of G which covers the original structure on E. The action of G on Belyi pairs structures. BelyT pairsthen then induces induces actions actions of of G G on on these structures. 3.1. Triangulations. Let Let us us form form aa triangulation triangulation7i7 of of E

Monodromy and Galois groups Monodromy

33

by choosing three vertices 0, 1 and and oo, by choosing three vertices 0,1 oo, and three three edges edges along along the line-segments line-segments in are in RR joining joining these these vertices, vertices, so so that that there are halftwo faces, corresponding and lower lower halftwo triangular triangular faces, corresponding to to the upper and planes S ais BelyT a BelyTfunction, function, then then/3'('7) j3~l{T\) is ais a planes of of C. C. If (3 : :XX—+—> is triangulation T X. Since Sincethere thereisisno nobranching branching away away from the triangulation I ofofX. vertices, each of the two two faces of 7; 7i lifts to N N triangular triangular faces faces on T V = deg(/3), and similarly each of the three edges lifts X, where X, where N = deg(1@), similarly each of the three edges lifts 7Vedges edgeson on X, X,so so'T T has has2N 2N faces faces and and3N 37Vedges. edges.Now Nowsuppose suppose to N over a vertex v = = 0, 0,1 oc of 7i, the N TVsheets sheetscome cometogether together that over 1 or oo of'Ti, . in cycles of lengths n V f i,..., nvA + -\ N); +\- nVikv ==N); , nVjkv (where n,,1 l vertices since vv has has valency valency 2, 2, itit follows followsthat that j3~ ,@'(v) (y) consists consists of of kv of valencies Using this this triangulation triangulationTI of X, of valencies 2n0,1,.. 2 n ^ i , . . . ,,2nv,kv> Using X, an one can can compute the genus genus gg of of d the one compute the Euler characteristic characteristic x and X: .

.

.

= (ko + k1 + k00)-3N 2—2g =X = 2-2g =

+ 2N== ko + ki + k00-N. (1)(1)

ofthe the faces facesofofT: I: one There is aa natural 2-colouring 2-colouring of one can label the the faces or — — asas they theycover cover the theupper upperororlower lowerhalf-plane half-plane of ofC, C, faces + or so each each edge edge separates separates faces faces with with different different labels. labels. Similarly the so vertices can can be 3-coloured, vertices 3-coloured, or labelled labelled 0, 0,11 or oo, 00, depending depending on on which vertex vertex v of 7; which 7i they cover, each edge edge joins vertices vertices cover, so so that that each with have with different different labels. labels. The The three three vertices vertices on on each each face faceofofTI have different face is called called positive different labels, labels, and and the face positive or or negative negative as its (induced by by /3 /3 from from the the orientation orientation of of E) E)corresponds corresponds orientation (induced to the cyclic cyclic order (Oloo) (Oloo) or (oolO) (00IO) of these labels. For x i—* For instance, the Belyl BelyT function function /3 (3 : S —+ -» E, S,# »-» xn in in ExamExampie of §2 §2gives givesrise risetotoaatriangulation triangulationTI of ple 1 of of S with two two vertices vertices of valency valency 2n 2n at at 0 and and oc, valency 22 at at the n-th 00, and iin vertices of valency n-th exp(2nij/n) ofof11(where roots exp(2irij/n) (wherethere thereisisno nobranching, branching,so son1,3 ni}j = 11 for j). There are for each each j). edges, one joining joining each exp(27rij/n) exp(27rij/n) are 3n edges, to 0 and one joining joining itit to oc, 00, and and nn joining joining 00 to to oo 00 (in (in this this case, case, in general, general, the the edges edges are are all alleuclidean euclidean line-segments line-segments though not in these edges edges enclose enclose 2n and in C); these 2n triangular triangular faces, faces, nn labelled labelled ++ and n labelled —. In Example 2, 2, the the BelyT BelyT function (#,y) y) \-+x'1xnofof labelled—. function j3 /3: :(x, 2 degree nn2 gives curve Xn with 3n degree gives aa triangulation triangulation of of the Fermat curve 2 2 vertices (all of valency valency 2n), 3n edges, 2n triangular faces. faces. vertices 2ii), 3n2 edges, and and 2n2 :

34

Careth Gareth Jones Jones and and Manfred Manfred Streit Streit

graph theory as aa minimum-genus onThis arises in topological topological graph minimum-genus orientable embedding embedding of of the the complete complete tripartite tripartite graph Kn^n [RY]: [RY]: there are three sets sets of of nn vertices, vertices, labelled labelled 0, 0,11 and oo, and and each pair with distinct labels are joined by a single edge. labels are joined by a single edge. 3.2. Bipartite maps. IfIfwe wedelete delete the the vertex vertex at oo oo and and its its two incident incidentedges edgesinin7i, 7, we are left with a map B\ consisting two left with embedded in in S. E. This has one edge edge I/ = [[0, of a graph Q\ embedded 0 ,11 l ] Cc R joining vertices 00 and and 1, and a single face SE \\ /I.. Our Belyl single face Belyi joining two vertices function function /3 (3 lifts lifts B\ to a map map 8 B = /9'(Bi) fi~l(B\) on X, which which can can deleting its vertices labelled labelled co oo and their obtained from from T Y by be obtained by deleting 1 = /3_1(I) is incident edges. edges. The incident The embedded graph Q = /?~ (/) is bipartite: its vertices vertices can can be be coloured coloured black black or or white white as as they they cover cover 00 or or 1, so that each each edge edge of Q joins joins vertices vertices of of different different colours. There There are are . . , no,fco k0 A ?o black black vertices vertices and andk1 k\ white whitevertices, vertices,of ofvalencies valencies no,i,. no,i,..., and n1,1,... respectively. There There are N and 7ii,i,... ,711^ respectively. TVedges edges in Q (each , covering covering/ I), ) , and and B 8 has has koo faces faces with with 2noo ? i,... ,272^^^ sides. sides. As As in §3.1, the genus of X X are §3.1, the genus and characteristic characteristic of are given given by (1). Just as as one can obtain bydeleting deleting vertices vertices and and edges, edges, obtain B from from T I by one can obtain obtain ITfrom from 13 B by by reversing reversing this extra one this process: process: an extra vertex (labelled oo) is placed placed in each each face face of of #, B, and and this is joined by by non-intersecting non-intersecting edges edges to to the the vertices vertices of of BB incident incident with with that face. The resulting resulting triangulation called the thestellation stellation of of B. B. face. triangulation T I isiscalled

For instance, the Belyl Belyi function function /? /3:: E £ -> £, E, xx i—+ h-> xn in Example 1 of §2 yields mapBB with with one oneblack black vertex vertexof ofvalency valency yields aa bipartite map n at at 0, 0, joined joined by by aa single single edge edge to each each of the n-th n-th roots roots of of 11 (which are coloured colouredwhite). white). This This is is an example of aa plane plane tree (which are example of (an embedding of a tree Q in in C), C),aasituation situationwhich whichalways alwaysarises arises if if X the Belyl Belyi function /3 j3 is is aa polynomial; polynomial; we wewill will examine examine X = SE and the this connection with trees trees more more generally generally in in §6. §6. Example Example22of of§2 §2 gives gives an an embedding embedding of of the the complete complete bipartite bipartite graph Kn%n in the Fermat curve curve Xn: there are n black black and and nn white white vertices, vertices, each each black by aa single single edge, edge, giving a total total of of n2 n2 black and white pair joined by edges; faces, each each of of them aa 2ri-gon 2n-gon formed formed from from edges; the the map has nn faces, 2n faces described in in §3.1. §3.1. For For more more faces of of the the Fermat Fermat triangulation described examples arising from from Belyl Belyi pairs, pairs, see see [Jon]. [Jon]. examples of of bipartite bipartite maps arising

3.3. Hypermaps. 3.3. Hypermaps.The Thedual dualofofthe thetriangulation triangulation IT inin§3.1 §3.1 is is

Monodromy and and Galois Monodromy Galois groups

35

a trivalent trivalent map map on on X, A',together together with witha a3-colouring 3-colouring of of its its faces faces with the labels labels 0, 0,11 and types of of faces faces form with and oo. These three types form the hypervertices, hyperedges hypervertices, hyperedges and and hyperfaces hyperfaces ofof aa hypermap hypermap % 1t on A', which which can can equivalently equivalently be the X, be obtained obtained by by using using j3 ,8 to to lift lift the hypermap 1-Li(with (withone onehypervertex, hypervertex,one onehyperedge hyperedgeand and trivial hypermap one hyperface) X. The The Walsh Walsh map map W(1-t) W(U) associated associated one hyperface) from from SE to to X. with 1-i % (see [Wai]) is the bipartite map B we have just (see [Wal]) is the bipartite map 8 have just described, faces of 8B corresponding corresponding with black vertices, white vertices and faces hypervertices, hyperedges hyperedges and and hyperfaces hyperfacesofof1-i. Ti. For good For a good to hypervertices, survey of [CM], and relevance to Belyl BelyT survey of hypermaps, hypermaps, see [CM], and for for their relevance functions see see [JS2, [JS2, JS3]. JS3]. functions

3.4. Maps. Maps.For Forour ournext nextexample examplewe weneed need the the idea idea of of aa clean clean function with with the property or pure pure BelyT function: function: this is a BelyT function so that that (in (in the the notation notationof of§3.1) §3.1) nij = 22 for all j = 1 , . .. . ,, fci, so for all j 1,. the sheets together in in pairs over sheets come come together over 1 (and hence hence the degree degree N = 2ki 2k1 must N is easily easily seen —> Sis is must be be even). even). It is seen that that ifif (3 : X —+ any Belyl function then 7 == 4/3(1 — E is BelyY BelyT function X —> E aisclean a clean BelyT — j3) :X .

:

function function of of degree degree 22deg(/?); this is is formed formed by by composing composing ,3 (3 with with quadratic polynomial polynomial q q: E : S—÷ —> E,S, zzi—p H* 4z(1 4z(l — —z)z)(which (whichis is the quadratic itself a clean Belyl BelyT function). function). itself

Given any any clean Given clean Belyl BelyT function function 'y 7 : X —÷ —> E,S, we we can can construct construct X. Let Let us us first first draw draw aa map map M1 A4\ on on ESconsisting consisting of of aa a map on X. single vertex "free edge" "half-edge ") along single vertex at at 0, and a single "free edge" (or "half-edge the real line-segment line-segment /I = [0, [0,1] point 11 (which (which is is 1] from from 00 to to the point not itself a vertex); vertex); the rest rest of of ES forms forms the the single single face face of of M1. M\. not itself 1 We then (A4i) on on X. X. Since Since the the We then define defineAiMtotobebethe themap map7~-y'(Mi) 1 vertex 0 has valency valency 1, 1, 7~ (0) consists of of k0 ko vertices vertices of of valencies valencies (0) consists n (j = 1,... , ko). o,j {j = l,...,fco)Since nij = 2 for all j , the half-edge 2 for all j, half-edge in 7iO,j M1 N/2 edges M\ lifts to k1 hi = N/2 edges on on X, X, each containing an element of 1 7~ (1). Since Since the the only only branching over the unique face of of A4\ M1 is at 00, there are ^ faces faces in 1 , . .. . ,fcoo),each oc, in M. M with riooj sides sides (j(j = 1,. containing an element of 7~ 1 (oo). In containing 4/3(1 — j3) In particular, particular, if 7y == 4j3(1 for some is the bipartite map map B some BelyT Belyl function function /3 on on A, X, then then A4 M is 8 l formedfrom from(3/3inin §3.2: §3.2:this thisisisbecause becauseq~q'(it4i) formed (M\) =— B\, so so qqlifts lifts vertex 00 of of M\ M1 to to the two the single single vertex two vertices 0 and and 11 of B\, and then i3 then (3 lifts lifts these to the the black black and and white white vertices vertices of M A4 == 8.B. .

36 36

Gareth Jones Jones and and Manfred Manfred Streit

The combinatorial combinatorial structures structuresobtained obtainedfrom fromBelyT Belyi functions as as above are natural above are sometimes sometimes known knownasas dessins dessinsd'enfants. d'enfants. The The natural action of G on on BelyT Belyi pairs various action pairs induces induces actions actions of of G G on the various types of dessins dessins representing representing these and explicit explicit examples examples these pairs, and of these actions actions are are given given in in §7. §7. As shown in Appendix, of these As shown in the Appendix, one can can give give algebraic algebraic definitions definitidnsofofthe the degree degree NN and and the parone partitions 7?î + • • •+ + nVtkof associatedwith withthe thecritical criticalvalues values v ofNNassociated vv = = 0, 0,l,oc, such a way way that these these are are invariant invariant under under this 1, oc, in in such action follows that that GGpreserves preservessuch such numerical numerical paraparaaction of of G. ItItfollows meters genus, the the numbers numbers of of vertices, vertices, edges edges and and faces, faces, meters as the genus, the valencies valencies of the vertices, etc. For example, G leaves invariof the vertices, For example, G leaves invarithe set set of of clean clean BelyT Belyi functions ant the functions and and hence hence acts acts on on the set of M. in in §3.4. §3.4. Similarly, Similarly, since G preserves the genus of maps M of XX and branching-pattern over set of of plane plane trees trees the branching-pattern over oo, oc, itit acts acts on on the set with BelyT Belyi polynomials /3 : ES —> £inin§3.2. §3.2.Other OtherG-Gassociated with invariant properties include include coverings coverings and orientation-preserving orientation-preserving groups. automorphism groups.

Although G preserves Although preserves all the above properties, it nevertheless nevertheless acts faithfully in the sense faithfully in sense that each each non-identity non-identity element element of of G of the same transforms some dessin into a non-isomorphic dessin of type. This restricts the action of type. This remains remains true true even even when when one one restricts action of G to to very very simple simple dessins: dessins: for for example, example, Schneps Schneps [Schi] [Schl] has has used used results faithfully results of of Lenstra Lenstra on on polynomials polynomialsto to show showthat that G acts faithfully on plane trees. This This allows allows one to study G G in inaavery veryexplicit explicit way, way, 1)0th visually and computationally, and it motivates both motivates the the choice choice of of plane trees for our examples examples in in §7. §7. Before Before this, this, we we will will show show how to use permutations to pairs; itit is possible to represent represent BelyT Belyi pairs; possible to do in terms terms of of branched branched coverings, coverings, without reference reference to to this directly in combinatorial structures, conceptually it is is perhaps perhaps easiest easiest combinatorial structures, but hut conceptually to understand the the connection connection in terms terms of of bipartite bipartite maps. maps. §4. Belyi pairs §4. BelyT pairsand and permutations permutations

Let (X, [3)be be aa Belyi BelyTpair, pair,and andlet letBB be he the the bipartite map Let (XJ3) map l (3~ (Bi) on on X X described described in §3.2. §3.2. The The N N sheets sheets of of the the covering covering [3 : X —* —>E £can canhebeidentified identified with withthe theset setE Eofofedges edgesofofthe the 1 bipartite graph one edge edge lying lying on on each each sheet. sheet. The graph Q = —/3_1(J), f3~ (I), one

Monodromy Monodromy and Galois Galois groups

37

positive lifts, via positive orientation orientation of of £ lifts, via /?, to an an orientation orientation of of the the surface X, and and this this induces induces aa cyclic cyclic ordering ordering of of the edges edges around surface each vertex of Q. Each edge e £e E is incident with a unique black vertex in /31(O) /3~l(0) and aa unique unique white white vertex vertexin in/3_1(1), /?~1(1), and so the cyclic orderings cyclic orderingsaround around the the black black and and white white vertices vertices of of Q c form and gi g\ of of E. E. These These the disjoint cycles of of aa pair pair of of permutations permutations go and permutations describe describe how how the sheets sheets are are permuted permuted by by using using /3 j3 to to permutations lift rotations in in £ around 0 and 1, 1, or or equivalently equivalently how the edges edges lift around their their incident incident black black and and of Q are permuted by rotations around vertices; the cycle-lengths cycle-lengths of therefore the white vertices; of go go and and g\ g1 are are therefore valencies of these vertices, forming the partitions valencies of these vertices, forming the nv,\ -\ nv,i.0 + + \-n Vikv of N associated with the critical critical values values vv = 0 and 11 in in §3.1. §3.1. Let Let C denote gi) generated gogo and G denote the the subgroup subgroup (go, (go->gi) generatedbyby and g\ininthethe E symmetric group of all all permutations permutations of of E\ E; since c is group S = SN SN of since Q connected, G is is transitive. We connected, G We call call G G the the monodromy monodromy group group of (X, /3), (A", /?), or or of of 8, B, since since itit isis the the monodromy monodromy group group of of the the branched branched covering covering /3: /3 X : X—÷ —>• S.We Wewill willshow showinin§5 §5that thatevery everytwo-generator two-generator transitive permutation permutation group group arises arises in in this thisway way from from some some Belyl pair.

The following following result is well-known, well-known, and is is easily easily proved: proved:

Lemma. Let /3) and (X',fl (X', f) be Belyr Let(X, (X,/3) Belyi pairs, bipartite pairs, with with bipartite maps B 8 and gi) and maps and 8', B', and andwith withmonodromy monodromy groups groups C G ==(go, (go^gi) &nd f C' G = = (g'0,g[) in SN. SN- Then Then the thefollowing following are areequivalent: equivalent: f /3) and (X', /3') a) the the Belyr Belyi pairs (X, (Ar,/3) (X',/3 ) are are equivalent; equivalent; 1 b) the bipartite bipartite maps maps 13 B and 8' B are areisomorphic; isomorphic; f c) the tie pairs pairs (go, (g0, ggi) and (gb, (g Q, g[)are areconjugate conjugateininSN, SN,that thatis,is,some some x) and l for ii = 0,1. 0, 1. gE G SN satisfies gig = g\ for satisfiesg~g'gjg

Notice that (c) that C Notice that (c) implies implies that G and and G' Gf are areconjugate conjugate in in SN, so in in particular so particular they are are isomorphic. isomorphic. Our Our definition definition of G may may seem to to give give undue undue prominence prominence to to the critical values 0 and 11 of seem /3, whereas Belyi's Belyl's Theorem Theorem gives givesequal equal status status to oo. In fact, /?, whereas oo. In fact, 1 the permutation of the element element g^ := (ôî)" G G describes describes the e C sheets induced induced by by aa rotation rotation around sheets around oc, oo, each each cycle cycle of length nn corresponding more correspondingtoto aa 2/7-gonal 2n-gonal face faceofofB. B. One One can can obtain aa more symmetric representation of Belyi Belyl pairs pairs in in terms terms of of permutations

38 38


by using using the the triangulations triangulations TI ininplace by placeofofthe thebipartite bipartitemaps maps1.3. B. to be be permuted permuted now now consists consists of of the positive positive triangles triangles of The set to T, and the permutations permutations gv gv are obtained obtained by using using the orientaorienta'T, to rotate rotate these these triangles triangles around around their their vertices vertices labelled labelled tion of X X to weidentify identifyeach eachpositive positive 0,11 and and oo, oo, so so that gogigoo==1.1-IfIfwe vv = 0, with its its unique unique edge edge labelled labelled 01, 01, then then go goand and g\ are areidenidentriangle with tified with the permutations of E defined earlier, so (• G°,g G°\g s—+ \-t gfg'and and a a EGG, anan isomorphism a bijectionEE—+ —>EU, E e, e H-> suchthat that(eg)' {eg)' e'g'forforallalle eE £E Eand and bijection e' e' such = =e'g' a equivalently, one E with {{ 1,. 1 , . . .. ,, N }} gJ GEGG;; equivalently, one can can identify identify E and EU a so that G and G are conjugate subgroups of SN. We will also G and GU are conjugate subgroups SN. We will so TVassociated associatedwith withthe thecritical criticalvalues values show that partitions of of N show that the partitions — 0,0,1 and oo oo are are invariant invariant under under G, G,sosothe thecorresponding correspondinggengen1 and v= a erators gv and g° of G and G have the same cycle-structures of C and G° have the same cycle-structures therefore conjugate conjugate in in SN SN for each v. However, However, they need and are therefore be simultaneously simultaneously conjugate, conjugate, by the same same conjugating conjugating perpernot be CU, there need mutation, so that although although G G = G*, not necessarily need necessarily carryingeach eachg,, gvtoto g%.Thus Thus be an isornorphism isomorphism C G —» G°carrying (X,(X,/3) /3) a and (.Y ,/3^) this possibility possibility and (XU, /30)need neednot notbe be equivalent, equivalent,and and itit is this which allows act non-trivially non-trivially (and (and indeed indeed faithfully) faithfully) on on which allows GG to act the equivalence equivalence classes classes of Belyl Belyi pairs. pairs. .

permutation group associated There is another closely-related closely-related permutation with aa BelyT /3), namely its cartographic group C. C. This Belyi pair (X, (X,/?), cartographic group This depends on a method depends on method for for representing representing maps maps by by permutations permutations which was developed developed in the the 1970s 1970s by by Malgoire Malgoire and and Voisin Voisin [MV] [MV] and by [JS1], though though the basic basic idea idea goes goes by Jones Jones and and Singerman Singerman [JS1], back back at least least as as far far as as Hamilton Hamilton [Ham]. [Ham]. The cartographic cartographic group group is rather more more general general than the the monodromy monodromy group, since since it can be defined forany anyoriented orientedmap mapA4, M, whether whetherbipartite bipartite or or not. not. If If defined for the embedded embedded graph Q has N edges, then the set fi permuted by

Monodromy Monodromy and Galois Galois groups

39

of the 2N darts C consists consists of darts (or (or directed directed edges) edges) of of M; M; we wedefine define n C = (ro,ri) (r o ,ri) < 5 , where where rr0 uses the the orientation orientation of of XX to rotate 0 uses darts around the the vertex vertex to to which which they they point, point, while while r1 ri is is the involution which reverses (so r2 T2 :=(rori)' (ôn)" 1 reverses the direction of each dart (so rotates darts around connected, C C acts acts tranrotates around faces). faces). Since Since Q is connected, Indeed, C can be sitively on ft. Indeed, be identified identified with the monodromy monodromy W(AA)formed formed by by inserting inserting aa new new vergroup of of the the bipartite bipartite map W(M) sothat thatdarts dartsofofM A4correspond correspond tex of valency 2 in each edge of Q, so of W(AA), W(M), with to edges edges of with r0 r0 and and r1 r\ corresponding corresponding to rotations of edges around old and and new new vertices. vertices. (If (If we we regard regardM. M as a edges around the old hypermap then W(Ai) is its Walsh map, described in §3.3.) hypermáp then W(M) is its Walsh map, described in §3.3.) Any BelyTpair Belyl pair (X, (X,,@) determinesaabipartite bipartite map B — = (5~l(B\)\ /?) determines as oriented map, cartographic group (ro,ri) ri) < as an an oriented map, BB has has a cartographic group CC == (ro, Sn, which we also group of of (X, (X,/3). /?). also define define to to be the cartographic group shows that (C, ft), like the monodromy group group Our main theorem shows that (C, (G, £), invariant under G. Unlike Unlike C, G, which which may be primitive, (C, E), is invariant C is is always always imprimitive, preserving aa non-trivial non-trivial equivalence equivalence rerelation two darts darts of of BB are areequivalent equivalent if they point to ververlation on on ft: ft two of the the same colour, colour, so so that that r0 tices of ro preserves the two two equivalence equivalence of T NVblack black and andNTVwhite whitedartswhile darts whiler1r\transtransclasses ft0 and fti of poses them. them. Thus poses Thus C C acts acts imprimitively imprimitively on Il, ft, with with two two blocks blocks NVblack ftx of T black and and NTVwhite whitedarts, darts,so soCChas hasa anormal normal ft lb0 and lbi 1 subgroup D = (ro, (ro,^ ) of index preserves the colours, colours, subgroup D= index 2 which preserves while the elements of C \ D transpose them. One easily checks while the elements of them. One easily checks 1 r0 and and TQ permute the the black black darts dartsinin11o ft0 as go go and g1 g\ perthat r0 underlying edges, edges, whereas and mute their underlying whereas they they act on ft 1 as g\ gi and respectively act on E; thus thus DDacts actsasasGGon onboth bothblocks, blocks,these these go respectively This two actions actions differing differing only only by by aa transposition transposition of generators. This gives an 00}. surface, with automornon-compact, simply-connected simply-connected Riemann surface, phism group Aut(U) Aut(W) = PSL2(R) PSL2(R) consisting of the Möbius consisting of Mobius transformations transformations z

az + b cz + + dd

where a, 6, b, c,c,dd G E R and where and ad ad——bebe== 1.1. The Themodular modulargroup groupisis the subgroup PSL2(Z) consisting of those transformations subgroup F = PSL {7i) consisting of those 2 with a, 6, c, d E£ ZZand andadad——bcbe==1,1,and andthe theprincipal principalcongrucongrua, b, ence subgroup subgroup F(2) F(2) of of level ence level 22 is the normal normal subgroup subgroup of of index index 6 in for which which aa and and d are in FF consisting consisting of those elements elements for are odd odd while and c are are even even (this (this is the kernel of the reduction while 6b and kernel of reduction of F mod in, n) is the subgroup of mod (2)). AA triangle triangle group group A = A(/,ra,n) subgroup of Aut(U) 2ir/m and Aut(ZY) generated generated by rotations rotations through through angles angles 2ir/1, 27r//,27r/m and 2n/n about about the thevertices vertices of of a hyperbolic triangle with internal an2ir/n 7r/rn and ir/n, 7r//, n/rn 7r/n, where where 1, /, m gles ir/1, in and and nn are are integers integers greater greater than 1 1 (such (such triangles ++ra" 1). triangles exist exist in inUUififand andonly onlyifif Z" 1_i m11 + n~l W/A * S in (b), in (c), or

X x

^ UJM -» w/r(2) * Es

in (d). (d). Here Here the arrows arrows represent represent the natural projections, projections, inin-

L < FF and duced by the inclusions inclusions K < A, L F(2), while while the and M •E,S,and andbybythe theF(2)-invariant F(2)-invariantlambdalambdamodular function function J J: U: U function A : :UU—+—>• S.In In each thethe degree of of /3 is equal to to thethe eachcase, case, degree is equal relevant subgroup. subgroup. index of the relevant

The natural action The action of of G G on on Belyl Belyi pairs pairs now now induces induces actions actions of (b), (c) (c) and of G on on the the various various sets sets of of subgroups subgroups inin (b), and (d), (d), or or more on conjugacy of such more precisely, precisely, on conjugacy classes classes of such subgroups, subgroups, since since conjugacy conjugacy corresponds corresponds to equivalence equivalence of Belyl Belyi pairs. One can use this version of Belyl's Belyi's Theorem to to show show how how pairs of permutations give pairs. The universal give rise to BelyT BelyT pairs. universal covering covering space of E0 1, oo} oo} isis the the upper upper half-plane U, with the {0,1, half-plane £/, the Eo = E \\ {O, projection U —>• S ogiven givenby bythe thelambda-function lambda-function A. The Thegroup group of covering covering transformations transformations of A isisthe theprincipal principalcongruence congruencesubsubgroup group F(2) of F, a free group of rank 2 generated by the the Möbius Mobius transformations z—2 and T1:zF-+93. To: and Ti:z

^^27TT

î^3-

(One can therefore identify F(2) F(2) with with the fundamental therefore identify fundamental group of which is freely generated by the homotopy freely homotopy classes classes of of loops in So,

42

Gareth Jones Jones and Manfred Manfred Streit

C which which wind wind once around 00 and and 11respectively respectively in in the the positive positive direction.) If permutations go, go^gi transitive subgroup subgroup direction.) If gi generate a transitive C G of of SN, SNJ then thenthere thereisisananepimorphism epimorphism1'(2) F(2)—+ —>G,G,T1T'—+ E Sinduced inducedbybyA,A,then then j3) isis aaBelyT Belyi pair with monodromy monodromy group groupG. G. (X, /3) Conversely, given given aa Belyl Conversely, Belyi pair (X, (X, /3) (3) one one can can obtain obtain its itsmonmonodrorny group group G = (go^gi) by taking m g1 to be the perodromy go and g\ to per(go,gi) mutations of the cosets cosets of F(2) induced induced by by the the generators generators of M M in F(2) To be identified identified T0and and T\. T1. Similarly, Similarly,the the cartographic cartographic group group C can be with the action action of of the the congruence congruence subgroup subgroup F0(2) Fo(2) on the cosets cosets of This group group Fo(2), sometimes sometimes called GrothendieckJs carcalled Grothendieck's of M. M. This tographic group, even; it group, consists consists of of those those elements elements of of FF with c even; has index 3 in F, it contains contains F(2) F(2) as as aa normal normal subgroup subgroup of of index index of orders orders oo cc and 2, and it is is aa free free product product of of two two cyclic cyclic groups groups of 2, whose generators z

T0o :: zzi.—* =T f/U0 H> ——— 0 = —2z+1

and

z—1 , 2z—1

U1 Ui ::zzi—+ *-+

fixing 00 and and 1, 1, induce induce the the permutations permutations ro r0 and and nr1 of of the the darts. fixing In this action, action, F(2) F(2) acts acts asasthe thecolour-preserving colour-preserving subgroup subgroup D D of its generators generators T0 To and and T1 7\ (which (which are conjugate conjugate of index index 2 in C, its 1 under U1) U\) inducing r0 and TQ . inducing the the permutations r0 The various various combinatorial combinatorial structures associated associated with with aaBelyT Belyi pair (X, (X, /3) f3) in §3 obtained as quotients quotients of of similar similar §3 can can now now be be obtained structures For example, example, the structures on on UU or or U. U. For the triangulation triangulation TT of of X X has form T/L, where L subgroup of (c), and and has the the form 'T/L, where L is the subgroup of FF in (c), T is the the universal universal triangulation triangulation of thishas hasvertex-set vertex-setQQ= = of U: U: this Q U {oo}, {oo},with with vertices vertices a/b a/b and andc/d c/d(in (inreduced reducedform) form)joined joined by by aa hyperbolic geodesic geodesic (euclidean semicircle or half-line) half-line) if and only ad—bc if ad — be==+1, ±1,sosothat thatAutY Aut T==F.F.Similarly, Similarly,the thebipartite bipartitemap map B where M B has has the form form B/M B/M where M is as in (d) and B is the universal universal map: again this lies on but ininthis thiscase casethe thevertices vertices bipartite map: on ZY, U, but are the elements a/b E€ Q with b6 odd, coloured black odd, coloured black or white as a is is even even or odd; odd; as as before, before, the the condition condition for for an an edge edge between between

Monodromy and Galois Monodromy Galois groups

43

a/b and = ±1, ±1, so the automorphism group of and c/d is is that that ad—bc ad —be— B (preserving (preserving vertex-colours) vertex-colours) is F(2).

§6. Plane trees and Shabat §6. Plane Shabat polynomials polynomials In this section, we will will consider consider the the simplest simplest situation in which one can find find aa faithful faithful action action of ofG. G. The simplest Riemann surface surface A' is is the the Riemann Riemann sphere sphere E, S, the theunique uniqueRiemann Riemannsurface surfaceof ofgenus genus X automorphism group group 0. This has automorphism g = 0.

Aut(S) = PSL2(C) PSL2(C)

= {T : z »->• —

cz + d

| a,6,c,d G C, ad—bc=1}. ad — be — 1 }.

The meromorphic functions rational functions functions The meromorphic functionson on SE are are the rational P(z)/Q(z), P(z)/Q(z), where where P and and Q Q are polynomials with complex complex coefcoefficients. Perhaps the simplest simplest rational rational functions functions are are the thepolynopolynomials themselves, which which can can be be characterised characterised as the meromorphic meromorphic functions with functions with a unique unique pole, pole, located located at at oo. oo. A polynomial polynomial P has oo critical value Belyi oc as as a critical value (unless (unlessdeg(P) deg(P) /2l)/9 in cases A and and B, B, and and a = = (—1 ±iy/7)/4 in cases C and D; in each each case, in cases case, the the non-trivial non-trivial automorphism automorphism of of the field of definition to an an definition extends extends (in uncountably uncountably many ways) to element &re conjugate conjugate to each other, as as are areCC element of of G, G, so A and B are and D. Thus G G has hastwo two orbits orbits of of length length 22 on on these these four four trees. trees. and D. One algebraicallythat that there there are no other plane One can confirm confirm algebraically plane trees of this this type. The trees Theelements elements go go e€S7 Srwith with cycle-structure cycle-structure ao == 4,2, 4,2,1 areall alleven, even,and andform form aasingle singleconjugacy conjugacy class class in inA7 Ar 1 are (consisting of its elements (consisting of elements of order order 4); 4); similarly, similarly, the theelements elements g1 with cycle-structure ,8 = 2,2, 1, 1 form a single class X X S induces a chain of field extensions —*—>• induces

M(X) D M(X) M(X) DDM(E) M(S)=- C0) - C(t) C(t)

(1) (1)

(of degrees degrees equal equal to to those those of of the the corresponding (of corresponding coverings). coverings). Since (3 isisaaregular regular covering, covering, M(X) A4(X) isisaanormal normalextension extension of of M(E), .M(E), /3 with Galois group G1 Gal (M(X)/M(E)) = G: the action Gi = Gal(M(X)/M(Z)) action of of G on on X A"asasthe thegroup groupofofcovering coveringtransformations transformations of of J3induces inducesanan C meromorphic functions, Under action on meromorphie functions, with with fixed-field fixed-fieldA4(E). M(E). Under Galois correspondence, correspondence, the correthe Galois the intermediate intermediate field fieldAi(X) M(X) corresponds to the subgroup Hi = Gal (A4(X)/A4(X)) of index iVin in sponds H1 (M(X)/M(X)) of index N Gi which There are areNTV coverings —>E,E,allall G1 which fixes fixes itit point-wise. There coverings X X—÷ conjugate under which have minimal regular lifting lifting /?, conjugate under G C to /?, which V distinct distinct monomorphisms monomorphisms from from M(X) A4(X) and these correspond correspond to to TN into M(X), fil(X), all allconjugate conjugate under under C1. G\. In In this thisaction action of of G1 Gi the stabiliser monomorphism induced induced by by /3 /3 is H1, H\, so biliser of of the monomorphism so there is aa permutation-isomorphism (G,1?) il) = (G1, (Gi,fii), where fti is the set set permutation-isornorphism (C, of cosets of H1 of Hi in C1. Gi.

BelyT's Theoremimplies impliesthat thatthe thesurfaces surfacesXXand andX, X, and and the Belyi's Theorem coverings /3 (3 and and /3, /?,are are all all defined defined over over Q, Q, so sothe thefield-extensions field-extensions in (1) can be be obtained obtained from from aachain chain of of field-extensions field-extensions D MQ(X) D

Mqp) ==4($) W) * Q(P'(Q) P*(Q)(strictly (strictlyspeaking, speaking,this this is the to the of the restriction restriction of of $ to the subset subset C Cof ofQ-rational Q-rational points points of X, A", where where X isisembedded embedded in in Pm(C)). P m (C)). By Byabuse abuseofoflanguage language 2 we identifyCC with with its its plane plane model model C CC we may identify C P2(Q), P (Q), which which always exists, is defined defined by byan anequation equation always exists, and and which which (like (like X) X) is f(x,y,z) = 0. Here / is in Q[x, y, z], z], f(x, is a homogeneous polynomial in Q[x, y, z) = 0. f y, is given by [x, and 33 is [x, y, y, z] F-+X/Z y-+ x/zwhere wherey/z y/z a primitiveelement element is isa primitive of Q(x/z,y/z) Q(x/z, y/z) == Mj^X)/Q0) = = Q(x/z). Q(x/z). Any automorphism automorphism a a EG GGsends Any sends f/totoa ahomogeneous homogeneouspolynopolynoa mial Q[x,y, y, z],by z], byacting actingon on its itscoefficients. coefficients. ItItthus thussends sends C C E Q[x, mialf r G a a to aa curve together with C P2(Q) defined by fa(x,y,z) = curve C P (Q) defined by f (x,y,z) 0, 0, with a covering covering fi° : Ca —+ —>•P1(Q), P J (Q), [x,y,z] [x,y,z] i—+ h->-x/z. x/z.This Thiscorresponds corresponds to an anextension-field extension-field :

D A4Q(E) = =

^

since this action of G preserves intermediate fields and and their Galois groups we get aa chain chain of of extensions extensions Galois groups and

D M^xy 2 MQ(£) = (3) M^{xy D = Q(/n = Q(0, (3)

where the outer extension where extension is is normal, normal, with with Galois Galois group group G3 G3 = C2; CJ2\ in fact, fact, since since G G commutes commutes with the theactions actions of ofthese these two

Gareth Jones Jones and Manfred Manfred Streit Streit

60

Galois groups, groups, a induces a permutation-isomorphism (G3, Galois {Gz,Sl-$) = (G'2 ,n 2 ) where ft3 is the set set of of cosets cosets of H3 H3 = Gal (A4 (G2, in C3. ) in G3.

By extending in (3) extending the constant-fields constant-fields in (3) from from Q to C C (again (again preserving fields and and their Galois groups), we get get a preserving intermediate fields chain of field-extensions M(X)a

M(X)° D M(E) = C(/T) S C(t). C(t). D M(Xy D M(E)

(4)

before, the the outer extension is normal with Galois As before, Galois group C4 G4 = G3, G the set set of of cosets cosets of of H4 H4 = Gal Gal (M(Xy/M(Xy) 3, and if Q4 is the in C4 (G4, ft4) = (C3, (G3, ft3). The finite extensions extensionsof ofC(/3U) C((3G) ^ G4 then (C4, C(t) in C(£) in (4) (4) correspond correspond to toaachain chainofofcoverings coverings X* -> I a ^ E

of X —>- E; the of the same same degrees degrees as as the theoriginal original chain chainXX—p -> X outer covering —+ ES isisthe covering /3 a : Xa —> theminimal minimalregular regularlifting liftingofof a the second unbranched outside —÷ E, and /J^ is unbranched second covering covering /3 : X° -> {0,1, {0,1, 1, oo}, oo}, so soitit isis aa Belyl Belyi function. function. By imitating imitating {0, 1, oo}^ = {0, the arguments one can can show showthat that the monodromy arguments we we used for /3, /?, one monodromy :

:

group of /3a is permutation-isomorphic to the action action of of the Galois Galois group so the isomorphisms ft1) = (C3, group C4 G4 on on flj, so isomorphisms (C4, (Gîli) ((^3,^3) = y f (G,E) now that it is (G 2,n2) ^ (G1j11) (G i,fii) ^ (G,1) (G,O) ^ (G,E) now show show that is permutation-isomorphic to (3. to the themonodromy monodromygroup groupof ofj3.

Invariance of au tomorphism groups Invariance automorphism

A Belyl E determines Belyi function function /3 fi :: X X —> S determines aa bipartite map map 1 r # = f3~ (Bi) on X. A . The The(orientation-preserving) (orientation-preserving) automorphism automorphism group Aut(B) of B group of of permutations group Aut(B) of B is the group permutations of of the the darts of B B which which commute commutewith with the the permutations permutations ro,ri r0, r1 and and rr2. 2 . Grouptheoretically, this centraliser of of the the cartographic cartographic group group theoretically, this is is the centraliser C= = (ro, (r o ,ri,r2) r1, r2) in S2N, S2N, and and topologically, topologically, it is the the group group of of covcovSimilarly, subgroup ering transformations ering transformations of of q oo /3 /3 :XX —> E. E. Similarly, thethe subgroup Auto(B) of Aut(B) Aut(#) preserving preserving the two two vertex-colours vertex-colours is the cencenAuto(B) traliser of the monodroiny monodromy group STV, or or equivalently equivalently the traliser group G C in SN, group By the Theorem, group of of covering covering transformations transformations of of /3. /?. By Theorem, G G

Monodromy and Galois groups Monodromy

61 61

preserves the the conjugacy conjugacyclasses classesofofCC in in S2N 82N and and of of G C in preserves in SN; Sjq\ since conjugate conjugate subgroups subgroups have conjugate conjugate centralisers, itit follows since follows that both under G G (up to that both Aut(B) Aut(#) and and Auto(B) Auto(#) are invariant invariant under to permutation-isomorphism). By contrast, the full full automorphism automorphism group group (including (including orientaorientation-reversing elements) elements) and and its colout-preserving subgroup need tion-reversing §7, for for instance, instance, FigFignot be invariant under G. G. InInExample Example55ofof§7, showsthat that in the G-orbit of ure 9 shows of length length 3, 3, one one tree has has aa Klein Klein four-group of automorphisms automorphisms (generated by four-group of by two two reflections), reflections), while the have only Similarly in the the while the other other two trees have only rotations. rotations. Similarly orbit orbit of length length 6, two two of of the trees trees admit admit reflections, reflections, while while the the four dQ do not. not. This This is analogous analogous to situation in in clasclasother four to the situation sical Galois Galois theory, theory, where where one one cannot distinguish sical distinguish algebraically algebraically between real and imaginary roots of between real of an an irreducible irreducible polynomial. polynomial. Invariance Invariance of of branching-partitions branching-partitions

At each of the possible critical values values vv == 0, and oo, each of possible critical 0,11 and 00, a Belyl Belyi function fi : X —÷ —> Edetermines determinesa apartition partition nv,i +

+h nv%kv == NN

(5) (5)

of of /? /3 at at v; of its degree degree N, AT, which which describes describes the branching branching of v\ here here k,, is the number of points in the fibre above v, and n,,,j is kv is the number of points in fibre above v, nVyj the number of sheets sheets of of the the covering coveringwhich whichmeet meetatatthe the j-th j-th point number of l E /3'(v). (Thereisisnonocanonical canonicalordering ordering for for these these points, points, v.j G [3~ (v).(There unordered partition.) We We need need to to show show so we we regard regard (5) (5) as as an unordered that the the action action of of G G on on Belyl Belyi pairs preserves preserves this this partition. partition. For notational critical notational simplicity, simplicity,we wewill willrestrict restrict attention attention to to the critical value vv == 0, writing writing (5) in the form value form

fll+"+TlkN,

ni + ---+n* = JV,

(6)

where k == k0 and n,j == noj; the thearguments arguments at at vv==11and andvv==00 00 fc0and are identical, except for an obvious obvious change change of of local local coordinates. coordinates. Since full [Mat], we will sketch the Since full details details can can be be found found in [Mat], will just just sketch proof.

Within the meromorphic meromorphic function function field M(E), .M (£),there thereis isa sub— a sub(the valuation ring K = — 7£o(S) (the valuation ring ring at 0), 0), consisting consisting of of those those

Gareth Jones Jones and Manfred Manfred Streit

62 62

meromorphicfunctions functions/ f which which are are finite finite (have (have no no pole) pole) at 0. meromorphic 0. Within this valWithin this ring, ring, there there isis aa maximal maximalideal idealXI = 2o(S) 10(E) (the valuation place, at at 0), 0),consisting consisting of of those those meromorphic meromorphic uation ideal, or place, functions / with a zero at 0; this is the kernel of the evaluation functions f with a zero at 0; this is the kernel of the evaluation epimorphism C, f/ i—+»->f(0), /(0), and and each each r-th r-th power power ir Xrofof epiinorphism 7£ — —*> C, I consists consists of with a zero zero of 0. As of those those /f with of order order atat least least rr at 0. we proof of induces an anembedding embeddingof of we saw saw in in the the proof of the Theorem, (3 induces M(Z) in the field where we we identify identify each with field M(X), M(X), where each /f EG M(E) M(E) /? G A4(X). Underthis thisembedding, embedding, the the ideal ideal II decomposes decomposes M(X). Under /3 oo / E product as a product

f

k

j=1

where Xj is is the valuation in M(X) valuation ideal ideal in M(X) corresponding corresponding to to v3. Vj. (This simply says saysthat that aa zero zeroofof/ I G E M(E) M(E) at (This simply at 00gives gives rise rise to aa zero at each each t>j,the theorder orderof ofthis thiszero zerobeing beingmultiplied multiplied zero of of f3 /3 o /f at by rij] in words, we we find find local local coordinates coordinates around v3 Vj by in other words, roots.) Just taking 7ij-th roots.) Just as as in in the theproof proof of of the the Theorem, Theorem, this decomposition decomposition is obtained by extension of of constant-fields from aa similar decomposition decomposition

11)723 where XQ and (XJ)Q are are the the valuation valuation ideals ideals in A^g-(S) and .MQ:(A') at 00 and and Vj. (Notice Vj is a Q-rational point of (Notice that that v3 point of X since since v3 G ^ ( O ) , /? Belyi function G Q.) The N /3 isis aa Belyl function and and 00 E e //3'(O), Q.) The action of G preserves preserves these these decompositions decompositions of valuation ideals, ideals, action and in particular particular preserves preserves the multiset multiset of of exponents exponents nj, so the unordered partition partition (6) is invariant invariant under under G, as required. unordered required.

This This result result means means that G G preserves preserves the the types types of of the the various various combinatorial combinatorial structures on which which it acts; for for instance, instance, valencies valencies of vertices vertices are are preserved. preserved. It also means means that, that, as of as we we have assumed throughout this this paper, of a surface throughout paper, G preserves preserves the genus genus of surface X, since this can be expressed in terms of the invariant since this can expressed in terms of invariant quantities an k0, ofof §3.§3. fco,k1, &iik00 fcooand d NN by byequation equation(1)(1)

Monodrorny Monodromy and Galois Galois groups groups

63

References References

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[JS1] G. G. A. A. JONES JONES and and D. SINGERMAN, 'Theory of of maps maps on [JS1] SINGERMAN, 'Theory orientable surfaces', surfaces', Proc. Proc. London Math. Soc. orientable London Math. Soc. (3) 37 (1978), (1978), 273—307. 273-307. [JS2] G. G. A. A. JONES [JS2] JONES and D. D. SINGERMAN, SINGERMAN, 'Maps, 'Maps, hypermaps hypermaps and triangle groups', and groups', in in The The Grothendieck Grothendieck Theory Theory of of Dessins Dessins d'Enfants L. Schneps), Math. Soc. d'Enfants (ed. (ed. L. Schneps), London London Math. Soc. Lecture Lecture Note Note 115—145. Ser. 200 (1994), (1994), pp. 115-145. G. A. JONES and D. [JS3] D. SINGERMAN, SINGERMAN, 'Belyl 'Belyi functions, functions, hy[JS3] G. permaps and Galois groups', Bull. London London Math. Math. Soc, Soc., to to appear. [MV]J.J. MALGOIRE MALGOIRE and and C. VOISIN, [MV] VOISIN, 'Cartes cellulaires', cellulaires', CaCahiers Mathématiques Mathematiques de de Montpellier Montpellier 12 12 (1977). (1977). [Mat] B. B. H. MATZAT, MATZAT, Konstruktive Konstruktive Galoistheorie, Lecture [Mat] Notes in Math. Math. 1284, 1284, Springer-Verlag, Springer-Verlag, Berlin, Berlin, 1987. 1987. [Neu}J.J. NEUKIRCH, NEUKIRCH, 'Uber 'Uber die die absoluten absoluten Galoisgruppen [Neu] Galoisgruppen algebraischer Zahlkörper', Zahlkorper',Astérisque Asterisque41—42 41-42(1977), (1977),67—79. 67-79. [Rey]E. E. REYSSAT, REYSSAT, Quelques Quelques Aspects Aspects des Surfaces Surfaces de Riemann, [Rey] Riemann, Birkhäuser, Boston / Basel // Berlin, 1989. Birkhauser, Boston [RY]G. G. RINGEL RINGEL and and J. J. W. T. [RY] T. YOUNGS, YOUNGS, 'Das 'Das Geschlecht Geschlecht des. des vollstãndigendreifarbbaren dreifärbbarenGraphen', Graphen', Comm. Comm. Math. Math. Helv. vollstandigen Helv. 45 45 (1970), 152—158. (1970), 152-158. [Schi] L. L. SCHNEPS, SCHNEPS, 'Dessins 'Dessins d'enfants d'enfants on on the the Riemann Riemann sphere', [Schl] in The Grothendieck Grothendieck Theory Theory of of Dessins Dessins d'Enfants d'Enfants (ed (ed Schneps), Schneps), London London Math. Soc. Soc. Lecture LectureNote NoteSer. Ser.200 200(1994), (1994),pp. pp.47—77. 47-77. L. SCHNEPS SCHNEPS (ed.), The [Sch2] L. Tie Grothendieck Grothendieck Theory Theory of of Dessins d'Enfants, London d'Enfants, London Math. Soc. Soc. Lecture Lecture Note Note Ser. Ser. 200 200 (1994). (1994). [Ser]J-P. J-P. SERRE, SERRE, Topics Topicsin in Galois Galois Theory, Theory,Jones Jones and and Bartlett, Bartlett, [Ser] Boston / London, 1992. Boston [S\J] G. G. B. B. SHABAT SHABAT and and V. A. VOEVODSKY, VOEVODSKY, 'Drawing curves [SV] over number number fields', fields', in in Grothendieck Festschrift III III over Grothendieck Festschrift (eds.P. Cartier et at.), Progress in Math. 88, (eds.P. et al.), Progress in 88, Birkhäuser, Birkhauser, Basel, Basel, 1990, pp. pp.199—227. 199-227. [SZ]G. G. SHABAT SHABATand andA. A. ZVONKIN, ZVONKIN,'Plane 'Plane trees trees and and algebraic [SZ] (eds. H. Barcelo and numbers', in Jerusalem Gombinatorics Combinatorics '93 (eds.

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Permutation techniques techniques for coset coset representations representations of modular subgroups subgroups Tim Hsu

§1. Introduction recall that that dessins d'enfants may We recall may be beconsidered considered as asone oneof of the thefollowing following equivalent classes classes of objects. Let Let G be equivalent be an an appropriate appropriate "cartographical" "cartographical" dessins, GG = Fo(2) for pregroup. (We may take take G G = F(2) for group. (We may for general general dessins, preclean dessins, dessins,and and G G = PSL2 forpre-clean pre-clean dessins dessins whose whose valencies valencies at clean PSL2 (7L) (Z) for the "other" "other" type type of of vertex vertex divide divide 3.) 3.) 1. 1.

2. 3. 4.

Coverings ofthe the Riemann Riemann sphere, sphere, branched branched (in a manner Coverings of manner consistent consistent 1, with G) G) only onlyabove above{O, {0,1,00}, along with aa choice choice of of (unramifled) (unramified) basepoint. Drawings (structured (structured in in a manner consistent Drawings consistent with G) on on connected connected oriented surfaces. surfaces. Subgroups of of G. G. Transitive permutation representations of Transitive permutation representations of G, along with a choice choice of basepoint.

among these classes of objects objects have been explained The correspondences correspondences among classes of many times, times, so we refer refer the reader to many to Birch Birch [2], [2], Jones Jones and and Singerman Singerman [5], [5], for details, details, mentioning mentioning only only one one piece piece of of terminology: terminology: In and Schneps Schneps [7] [7] for describing basepointed describing the the correspondence correspondence between between (3) (3) and (4), we call the basepointed transitive permutation representation of G corresponding to subgroup transitive permutation representation of corresponding to a subgroup G\ thecoset cosetrepresentation representationof ofG1 G\ as asaasubgroup subgroupofofC.G. G1 < et par le stabilisateur stabilisateur H de l'un ses drapeaux. drapeaux. L'action v, droite) de G G sur sur les les drapeaux drapeaux est est alors alors equivalente equivalente aa celle celle de sur (a droite) de G sur G/H. Deplus plus (G, (G,H) H) est estlelequotient quotient du dudessin dessin régulier regulier (G, (G, 1) par le groupe G/H. De d'automorphismes H...cf. H...cL [Z]. [Z]. Dans d'automorphismes Dansnotre notrecas, cas,posons posonsHH==< =< a >= Z/2Z. Nous rappelons que le le N-ième TV-ieme polynôme polynome de Tchebychev Tchebychev TN Tpj est défini defini par par cos(N0). la relation TN(cos(O)) TN(cos(0)) = cos(NO).

Lemme. Une 7L/2Z) est Lemme. Unepaire pairededeBelyi Belyiassociée associeeau audessin dessin(SN, (Aw,Z/2Z) est donnée donnee par par (P',TN) oii ouTN T/v est estleleN-ième N-iemepoiynôme polynomede deTchebychev. Tchebychev.

Demonstration. On a le Demonstration. le "diagramme" "diagramme" commutatif commutatif

Z/2Z)

LXN

I

I

Z/2Z

(1) (1)-

Z/27Lest estlalaprojection projectionnaturelle. naturelle. D'un point ou AJV —+ —>- Z/2Z point de devue vue"graphique" "graphique" obtient (dans la figure on a indiqué cas N N = 5) on obtient indique le cas 5)

p Finalement, vu que tous ces ces dessins ont genre 0, on arrive au diagramme diagramme commutatif p1

TN TN! p1 P1

4P2

p1

F FNj N p1

Tout d'abord, FN Tout Fjsradmet admetcomme commemodèle modeleFN(Z) Fjy(Z)==ZN ZN(en (enayant ayantimpose impose FN(±1) FJV(ILI) == ±1, ±1,ce cequi quin'est n'est pas pas une une limitation...cf[Sh]). limit at ion... cf [Sh]). Pi pi est est un un revêtement revetement double de On double de la la sphere sphere de de Riemann Riemann ayant ayant deux deux points points critiques critiques(Z (Z == ±1). On

Dessins d'enfants en genre Dessins d'enfants genre 11

85 85

deduit p2(Z) = — \(Z + —). Par Ia la commutativité commutativite déduit facilement facilementque quep\{Z) pi(Z) = p2(Z) ZJ du diagramme du diagramme on on obtient obtient

=

p(Z) = Re(Z) Re(Z) = cos(O) si Z Z = e16, eie', p(Z) cos(0) et TN(cos(O)) TN(cos(0)) = cos(NO). cos(N0). En particulier, particulier, si TN est done le N—ième polynôme de Tchebychev. 0 T/v le N—ieme polynome Tchebychev.

1.3. 1.3. Modèles Modeles pour les les éléments elements de de FN. TN* Revenons finalement aux aux dessins dessins presentes présentés au §1. § 1.Soit SoitDDk E 1N TN etetposons posons Revenons finalement k G on en déduit Du fait fait que que 00 < < k < ^, on deduit que que Xk = Ak = Re(etn'N) = = cos(Tr^). cos(lr*). Du < 1. 1. On peut alors alors considérer considerer la la courbe courbe elliptique elliptique réelle reelle définie definie par Afc < o0 < Ak 2 2 — 1)(X C -l)(X-X — Ak)k) Ck k:Y ==(X2(X

Si TT(X, 7r(X, Y) Y) — = XXest estlelerevêtement revetement canonique canonique de deCk, Ck, alors alors (par (parconstruction) construction) fc (Ck, est une une paire paire de de Belyi, Belyi, ou oü f1k ir. De De plus le (C = (—l) T/v7r. ledessin dessin k,fk)1k)est k = correspondant est un un élément element de de .FN TN (en (en effet effet .FN T^ est une une classe classe de de valence valence [2]N[4]1 Ton voit voit immédiaternent immediatement que et l'on que les les partitions partitions induites sont bien [2] ^[4]* et et 1 [2n]1). La dynamique dynamique du revetement revêtement ir [2n] ). La TT montre montrefacilement facilement que quececedessin dessinest est a ainsi ainsi le bien D bien Dk. k. On a — 1)(X Lemme. Considérons courbe = (X (X22 — Considerons lala courbeelliptique elliptique Ck Ck : Y2 = 1)(X — — Ak) Xk) k (_1)kTNlr Soit de de plus pins ffk = (—l) TN7r ou oil n ir est le le oil ou kk E G DN T>N et etAk Xk = = cos(ir*). COS(TT^). Soit k =

revêtement canonique canonique de de la Ia courbe. courbe. Alors Alors la paire (Ck, revetement (Ck, f1k) Belyi et le le k) est de Belyi dessin DkEG.FN. TN • dessin correspondant correspondant est est Dk

Remarque. Remarque.SiSileledessin dessinest estdonné donne par par (G, (G,H), H), alors alors [G [G :H] H) == 2N 2N etet AN est quotient de G. G.

1.4. Action du groupe 1.4. groupe de de Galois. Galois. Le groupe II' = d'une façon groupe F = Gal(Q/Q) opère opere d'une fagon naturelle naturelle sur les paires de Belyi: (C, (C, f) / ) '—* i—> (C°', (C*a,1°)• f a ) . Pour Pourles lesdessins dessinsde deFN, TN,nous nousvenons venonsde depresenter presenter modele qui permet de décrire decrire completement Nous pouvons pouvons un modèle complètement cette action. Nous finalement demontrer démontrer la

Proposition. La famille (classe d(N) —— 1 1 (classede devalence) valence)FN FNsesedecompose decomposeenen d(N) orbites sous sons Vaction l'action de IT. IF. Soit Soit D eG .FN J~N etetposons posonsc(D) c(D) =— ^N\ oü ounn est estlele nombre d'aretes d'arêtes sur stir une des boucles. Alorsc(D) c(D) est un invariant galoisien nombre boucles. Alors

Leonardo Zapponi Leonardo

86

absolu (i.e. (i.e. D et Si J et D' D' sont sontconjugnés conjugues si si et et seulement seulement si c(D) == c(D')). c(D')). Si est l'invariant de la Ia courbe courbe associée D, alors I'invariant modulaire modulaire de associee aa D, alors Q(J)/Q est estune une extension abélienne abelienne de de degré degre ^(f) Demonstration. Dans fk), le revêtement fk est Demonstration. Dans le le modèle modele (Ck, (Cfc, fk), le revetement est défini defini sur Q. Q. L'action de IF se réduit alors a celle sur la courbe, et donc a l'action sur Ak. F se reduit celle sur la courbe, done Faction sur A&. Soit QN QN le le corps corps (reel) (reel) obtenu obtenu en en rajoutant rajoutant Ak Q (pour Soit A& aa Q (pour k EGDN). V^). Si st Q((N )/Q eest l'extension cyclotomique cyclotomique de de degre degré JV, N, alors, QN QN est une sousQ(Gv)/Q l'extension extension de Q(C2Ar) Q((2N) (en effet effet A Ak extension & est la partie partie réelle reelle d'une d'une racine racine2N-ième 2AT-iemede de l'unité) . En particulier particulier Ak l'unite) A& etet Am Am sont sont conjugués conjugues si si et et seulement seulement s'il s'ilexiste existe de Q(C2iv)/Q Q((2N)/Q transformant e"1""^ en el7r^. Ceci revient un autornorphisme automorphisme de p.g.c.d de a et de N) = (fe, (k, N) (oii a dire que (m, (ra, N) (ou (a, (a, b) 6) indique le p.g.c.d de b). b). alors l'entier l'entier c(D) Pour DD = Dk E€ FN Pour J~N définissons definissons alors c(D) = = sjJ*N\ -OnOnpeut peut remarquer k,N) c(D) = ,J*N, oü ou ddest estlelenombre nombre N) etetc(D) remarquer que que (k,N) (k, N) = (N — — k, = d'aretes sur une une quelconque quelconque des des deux deux boucles. boucles. d'arêtes sur Deux dessins dessins D D et D' sont c(D'). Deux sont donc done conjugués conjugues si si et seulement seulement si si c(D) c(D) = c(D'). absolu. On en En d'autres termes, termes, cc est est un un invariant invariant galoisien galoisien absolu. en déduit deduit imimse médiatement que mediatement que .FN TN se decompose en d(N) d(N) — — 11orbites orbites sous sousl'action l'actiondedeIFIT (ii (il faut éliminer eliminer le cas c = 1). Intéressons-nous finalementaa l'invariant l'invariant modulaire: modulaire: Pour Pour Dk on obtient Interessons-nous finalement .

J= .

J

2 (l-cos(2 (1 — 7 rA))

racine Si D eG FN, estesune TN, lelecorps corps Q(J) Q(J)est estcontenu contenudans dans Q(Cc(D))(oui ( OU CC(D) ^ une racine (c(D) primitive c(D)-ième l'unité). De c(D)-ieme de l'unite). Deplus, plus,l'automorphisme 1'automorphisme o•a qui transforme transforme galoisien, Cc(D)ene n C m est estleleseul seulqui quifixeJ. fixe J.On Onenendéduit deduitque queQ(J)/Q Q(J)/Qestest galoisien, de groupe groupe de de Galois Galois(Z/c(I?)Z)* (Z/c(D)Z) / ±± 1. 1. En En particuher particulier

[Q(J) : Q} =

(c(D))

Si est est la fonction indicatrice d'Euler. Pour le cas c(D) = 2= 2 la fonction indicatrice d'Euler. Pour le cas c(D) si c(D) ^ 2 (ici (ici

l'invariant l'invariant est rationnel). rationnel).

0

L'invariant c(D) c(D) mesure mesure done donc lala "complexite" "complexité"arithmetique arithmétiquede deJJ = J(D). L'invariant J(D). On peut en particulier demander quand quand J(D) J(D) est particulier se demander est rationnel. rationnel. Le Le premier premier cas c(D) = 22 et et done doncJJ = 1728 cas correspond correspond aa c(D) 1728 (on remarquer (on aurait aurait pu remarquer que c(D) = 22 implique que c(D) le même meme sur sur les les deux deux implique que que le le nombre nombre d'aretes d'arêtes est le boucles. boucles. Dans Dans ce cas le le dessin dessin admet admet un un automorphisme automorphisme d'ordre d'ordre quatre quatre qui fixe fixe un un point. point. Or courbe elliptique elliptique admettant admettant un tel Or iiil existe existe une seule seule courbe isomorphisme: 1728). En général, general, ceci revient aa résoudre resoudre qS(c(D)) <j>(c(D)) isomorphisme:JJ = 1728). = 2= 2 done c(D) et done c(D) £e {3,4,6}. Pour c(D) = 33 on on obtient obtient JJ = ^ 2 , pour c(D) c(D) = 4, J = 10976 10976 et enfin enfin pour 6, J = 54000. 4, pour c(D) c(D) = 6,

Dessins d'enfants d'enfants en genre 11 Dessins

87

Remarque. Remarque.Tous Tousces cesdessins dessinsont onttous tousune uneseule seuleface. face. En Enfait, fait,les lesdessins dessins propres ayant ay ant une une seule seule face face pour lesquels lesquels tous tous les les sommets sommets ont ont valence valence >> 11 sont de deux types: ceux ceux décrits decrits plus haut et et ceux ceux induisant induisant les les partitions partitions 2 N 3 1 [3]2[2]N3 Pv = Pv = [3] [2] ~ et et PF PF == [2N11. [2N] . Cette Cette distinction distinctionest estliée lieeaa la la decomposition decomposition cellulaire de l'espace surfaces de Riemann en en genre genre 11avec avec cellulaire l'espace des modules des surfaces un point marqué marque (cf (cf [ZI). [Z]). Les dessins dessins que nous avons étudiés etudies sont tous tous les les dessins genre 1 ayant une seule face dessins en genre face tels la courbe associée associee admette une une equation du type e3)e3on e1,e2,e3 E IR. equation du type V2 Y2 = = (X — — ej)(X e\){X ——e2)(X e2)(X — — ) ou ei,e 2 ,e3 e RLe Le cas general général est est bien bien plus plus delicat. délicat. Dans la deuxième cas deuxieme partie de l'article, Particle, nous nous allons etudier étudier des des dessins dessins qui, qui, tout tout en allons en ayant ayant deux deux faces, faces, sont sont intimement intimement [Z]). lies a ce problèrne probleme (cf. (cf. [Z]).

1.5. 1.5. Exemples. Exemples. Nous allons allons terminer terminer cette section en donnant quelques Nous quelques exemples exemples numérinumeriques: -N — = 5.5. T*> contient deux dessins et possède uneune seule orbite sous l'action -N contient deux dessins et possede seule orbite sous Faction —1 1==1). 1 ).Les Lesdeux deuxdessins dessins seront seront donc doneconjugués conjugues etetlelerésultat resultat de II' T (d(5) (d(5)— est présenté presente dans la la figure figure suivante. suivante.

J=

71224 + 26664A/5

J =

71224 - 26664V5

-N = 10. -N 10. Dans ce ce cas cas .T10 T\o contient contient 55 dessins. dessins. d(10) d(10) = 44 et donc done on on retrouve 3 orbites. Tout d'abord d'abord on aale dessin D$ D5(C(D) (C(D) = 2), Tout le dessin 2), qui, pour des raisons de symétrie symetrie 1728. correspondka JJ = 1728. correspond

Pour c(D) c(D) — = 55on on obtient obtient ^0(5) == 22dessins dessins conjugués. conjugues. Ce Cesont sontles lesmêmes memes Pour courbes que que dans dans le le cas cas NN = 5. courbes Finalement, les deux derneirs dessins dessins vérifient verifient c(D) c(D) = 10. Leurs Leurs invariants invariants appartiennent a une appartiennent une extension extension quadratique de de Q Q (figure). (figure).

88

Leonardo Zapponi

J = 211688 + 9 2 1 6 8 ^

JJ = 211688 -— 92168\/5;

en genre genre 1, points de torsion et formes §2. Dessins Dessins en 1, points formes modulaires. modulaires. Nous allons a present étudier valence pour laquelle etudier une une deuxième deuxieme classe de valence les dessins dessins sont sont en en quelque quelque sorte sorte une une generalisation généralisation de ceux decrits décrits precedemprécédemment - on rajoute une une "boucle". "boucle".Ici Iciililest estbien bienplus plusdifficile difficile de donner une une sosolution paramétrique; parametrique; on on pourra pourra néanmoins neanmoins introduire introduire un uninvariant invariantgaloisien galoisien calculable de utilisant des des méthodes methodes analytiques analytiques calculable de fagon façon combinatoire. combinatoire. En utilisant (en passant passant par le le revêtement revetement universel universel de la courbe associée associee au dessin) dessin) on donnera donnera ensuite ensuite une caractérisation caracterisation "modulaire" "modulaire" de ces ces dessins. dessins. Dans Dans son son Pakovitch [P1 [P] présente presente des resultats analogues, analogues, en utilisant utilisant des des article, F. Pakovitch des résultats methodes differentes. méthodes différentes.

2.1. Description combinatoire. Soit N Soit N > > 22 un entier. La Lafamille famille EN £/v est est formée formee par par les les dessins dessins propres propres ayant 2N 2N drapeaux (N sont propres) propres) et induisant ayant (N arêtes, aretes, vu vu que que les les dessins dessins sont les partitions les N 3 = [2] [2]N_3[6]1 Pv = - [6}1 et PF PF = [N]2 [N]2

En d'autres termes, termes, D EG £N sommet de valence valence 6, tous les autres autres EN a un sommet sommets de valence 2 et deux faces faces de de valence valenceN. N. Ici aussi 8N £N est une classe valence. Ces dessins dessins sont sont obtenus obtenus en en prenant le dessin regulier régulier de groupe de valence. G= = Z/6Z Z/6Z muni de la presentation vv == 1,1, 1/ = 3, 3, f/ = = 22 et en lui "rajoutant" "rajoutant" des sommets de valence valence 2 (figure). (figure).

Comme Comme précédemment, precedemment, tous ces ces dessins dessins possèdent possedent une une involution involution avec avec faces. L'ordre cyclique des drapeaux quatre points points fixes fixes qui permute les deux faces. sortant du sommet sortant sommet de de valence valence 6 induit induit un ordre ordre des des "boucles" "boucles" du du dessin. dessin. es est alors déterminé un triplet d'entiers positifs (N1, N2, N3) D D eG £N t alors determine par un triplet d'entiers positifs (Ni,N2,Ns) EN

Dessins d'enfants en Dessins d'enfants en genre genre 11

89 89

(avec N1 + N3 Ni + N2 iV2 4N3 =—N) N)modulo moduloune unepermutation permutationcyclique cycliquedes desindices indices (Ni d'arêtes sur est le nombre d'aretes sur la la i-ième 2-ieme boucle). boucle). On pourrait étudier TN'- Soit Soit en etudier eN £^ en en suivant suivant les les méthodes methodes utilisées utilisees pour .FN: effet E CN. L'involution fixelelesommet sommetde devalence valence66du du dessin. effet DN1 DNUN2,N2 ,N3 ,N3 € £N> L'involution a afixe Ni est alors ci a fixe sommet central de la la boucle boucle correspondante; correspondante; Si N2 est pair alors fixe le sommet s'il s'il est impair, impair, c'est l'arête l'arete centrale centrale qui qui est est flxée. fixee. Le Ledessin dessin quotient quotient que que l'on arbreayant ayantun unseul seulsommet sommetde devalence valence 3. 3. IiIIest esttoujours toujours Ton obtient est un un arbre pre-propre (propre si N1, N\,N2 pré-propre N2 et et iV N33 sont sont pairs).

Les paires de de Belyi associéesaa ces ces arbres arbres sont sont du du type (P^p) p) ôu Les paires Belyi associees ou p est est un polynôme de degre degréNN ayant ayant deux deux valeurs valeurs critiques critiquessur surC. C. p est appelé un polynome de appele polynome de de Tchebychev Tchebychevgénéralisé generalise[Sh}. [Sh]. Supposons Supposons qu'on polynôme qu'on connait connaIt p, p, et considérons les points points ei,e2,C3 e1, e2, e3relatifs relatifsaux aux"extremites" "extrémités"de de 1'arbre, l'arbre, et e, considerons les e, le le valence 3. Soit Soit finalement C Ia la courbe elliptique elliptique définie definie par sommet de valence

C:Y2= (X - e)(X - ex)(X - e2)(X - e3) C : Y2 =(X—e)(X—ei)(X—e2)(X—e3) (C,poir) est alors et ir(X, X le revêtement n(X,Y)Y) = = X revetement canonique. canonique. (C,pon) alors une une paire paire (avecdes desmodifications modificationselementaires élémentaires ilil est Belyi associée associee aa DNUN,N2 de Belyi (avec 2,N3 ,N3 possible modele non singulier de C). possible de de donner donner un modèle Obtenir des polynômes généralisésn'est n'est pas pas simple Obtenir polynomes de Tchebychev Tchebychev generalises simple et en en general definis sur numerique on renvoie général ils us ne sont sont pas définis sur Q (pour un exemple numérique la [S], p. p. 75). C'est C'est en jouant une autre autre carcarLalectrice lectriceou ou le le lecteur lecteur a [S], jouant sur une acteristique suivant) actéristique de ces ces dessins dessins (qui (qui sera sera introduite dans le paragraphe suivant) que que nous nous pourrons pourrons obtenir obtenir des resultats résultats plus tangibles.

2.2. 2.2. Dessins orientables. définir le le concept concept de de dessin dessin orientable, orientable, qui Avant de continuer nous allons definir permet de simplifier les calculs. calculs. En En effet, effet, pour pour un tel dessin, permet simplifier les dessin, l'application l'application Belyi est est la composition de Belyi composition de deux revêtements. revetements. Un dessin dessin oriente orienté est simplement simplement un dessin pour lequel chaque arête arete aa été ete orientée. Evidemment, nous pouvons orientee. pouvons choisir choisir 2 n orientations différentes differentes (oii (ou n est est le le nombre nombre d'arêtes). d'aretes). Une Une orientation orientation est est dite diteadmissible admissiblesisi toutes toutes les les aretes une même meme face face sont sont orientées orientees de de façon fagon cohérente. coherente. On On arêtes appartenant aa une demontre démontre facilement facilement que que sisi elle elleexiste, existe,une une telle telle orientation orientation est est unique unique (au signe pres). près).

90


Un dessin dessin est est orientable s'il existe Un existe une une orientation orientation admissible. admissible. Ce Ce concept concept est equivalent equivalent aa celui celui de de dessin dessin "bicoloriable", "bicoloriable". v, 1,f f > et B Soit dessin de groupe groupe cartographique cartographique G B < G le Soit D un dessin G =< v,l, resp. /f)) est stabilisateur d'un drapeau stabilisateur d'un drapeau (notation: (notation: vv (resp. (resp. /,1, resp. est l'élément Pelement de resp. aux G correspondant correspondant aux aux sommets sommets (resp. (resp. aux arêtes, aretes, resp. aux faces)). faces)). On On demontre le critère critere suivant suivant [Z]: [Z]: démontre alors le

Critère. Un Critere. Undessin dessinest estorientable orientablesisietetsettlement seulement s'il s'ilexiste existe un un homomorhomomorphisme ^ Or Z/2Z G tel que Or(v) Or(v) = Or(l) = —1, -1,

Or(f) B C C—* —^» PÎalafonctionelliptique fonction elliptique Si z+ = correspondante (o'à correspondante (ou (3 == p/3o). pfio). Si z^ — -^ +4- -^r est est lelezero zero de de /3Q dans dans lele 13o

1

parallélogrammefondamental, fondainental, alors parallelogramme

/I — _ Z+)N +\N I

i3(z) — 0

—

Demonstration. Dans la suite, suite, lele parallélogramme parallelogramme fondamental fondamental sera sera celui celui déterminé par par les les points points 0,1,r 0,1 ,retet 141 +r.r, En ayant suppose sommet de determine suppose que le sommet valence 66 corresponde corresponde aa l'element l'élément neutre neutre de C, on en déduit 1. valence deduit que que /?o(0) = 1. D'apres la proposition précédente, precedente, le le diviseur diviseur de de /?o est donné donne par par div(/30) div(/?o) = — N[—z+]. plus, = a += br. le théorème on on A^[z+] — N[—z"1"].DeDe plus, 2Nz~*~ a 4-En br.appliquant En appliquant le theoreme obtient 13o(z)=c z+ — o(z + + z+)"- 1 a(z + z+ - aa — - bT) br) ~ C a(z oh c E C est est une une constante. constante. Or o{z a(z+z+ —a—br) = oiicGC + z+-a-br) a(z+z+)e-2z(am+br)2)+d oh d est une r, a, b etet N. N. On ou une constante constante dépendant dependant de deij1, rji,rj2,T,a,b On arrive arrive donc done a l'expression +\N — Z j

/3(z)

— —

Finalement,(3Q(0) /3o(0)= =11 =» c Finalement,

I z+\N +\N a(_

= (-l)ce —eed =

= 11 car oa est est impaire. =

N On en deduit déduit immédiatement (—l)"e" immediatement que c = (—l) e~d et le le lemme est demontre.O

Avant Avant de continuer continuer nous allons montrer le le lien lien entre entre /3o et la la fonction fonction de Ch.6): hauteur locale locale de de Néron Neron (cf. (cf. [Si] Ch.6):

Corollaire. Corollaire. log log/3o(z)l \/30(z)\ = N N [)t(z [X(z +4- z+) - A(z X(z —- z+)] ou X est est la lafonction fonction de hauteur locale localede deNéron. Neron. Demonstration. L'expression présentée dans le lemme precedent précédent peut L'expression de de /3o /?o presentee peut être réécrite etre reecrite comme comme

=

N

[a(z cr(z

+ z+)

oh ij ou 7]est estlalafonction fonctiondedequasi-période quasi-periode([S] ([S]p.p.465). 465).EnEneffet effet 2z(ar]i+ + 67/2)= = 2Nz (i)) == 2Nz 2Nz ((*( K (I) + ^C 2Nz (^7,(1) + ^ , ( r ) ) = 2Nzij2Nzrj(^). *( (§)) Or, i(z + z+)7?(z + z+) -— \{z -— z+)r)(z -— z+) = zr,{z+) + z+V(z) (par R-linéarité) et Z+TJ(Z) -— ZJ](Z+) = il ii avec avec /I G ER R-linearite) R ([Si] ([Si] prop. 3.1, 3.1,p.p. 465)=* l

-{z + z+)n(z + + z+) — - l-{z -— z+)V(z -— z+) = 2zr}{z+) + il U=>

Leonardo Zapponi Zapponi

94

=

TN N

o(z — — a(z

e_21

+

Finalement, une expression pour la Ia fonction fonction de de hauteur locale de Néron expression pour Neron est

donnée donnee par ([Si] ([Si] Th. 3.2, 3.2, p.466) p.466) X(z) = —log -log log

= N [.A(z +

—

A(z

0

—

Corollaire. Soit Soit (C/A, (C/A, /?) une paire de Belyi Belyi associée associee a un dessin dessin de de EN telle que le sommet sommet de valence 6 corresponde correspondean aupoint pointzz==0.0.Alors Alors

T\[-i,

1]) + z+)} {z E i]) = {* e cC I| x(z -— z+) == A(z x(z +

Demonstration. Immediate en utilisant le le corollaire corollaire précédent. precedent.

0

Ce dernier dernier corollaire corollaire permet permet de de donner donner une une description description en en termes termes de de la Ce fonction de hauteur locale de Néron K immergé fonction Neron du complexe complexe K immerge dans la courbe courbe elliptique, determinant la structure comme une elliptique, determinant structure de dedessin. dessin. En voyant voyant A comme "distance", KK est est alors alors l'ensemble l'ensembledes despoints points"equidistants" "équidistants"dedez^z+etetz~\ z. "distance", Nous allons allons a present present exprimer l'invariant Nous Tinvariant rip de fagon combinatoire: SN, alors Proposition. Soit D = DN1,N2,NS i?^ E EN, 2N

(N1+N2,N2+N3,N3+N1) Demonstration. Choisissons admissible dede DN1 ,N2 ,N3 (parmi -DATI,^,^ Demonstration. Choisissonsune uneorientation orientation admissible les deux possibles). les possibles). Les Les "boucles" "boucles" deviennent deviennent alors des des chemins chemins fermés fermes comme dans dans la Ia figure figure qui qui suit suit ) tels que orientes 71,72 (numerotes comme 72 et 73 (numérotés orientés 'yi, 1, TTI(C) iri(C) est commutatif). 7i + 72 72 + 73 73 = 0 (en genre genre 1, commutatif).

De plus plusix1 7Ti(C) =< = < 71,72 Ton peut peut choisir choisir une valeur valeur de T r G telle E C telle 'yr, 72 > et l'on

que dans le le parallelogramme parallelogramme fondamental fondamental 71 et 72 72 soient respectivement

Dessins d'enfants d'enfants en genre Dessins genre 11

95

1] (les homotopes a Ji(t) = tr et 72(£) — 1I — —t,t, avec avec rrEE[0,[0,1] (lesdeux deuxcôtés). cotes). homotopes Dans ce ce cas pour 73 on obtient 573(t) Dans 73(t) = rr + t(1 £(1— —r)t r)t (la (ladiagonale diagonalenene passant pas par l'origine). de plus plus que que pour pour ce ce choix choixde derr on on ait ait passant l'origine). Supposons Supposons de r+ +— — _o_ 1 b __s_ -i2NT 2N "•" 2N ' ' * ~ 2N

dX méromorphe uw = Considérons sur P^ le différentiel Considerons differentiel meromorphe = ^ - T T et le le chemin chemin 1 exit -y(t) = 1]. Posons [0,1]. Posonsenengénéral general jn(t) = e™*, * 6 [0, n].-y7peut peutêtre etre 7(t) = e™ , t, tEe[0, releve relevé via /30 de trois façons fagons différentes differentes sur C en choisissant choisissant l'origine Torigine comme comme On peut alors relever 7 ^ point base. Supposons Supposons de de relever relever 'y 7 le long de 7y1. On et son son extrémité extremite est est ~ (point fixe de "boucle" par l'automorphisme Tautomorphisme du (point fixe de la "boucle" dessin). alors dessin). On a alors

= exp exp

(ai]i

= exp

(I + bij2

= —exp -exp ( ^ ± ^ r

N

= r— - 2ij2z 2*z+) == —exp -exp

)

+ N

r—

a + br N

=—ein N — z+) 712 OnOn obtient donc car cr(| + + z+) z+) = = e27?2Z cr(| — z+)etetij1r ?/ir—— 772= iir. = ^TT. obtient done

= in-^

=i7T mod(27ri) =»aa = NiV++N1 iVi mod(2N). mod(2AT). iir ++ i7r^- mod(2iri)

De fagon façon tout a fait mod(2N). L'ordre fait analogue, analogue, bb = N ++ N2 N2 mod(2AT). L'ordre de z + est alors égal + N1, NT+-fN2, egal a ^ oonu d == (N (iV 4JVi, A 7V22N) ,2iV)etetonondémontre demontrefacilement facilement que(N+N1,N+N2,2N)=(Ni+N2,N2+N3,N3+Ni). que (N + NUN + N2,2N) = (Nx + 7V2, AT2 + 7V3, AT3 -f AT^. 0

Remarque. Dans Remarque. Danslalademonstration demonstrationonondonne donneexplicitement explicitement les les valeurs valeurs de combinatoires. Le point de torsion est a et et bb en en fonction fonction des données donnees combinatoires. est donc done uniquement determine, déterminé, en ayant choisi une orientation admissible. uniquement admissible.

2.5. 2.5. Formes Formes modulaires. modulaires.

2.5.1. 2.5.1. Inversion du problème. probleme. Soit C = C/AT 7i. Considérons C/A r une courbe elliptique avec r E %. Considerons 0 < a, a, b
= Z/nZ}. resultat 0(n) = 2 classiqueaffirme affirmeque quelaIacardinalite cardinalité Ro(n) égale classique dede Ro(n) estest egale a na n2 Yipfli(l \n(^ —— ^ ) * une action action a droite droite de F(l) F(1) sur Ro(n) On définit definit une Ro(n) en posant (a,b)M = = (a',l/) (a',b') (a,b)«

on ou

= tM

(b)

avec M E matrice transposee transposée de M). Posons £ F(1) F(l) (ici (ici ltM M indique indique la matrice Posons

H(a, b) b) == Stabr(l)(a, b) H(a, b) b) est est un unsous-groupe sous-groupe de de congruence. congruence. En Eneffet effet on on vérifie verifie immédiatement immediatement l'inclusionF(n) I'(n)Cc H(a,b) H(a,b) (oà —+ SL2(Z/n7L))). rinclusion (ou F(n) = ker(I'(I) ker(F(l) -+ SL 2 (Z/n Pour bb= on aaH(a, H(a,0)= F1(n) ou on Pour = 0 on 0) = H(1,0) H(l, 0) = Fx(n)

modn} mod nl H(a, par un élément H(a,b)b) est alors alors le conjugué conjugue de F1(m) Fi(n) par element de ['(1) F(l) (en (eneffet effet l'action est est transitive). transitive). Faction Considérons Considerons sur Ro(n) la la relation relation d'équivalence d'equivalence (a, (a,b) b) ~ (—a, (—a, —b). —b). ['(1) sur R(n) R(n) = F(l) opère opere sur = Ro(n)/ (Faction est bien bien définie) definie) et et nous indiquons Ro(n)/ ~ (l'action par [a, [a, b] d'équivalencedede(a,(a, b] =={(a, {(a,b); 6);(—a, (—a, —b)} Ia la classe classe d'equivalence b).b). Trivialement, H[a, H[a,b] Stabr^([a,b])b]) =< —<Stab(a, Stab(a,b), -I >. >. Tout Tout ce cequi quiaa Trivialement, b] — = Stabr(l)([a, b), —I ete dit pour H(a,b) peut se transposer a H[a,b). En particulier c'est été pour H(a, b) se transposer a H[a, b}. En particulier c'est un sous-groupe < F(n), —/ >C if [a,b]. 6]. Comme Comme avant, avant, sous-groupe de de congruence congruenceet et F[n] ['[n] ==< I'(m), —I >c H[a, pour o naaH[a,0] H[a, 0]==H[1,O] ff[l, 0]=- r1[n] T^n]=< =. Fi(n), -I >. pour & b==00on Soit ['(1) F(l) = = (J2 |J i r1[n}M, Fi[n]Mi une decomposition decomposition en en classes classes a droite du dugroupe groupe De cette decomposition decomposition on Fi(n), donnée donnee modulaire. De modulaire. on en en deduit déduit une pour pour Fi(n), par (r1(n)M, —Fi(n)M2). = (J |J(r U-r 1 (n)M iU 1 (n)M i ).

Posons = [r'(l), r1[n]] = Posons m = — j ] On a alors Fi(n)M2M = = Xi{M)T{l)M(JM(i) ou oii aa e Sm {1, 1} alors V^MiM Sm et Xi(M) €E{-1,1} vérifient verifient

Xi(M'M) =

Xi(M)x.M(i)(M')

Leonardo Zapponi

98 98

Z/27L défini Soit # n : rI'(1) ( l ) —* -> Z/2Z defini par * n ( M ) = HiXi(M). caractère. car act ere. En En effet effet

Alors # n est un

= = [Jxi(M) [JXaM(i)(M')

[J Xi (M) fiX3 (M') =

non trivial Remarque. Si \I>n est non trivial alors alors son son noyau noyau est est un un sous-groupe sous-groupe d'indice distingue) de de F(l). F(1). Or d'indice 2 (donc (done distingue) Or un un tel tel sous-groupe sous-groupe est unique unique et 2 [L] p.359). p.359). Une decomposition en classes classes aa droite égal egal aa H H = F(l) F(1)2 (cf. [L] decomposition en

est donnée par T(l) r(1) = H U HU donnee par HU ou UU = ( * j ). IiII suffira suffira donc done d'étudier d'etudier = l'action Faction de U U pour pour verifier verifier la trivialité triviality de de \I>n. Par exemple, exemple, ^4 est non non trivial. trivial.

2.5.3. Les Les fonctions fonctionsFa,b(T) Fo>6(r) et Fm(r). Fm(r). étudier a fond En reprenant les les notations du du 2.5.1 2.5.1 nous allons etudier fond la condition

(cf. 2.5.2). Posons Soit nn>> 11 et (a,b) Soit (a, b) e Ro(n) (cf. Posons Fa,b(T) =

(n(

+

—

2a(

(i))

—

Pour tous a,b a, 6EGZ,Fa+fl,b(T) Z, F a+n) 5(r)= =Fa,b+n(T) F a j b + n (r)= =Fa,b(T). JF^bW-Fa,b â,bnenedepend dependque quedede l'image de de a et bb dans Z/nZ, 1'image Z/nZ,elle elle est est donc done bien bien définie. definie. écrirons plus plus simplement simplement 7ij1(r) Dans la suite, nous ecrirons 71 (T) == C (2)eteîj2(r) V2(T)== C(i). (§)• ne Proposition. Fa,b(T) forme rnodulaire poids 1 powr pour H(a, b). i7Ia,6(T) e5est ^ ûne forme modulaire de poids H(a,b).

démontrer en Fa^{r) est holomorphe (ce n'est pas pas difficile difficile a demontrer Demonstration. Fa,b(T) utilisant des des critères criteres de de convergence). L'expression L'expression de la fonction (£ présentée presentee au 2.4 permet d'obtenir l'egalite Fegalite

= (Cr +

((Cr + D)z, r)

V

M

=

E

F(1)

Dessins d'enfants en genre Dessins d'enfants genre 11

99 99

En particulier,

(a +

:

= (Cr + D)(

(aD + bB + n

bA ± aC) n

De même, meme,

fç

= (Cr + D)(

=

=

=

+ D)

oh r/ est la fonction ou fonction de quasi-période. quasi-periode. Finalernent, Finalement, en combiiiant combinant ces expressions, expressions, on arrive a

Fa,b(MT)= (CT+D)Fa',b'(T) b'{T) avec avec

(b,) =tM(b)

En accord avec avec Faction l'action de de F(l) r(1) définie écrire (a', b') definie au 2.5.2 on peut donc done ecrire bf) == M (a, b)M. b) ononmontre (a,b) . En En prenant prenant M M eGH(a, H(a,b) montreque queFa,b Fa^ est est une une fonction fonction faiblement modulaire de poids 1, Pour terminer la demonstration faiblement de poids 1. Pour terminer demonstration ii il faut faut étudier etudier le comportement comportement a l'infini, l'infini, question question qui est reportée reportee au au prochain prochain paragraphed

Corollaire. Fm(r) Fm^(r). Alors Fm Fm est Corollaire. Soit SoitmmeG(Z/n7L)* (Z/nZ)* et et posons posons F m (r) = Fm,o(T). poids 11 pour pour Fi(n). une forme modulaire modulaire de poids Ti(n).

2.5.4. 2.5.4. Comportement Comportement a l'infini. l'infini. La fonction fonction theta. fonction 60 est est proche proche (comme (comme nature) nature) de la fonction fonction a. Elle est souvent La fonction utilisee pour des raisons numeriques. effet elle admet un développement developpement utilisée numériques. En effet en serie série a convergence convergencerapide. rapide. Nous Nous allons allons l'introduire l'introduire par l'expression l'expression suivsuivante (en (en produit produit infini): infini): n 27riz 0(z) 9(z, r) = 2cqhin(7rz) J J ((1l —- q e 6(z) = = 0(Z,T) ) f_f f j ( l —- qfle_27flz) qne~2niz) f_f 11>0 n>0

n>0 n>0

c = rin>o(l — —qTh) qn) et^ qq== e2tnT. Grace Grace aace ce produit produit on onpourra pourraétudier etudier le comportement comportement de deF11 Fn a l'infini. l'infini. En utilisant les p.60) on obtient: les propriétés proprietes élémentaires elementaires de 06 (cf. (cf. [C], [C], p.60) avec avec

(1) &(z,r) =


100 100

(3)

0(z —

(z) =

(2)

((z) =

0'(z)

—2iriz

+

relation permet permet d'obtenir La troisièine troisieme relation _i_ h a+br

n«^—) n Ti

a+br

)—

—

Qf ( a+br \

= n - ^ - ++ 2aij1 2aVl + 2bijjr 2bVlr 0

0)

2bi12 = n^£± 26%

+ 2bVlr - 2bm = n ^ ^ +2biri + 26™

car c a r ij1r rjiT — — 772 = =

ri

Fa,b(T) =

+

2b.

in

en produit infini L'expression de de 06 en infini amène amene aa l'egalité Tegalite L'expression

1 0'(z) — iri0(z) 7ri 0(z) — e

+1

—

qm

—

—11

m).O

+ 1— qme2lriz 1 q e

m).O

1

qme_21rzz

Eu En supposant que aa et etb6sont sont des des entiers entiers avec avec a, a,bb < n, n, ini

—

aq*

—

— 1

a —1 q

2

m>O 1

avec a = q*; g n ; la dernière derniere expression expression devient devient a = e 2 7 r l n. Posons Posons pp == e 2 7 r i ^ = alors 1

n— in

1+2b=n apb +— 11

—

- 22nn m>O m>0

1—

lpmn_b

+ 2n m>Q m>0

1—

+ 2b

apmn ^

2 = cCo0 + 4- clp cip + cc2p2 2p H+.....

on Ci €C. E Qab et et donc C2c»EGQab ou C.De Deplus, plus,a a G Qa& done Q a b . F npeut peutdonc doneêtre etreexprimé exprime en série serie de Puiseux: Puiseux:

Fa,b(Y)

forme modulaire et et la démonstrace qui montre que c'est effectivement effectivement une forme demonstration de la la proposition du du 2.5.3 2.5.3 est est terminée. terminee. Plus explicitement, explicitement, pour b6 = 00

F Fm(T) m{r) ==fl n^±l — - 2n £ - ^f c+ + 2n a—i a — 1

^ ^ 1— 1 — ao

k>O k>0

*

= -—in(co + c1q m(c0 + ci# + c2q2 c2g2 H+•••))

k>O fc>0

a_lqIc 1—

Dessins d'enfants d'enfants en genre Dessins genre 11 oà Co

1

=

—1

= cotg

et

Ck

101 101

= 4>.ldIksin

2.5.5. Quelques Quelques proprietes propriétés des des formes formes Fm. Fm.

Dans ce paragraphe nous allons nous limiter au au cas cas bb == 00 (et H(a, b) b) sera donc sous-groupe de de congruence congruence Fi(n)) F1 (n))pour pour etudier étudier certaines certaines proprietes propriétés done le sous-groupe des formes liées en en particulier particulier aux aux formes de Fo(n) formes FFm, formes de Fo(n) avec caractère caractere et m , liees courbes modulaires. modulaires. aux courbes En général, general, si Mk(Fl(fl)) Mk(Ti(n)) désigne designe le le C-espace C-espace vectoriel vectoriel des formes formes modulaires de poids poids 1 pour [K] Prop. Prop. 28, ulaires de pour Fi(n), Fi(n), on onaal'isomorphisme Tisomorphisme (cf. (cf. [K] 28, p. 137)

Mk(I'l(n)) somme directe directe etant étant étendue les caracteres caractères xx :: (Z/nZ)* la somme etendue a tous tous les (Z/riZ)*—* —>• C* etet Mk (n, x) designant désignant l'espace l'espace vectoriel vectoriel des des formes formes modulaires modulaires de de poids poids 1 et Mk(n,x) caractère caractere x pour pour Fo(n). On rappelle rappelle que Fo(n) est est le le sous-groupe sous-groupe de F(1) F(l) la

(a

mod n. On , 1 avec cc = 0 modn. On aa alors alors

formé par les matrices forme par matrices du type type (

Fi(n) en considérant considerant l'homomorphisme rhomomorphisme // : 1'o Fo(n) • (Z/nZ)* defini par par (n) — p(M) d. Pour tout on en en obtient obtient un de Fo(n) /x(M) = of. tout caractère caracterexxde de(7L/n7/4*, (Z/nZ)*, on F 0 (n) en x n posant x(M) = x(°0 == X/ xj4M). Mk(n, x) es est (^)- ^fc( ?x) ^ alors l'ensemble l'ensemble des éléments elements k /f EG Mk(Fl(n)) Mfc(Fi(n)) tels tels que /f(Mr) ( M r ) == x(M)(cr ++d)kf(M) d) f(M) VM VME6 Fo(n). F 0 (n). 2m > C* un caractere, x = X M le caractere induit pour F C' un caractère, le caractère induit pour Fo(n) x: 0 (n)etet = general posons en general

g(r) = >

(m,n)=1 (m,n) = l


102 102

on voit voit que queg(r)I(M)1 g(r)\[M]l = = x(M)g(r) Alors on x{M)g{r)=

c

^

mFdm(r)=

^

cd-imFm(r)

(n,m)=1 (n,m)=l

(n,tn)=1 (n,ra)=l

= Cdlm Vd E (Z/nZ)* = x~1(rn)En posant c1 ci = 11 on obtient obtient finalement finalement Cm cm = : Proposition. Soit Mi(n, x) == Proposition. Soitx:x(7L/n7/4* (Z/nZ)* —> C* un uncaractère. caractere.Alors Alors VnflflAfi(n,x) avec Cgx avec

gx(T) = (n,m)=l, (n,m)=1, 2m, 1 etetconsidérons Soient Soient maintenant Ti[n] =< —< Fi(n),—J >, i ==0,0,1 considerons les courbes modulaires courbes

Xi(n) = = Ti[n\\H (ici (ici M = = IiHUUQ Q U {ioo} {zoo} est estlalacompactification compactification ordinaire ordinaire du dudemi-plan demi-plan supérieur). Xo (n) parametrise paramétrise de façon naturelle les couple couple (£", (E, 5) S) ou E est superieur). Xo(n) fagon naturelle est une courbe courbe elliptique elliptiqueet et 5S est de E d'ordre n. une est un unsous-groupe sous-groupe de n. De De même meme Xi(n) X1 (n) parametrise paramétriseles lescouples couples(E,P) (E, P)ou oii EE est est une unecourbe courbe elliptique elliptique et P eEEEest estun unpoint pointd'ordre d'ordre(exactement) (exactement)n n(voir (voirpar parexemple exemple[K] [K]§5 §5p.153). p.153). La théorie theorie génerale generale des courbes modulaires modulaires affirme affirme alors que Xi(n) une (n) est une courbe projective projective sur sur C (et On indiquera = courbe (et même meme sur Qa&). On indiquera par Ki(n), (n), i = 0, les corps avec le corps des des 0,11 les corps des desfonctions fonctions respéctifs. respectifs. Ki(n) s'identifie s'identifie avec formes modulaires modulaires méromorphes de poids poids 0 pour Ti(n). formes meromorphes de ['i (n) est (n), ce qui qui induit un Ti(n) estun unsous-groupe sous-groupe distingue distingue de de I'o r o (n), unrevêtement revetement galoisien

Xi(n) Xo(n) Xiin)-1* X0(n) /{±1}6. de degré groupe dede Galois isomorphe a a (Z/nZ)* degre \(n)et et groupe Galois isomorphe /{±1} 6 . Le corps K1 (n) est est donc K\{n) done une extension galoisienne galoisienne de Ko(n). Dans ce conFaction du du groupe groupe de de Galois Galois sur une une fonction f/ est est simplement simplement donnée donnee texte, l'action f(r)'—+ \—>f(Mr) f(Mr)pour pourtout toutMM Fo(n). formesmodulaires modulairesFm Fmsont sont par f(r) E6 Fo(n). LesLes formes particulierement intéressantes interessantes car elles elles réalisent realisent de façon fagon pratique cette cette exexparticulièrement tension:

Proposition. E Mi(To(n)) uneune forme Proposition.Soit SoitF F E Mi(Fo(n)) formemodulaire modulairequelconque quelconque(non(nonidentiquementnulle) nulle)de depoids poids11pour pourFo(n). r0(n). Considérons <m m K0(n){frn) = = Ki(n) 0 Ko(n)(fm)

Remarque. Remarque.On Onpeut peutreformuler reformulerlalaproposition propositionprécédente precedente de de facon fagon plus explicite en remplagant remplacant Fm explicite en Fm par F^ et en posant posant F2

F= (m,n)=l, 2m1kO Efc>o «kqk oü ohqq = e2™. D'après D'apres l'expression Texpression de Fa,b(r) Fa,h(r) en série serie de Puiseux (cf. (cf. 2.5.4), 2.5.4), les lescôefficoefficients appartiennent tous cients appartiennent tous aa une une extension extension cyclotomique cyclotomique de Q Q (de (de degré degre n Nous pouvons donc considérer l'action de ). pouvons done considerer Faction Gal(Q /Q) (ou Q = Q(e"^)) (oii n n ). sur sur Fn induite par l'action Factionsur surles lescoefficients. coefficients.


(Z/nZ)* (o Identifions Gal(Q n /Q) avec (Z/nZ)* (a '—÷ i—> x(°) x{a) cyclotomique). On a alors cyclotomique). *F a , 6 (r) = Fx{a)a,b(r)

V Vo•a Ee

105 ou

est x est le X l e caractère caractere

Gal(Q n /Q)

En particulier l'action Faction preserve preserve les les formes formes du type type Fm. Fm. Nous obtenons obtenons une une action sur O n : Nous

0(T)

=

Si Q cnn est est un caractère 7L/27L. Si Gn est caractere Gal(Q n /Q) —* -> Z/2Z. est trivial, trivial, alors alors les les cocoefficientsa,k ak sont efficients sont rationnels, rationnels, sinon sinon iiilexiste existe une uneconstante constante Cn appartenant appartenant a une une extension extension quadratique quadratique de Q telle que Cna,k E GQ Q pour tout k. k. Ceci découle de de Faction l'action "globale" decoule "globale" de de Gal(Qn/Q) sur 0 n . Cn est effectivement effectivement calculable en determinant determinant a0 calculable en > 4) 4)

En développant en série developpant en serie les les deux termes, termes, on on voit voit que queles lescoefficients coefficients de de

peuvent être Pn peuvent etre obtenus obtenus en en resolvant resolvant des des equations equations linéaires lineaires aacoefficients coefficients rationnels (car (car les les coefficients coefficients de de A,J et 0 n sont rationnels). 0

2.5.8. Exemples. Exemples. Etudions le cas cas N N = 3; 3; le plus plus simple. simple, II existe existe un un seul seul dessin dessin et et pour des raisons de symetrie symétrie ilil correspond correspondaa lala courbe courbe d'invariant d'invariant modulaire modulaire JJ = raisons de = 0. D'un point de de vile vue "modulaire" "modulaire" on on retrouve retrouve la la démarche demarche suivante: suivante:


106

N1 = N2 = N3 = 1.1. On aura aura done donc (2,2,2) (2,2,2)== 22 => riD £(33 = {Di,i,i}; {.D 1|U }; Ni N2=N n D = 3. 3. 3 R(3) est compose de 4 elements. éléments. De plus \&3 est trivial et la forme modulaire i?(3) est non non parabolique parabolique (car (car nn est est impair). Par un est un théorème theoreme classique classique d'unicité, d'unicite, on en en deduit déduit immédiatement on immediatement que

e3(r) 9 3 (r) = cg2(r) cg2(r) et l'on retrouve JJ — = 0.0. avec Cc E £ C et Ton retrouve

Dans ce ce cas cas aussi On obtient N N = 4. Dans aussi nous avons avons un seul dessin: dessin: D1,1,2. £>i,i,2- On faut donc étudier 9e88 ((r). n = 88 et ii il faut done etudier r ) . C'est une une forme forme modulaire parabolique parabolique poids 24 24 ayant ayant un zero simple en en zoo. ioo. On On a alors de poids zero simple 6 8 (r) = aA 2 (r)(J(r) - J o ) 011 aa est une constante et Jo Jo est est l'invariant l'invariant modulaire modulairede delalacourbe courbeassociée associee au dessin. dessin. En développant V) on obtient developpant en série serie (Maple (Maple V) 6 8 (r) = 549755813888q(6561 549755813888^(6561 + 4-4358810q 4358810^ + • • •) 2

= aq(1 aA (r)(J(r) -— JJo) aq(l + (696— (696 - Jo)q JQ)q + • • •) o) = identifiant les on determine donc En identifiant les coefficients coefficients on done J0: Jo: 210646 = 6561 6561

Jo =

Remarque. dessin Remarque.LeLe dessinn'ayant n'ayantpas pasdedeconjugué conjuguesous sousl'action Factionde deIF T et et le le point point de torsion etant étant uniquement uniquement déterminé determine (en (en ayant ayant choisi choisi une orientation adad2 6 missible), on en déduit que la courbe elliptique d'invariant Jo = missible), on en deduit que la courbe elliptique d'invariant JQ ^ ^ pospossède un point de torsion d'ordre 88 défini sede un defini au pire sur sur une une extension extension quadraquadra-

tique tique de Q Q (ceci car ilil faut faut considérer considerer l'éventuelle l'eventuelle action action P P '—* \—> —P —Pdede F). r).

N= = 5. Id Ofl avec respectivement respectivement N 5. Ici on retrouve deux dessins dessins D1,2,2 ^1,2,2 eett D3,1 ^3,1,1, avec n = 10 deux dessins dessins ne ne sont sont done donc pas pas conjugués. 10 et n = 5. 5. Ces Ces deux conjugues. Etudions Etudions

Dessins d'enfants en Dessins d'enfants engenre genre11

107 107

de plus près e5 est pres D3,1,1: ^3,1,1: 65 est une forme forme modulaire modulaire non parabolique parabolique de poids 12 et Ton l'on obtient e5(T) — J0) 0 5 (r) = aA(r)(J(r)-Jo)

Comme dans dans l'exemple l'exemple precedent, précédent, en en developpant développant en en serie série on arrive aux Comme arrive aux égalités egalites e5(r) 9 5 (r) = = 5^(243 + + 154480q 154480? + • • •) a(1 ++(720— aA(r)(J(r) —- J0) J o ) = a(l (720 - Jo)q J0)q + • • •) et finalement

20480 20480 243

Dessins d'enfants d'enfants en §3. Dessins en genre 11 et et isogénies. isogenies. Un Unexemple. exemple. Cette dernière derniere partie est estconsacrée consacree aa la laréalisation realisation de de groupes groupes de deGalois Galois (infinis) comme commegroupes groupesd'automorphismes d'automorphismesde de groupes groupes profinis. profinis. Notre Notre but (infinis) décrivant le théorème est celui celui de de fournir fournir des desexemples exemples explicites explicites decrivant theoreme général general (action F2) présenté presente dans dans [I]. [I]. (action de T If' = = Gal(Q/Q) sur F2) 3.1. Réalisation Realisation de Tab = Gal(Qa6/Q). de Fab = Gal(Qab/Q). Ce premier premier paragraphe est equivalent au §1: Ii equivalent au II n'est pas d'un intérêt interet fonfonde mettre mettre en en evidence evidence certaines certaines caractéristiques caracteristiques qui qui damental mais permet de retrouvent dans des des généralisations generalisations ulterieures. F a 5 est bien connu connu et se retrouvent ultérieures. FOb est bien isomorphe aa Z*. Dans ce ce contexte, sa réalisation realisation comme groupe isomorphe z*. Dans groupe d'automord'interet (car (carpour pourlelefaire faire ililn'est n'estréellement reellementpas pasnécesnecesphismes perd un peu d'intérêt Nous allons neanmoins saire de passer passer par les dessins dessins d'enfants). d'enfants). Notis néanmoins decrire décrire une groupes diedraux telle construction construction qui qui met met en jeu les groupes diédraux et les points d'ordre la suite suite ililsera sera question question de de"groupes "groupesdiédraux diedrauxgénéralisés" generalises" fini sur C* (dans Ia de torsion torsion sur sur des des courbes courbes elliptiques). elliptiques). et de points de Nous avons vu dans le le 1.2, 1.2, que que la la paire paire de deBelyi Belyi(P', (P1fTh) , fn) (oü (oh fn = = |(zn + est un modèle au groupe diédral —•)) est modele pour le le dessin dessin régulier regulier associé associe au diedral = I

muni a, /1 = ba ba et et /f = b. b. Nous Nous indiquerons par muni de la presentation presentation vv = = a, indiquerons par le groupe d'automorphismes du revêtement ("deck transformaAut(/ n ) groupe d'automorphismes revetement ("deck transforma-

Leonardo Zapponi

108 108

tions"). tions"). Le Lechoix choix d'un d'un drapeau drapeaude deréférence reference amène amene alors alors aa un un isomorphisme7 isomorphisme7

An - A Aut(/n)

On peut poser ici a Faction Indiquons par parxx: IF de IF sur On a alors IT . alors

I

j

= CX((T)z =

4>{v)(z) = i = #«x)(z)

Finalement, on peut considérer considerer le groupe (profini) (profini)

Aoo = lim A n 2* Z/2Z ix Z

muni de la L, F provenant muni la presentation presentation canonique canonique V, F,L,F provenant de celles celles des des A n . L'action de F T est est alors alorsexplicitée explicitee par par les les relations relations

ivy L

i y yx{°) i__÷LX(a) i—

1F et permet de de caractériser caracteriser les les automorphismes automorphismes de A ^ provenant de l'action l'action 8 de FF 8. Ces Ces relations relations sont sont en en accord théorème enonce énoncé dans dans [I], et accord avec avec le theoreme de réduisent au cas se reduisent cas le le plus plus simple. simple.

3.2. Réalisation Realisationde deGal(Q(i)ab/Q). Gal(Q(i)a6 Nous allons allons aa present present nous nous interesser intéresser aa un un groupe groupe de de Galois Galois non-commutatif non-commutatif dans le but de rendre moins moms "triviale" l'action IT. La construction présenpresenFaction de IF. tee passe par l'étude Tetude des des dessins dessins obtenus obtenus en en composant composant une une application application de de Belyi avec avec une une isogenie. isogénie. Dans le cas present present nous nous limiterons aux aux dessins dessins réguliers en genre 1. reguliers 1.

Soit C la courbe elliptique d'équation d'equation

C:: Y2 == 4X(X2 C 4X(X2 —- 1)1) 7 88

qui induit une structure de de An-revetement. On remarquera que que Fab Tab —* —> Aut(Aoo)est estinjectif. injectif.


109 109

et considérons —* considerons le revêtement revetement f/ C: C —)• P^ défini defini par f(X, f(X, Y) Y) ==X2. X~2. Les points critiques 1 etetCX (C, /f)) est une paire de Belyi Belyi (f (/ critiques de def/sont sont0,0,1 oo done (C, a un zero d'ordre d'ordre 4 dans le point point a 1'infini l'infini de de C, C, un un pole poled'ordre d'ordre 44 en en XX = 0, 0, revêtement est galoisien doubles en en XX = ±1). ±1). Le revetement galoisien et le le et f/ — - 11 a deux zeros doubles groupe est cyclique, isomorpheaa Z/4Z Z/4Z et engendré groupe d'automorphismes d'automorphismes est cyclique, isomorphe engendre par par avec 0(X, 9(X, Y) Y) = (-X, (—X, iY)9. O-.C^C iY)9. La La paire paire (C, (C, f) / ) est estdéfinie definie sur sur Q. Q. 0 C -÷ C avec comme valeur valeur du du module module associe associé aa la courbe determine le Le choix choixrr = i comme Le le revêtement universel C -^» C. C. On obtient alors revetement alors l'isomorphisme Pisomorphisme analytique analytique C C = C/A oü ou A A = ZZ 0 i7L. iL. Dans le parallélogramme parallelogramme fondamental (déterminé (determine points 0, i,i, 1,1 1, 1++z), i),lelezero zerode de/f correspond correspond aa zz — = 0;0;zz== ^ est estlele par les points pOle, les deux deux autres autres points points de de ramification ramification sont sont zz = ^ et z = | . Nous pole, etet les retrouvons done les [2]. Sur retrouvons les points pointsdedeCC[2], Sur C l'automorphisme Tautomorphisme 06 s'exprime s'exprime simplement par 9(z) simplement 6(z) = iz iz et est est done done un unmorphisme morphisme (en (envoyant voyant CCcomme comme groupe abélien). abelien). Considérons present le revêtement Considerons a present revetement non-ramiflé non-ramifie (endomorphisme) (endomorphisme)

I

correspondant a la multiplication alors Sil'on Tonpose pose fn == fn, alors correspondant multiplication par iin eGN.N. Si la paire (C, (C, fn) est est de de Belyi Belyi et et nous nous obtenons obtenons un un nouveau nouveau dessin dessin Dn. Cette Cette méthode est effectivement methode effeetivement utilisable pour tout dessin dessin en genre 1 mais mais dans dans ce ce les choses choses se se simplifient simplifientparticulierement. particulièrement. Notre Notre but but est celui d'étudier cas les d'etudier a fond fond Dn. 2 L'application zeros d'ordre d'ordre 4 correspondant L'application f,-, fn aa nn2 zeros correspondant aux points points de de C[n], C[n], n2 zeros — 1 1possède possede exactement exactement 2n2 2n2 zeros zerosdoubles doubles etet n? zeros d'ordre 4; de plus plus fn — ceux-ci sont sont les les seuls seuls points points de ramification ramification de fn. ceux-ci 2 revêtement est galoisien de degre degré An 4n2. . Son groupe de Galois Galois Aut(/ n ) est Le revetement engendre par par 06 et Tn = {o a(P) = P oü engendre {a G Aut(C) | a{P) = P + P ou Pn EGC{n]}. C[n]}.On On E Aut(C) I + n = a évidemment evidemment Tn = C[n]. C[n). Pour décrire decrire Aut(/ n ) ilii est comode comode de passer par le le revêtement revetement universe!. universel. Nous allons allons identifier identifier C[ri] C[n] avec avec(7L/mZ)2 (Z/nZ) 2 (en (entant tantque quegroupes) groupes)enenconsidconsidérant l'application erant l'application (Z/Z)2 (Z/nZ) 2 —> C/A aa ((a,b) V\ '—*v — x b +i— {a,b) •—> fl—h i— Ti n n Dans cette cette optique, optique, 9(a,b) 9(a,b) = (—b,a). (—6, a). Pour tout tout P EG C[n], C[n}, indiquons indiquons par par ap 0p l'automorphisme de C Pour C défini defini par op(Q) = Q = ae{P) =» Tn < Gn. De De plus plus vp{Q) Q + P. On Onaaalors alors OcrpO1 6aPe~l =

On retrouve ici un des dessins dessins d'enfants d'enfants (le plus simple) présentés au §1. simple) presentes

Leonardo Zapponi

110

>— 1,1, et et Gn est donc done le produit semi-direct semi-direct de 6 > et Tn. On Tnn = aura alors alors (Z/n Aut(/ n ) ^ Gn = (7L/nZ)2

Z/4Z

C[nJ

Aut(C)

Remarque. Remarque.LaLacourbe courbeCCpossède possede des des isogénies isogenies non-triviales non-triviales (l'anneau (Fanneau des des endomorphismes est isomorphe isomorphe aZ[i]). a Z[i]). On peut donc endomorphismes est done appliquer la construction précédente -f ib. ib. precedente pour un un morphisme morphisme quelconque quelconque correspondant aa a = a + Si nous nous ne ne Favons l'avons pas pas fait fait e'est c'est que ces derniers peuvent être obtenus comme etre obtenus comme quotients de ceux décrits plus plus haut. En ceux decrits En effet, effet, si l'on Fon pose n == a2 a2 + b2, b2, alors revêteinent C le revetement C —^ C se factorise en C C —^>CC —^>C.C. Choisissonspour la presentation v = (0,0,1),1 (0,0,1), / = (1,0,2), (1,0,2), /f = (1,0,1) (1,0,1) Choisissons pour Gn Iaprésentationv et un un drapeau drapeau de deréférence reference b0 bo donné donne (sur (sur le le revêtement revetement universel) universel) par le le orienté de gauche gauche a droite (figure "segment" ]0, ]0, ^[[ oriente (figure suivante).

I

fn

0

Dans ce cas, &S

= =9(bo) 0(b0) =o092(bo)

bf0 I.bl

=c

oü CTQ est est la translation déterminée ou determinee par le le point point de de C[n] C[n]correspondant correspondant aa z = ^ sur le revêtement universel. revetement universel. définit ainsi, On definit ainsi, en accord avec le paragraphe précédent, precedent, un un isomorphisme isomorphisme

Gn A Aut(/n) induit une une structure de Gn-revetement, exprime exprimé par qui induit structure de par 9v gv = (f)(v) ==0,6,

9i = 0(0 =

2 oo02 et g1 (T gf == q5(f) (j)(f) O6

gvgigf = 1. 1. = acr0O o0 avec g vgigf =

On peut finalement finalement passer passer a l'action Faction de de IF. IT. (C, (C, fn) étant etant définie definie sur sur Q, Q, iine telle telle action Aut(/ une action n'intervient que que sur sur l'isomorphisme Pisomorphisme 40 : Gn -> —* n ).


111 111

Pour on aura auraainsi ainsiun unisoinorphisme isomorphisme Pour tout tout r eGIFF on T

(j): Gn —* -> Aut(/ n ) 99 i—> T(çb(g)) r((t)(g))

iY) => 'r(O) d'abord, 0(X, 0(X,Y)Y) = (—X, (-X,iY) r(0) = = ex(T) 0x(T) on ou x\ est le caractère caractere Tout d'abord, XA cyclotomique. On On peut peut aussi aussi ecrire écrire r{9) r(0) = cyclotomique. — 0x4(T) 9 ^ oü ou X4 x 4 : :IFF—* —>(Z/4Z)* (Z/4Z)*estest

l'homomorphisme correspondant aa l'extension l'extension Q(i)/Q. Q(i)/Q. rhomomorphisme correspondant Pour les translations, en en ayant ayant choisi choisi deux deux générateurs generateurs de de C[n] C[n](ce (ceque quenous nous 2 avons fait C[n] avec avec (Z/nZ)2) (Z/nZ) )on onobtient obtientl'homomorphisme rhomomorphismedede avons fait en identifiant identiflant C[nJ group es groupes IF —* F -> GL2(7Z/n7Z) GL 2 (Z/nZ) aT bbrT (aT dT

et sur C, l'action Faction de de FT sur surC[n] C[n] s'exprime s'exprime par r

(N 'N

.M\ M\ aTN + bTM .. —+t— = — ^— +1 + znn n n)

nn

r(up) = Pour Pour up

Proposition. muni de la v =v = Proposition. Soit Soit Gn = = (Z/nZ)2 (Z/nZ) 2 >\QZ/4Z TLj^TL muni depresentation la presentation 1), et fI = 1). Soit (0,0, (0,0,1), = (1,0, (1,0,1). Soit PPun unpoint pointde deC[n] C[n]vérifiant verifiant(*). (*).Alors Alorsl'action Vaction

de IT F sur par sur Gn est donnée donnee par ( fv v

|

if

VX(T)

^5T-1/X(T)3r

\f ou gT oü

y

est défini est defini comme en en (**). (**).

pair, N N= sont uniquement Remarque. Si nn est pair, = NT et M Mr = Mne sontpas pas uniquement T ne déterminés. Les éléments 91 et 92 ne dependent pas de T IF. g2 dependent de r G Deplus plus determines. Les elements g\ E F . De g±g2 = = 9291 g2gi et on on vérifie verifie immédiatement immediatement que {9r} {gT} définit definit un un cocycle; cocycle; on 9192 1 obtient donc done une une classe elasse de decohomologie eohomologie dans dans lelegroupe groupe H' i7(IF, (F,C[n]). C[n]). Pour obtenir un un énoncé enonce mettant mettant en enjeu jeudes desgroupes groupesprofinis profinis remarquons remarquons Gal(K/Q) oü reduit a l'action Faction de Gsl(K/Q) ou K est est Ia la tout d'abord que l'action Faction de F se réduit limite directe des corps Kn obtenus en rajoutant a Q(i) les coordonnées des rajoutant a Q(i) les coordonnees des points de n-torsion. n-torsion. On plus ([Si] chapII) II) que que K K = Q(i)ob* points On démontre demontre de plus ([Si] chap En posant posant

= limG lim n^Z2^9 G=

Z/4Z ^ f(C) x Aut(C)

et cii presentation induite, on en considérant considerant Ia la presentation on peut reformuler reformuler la la proposition proposition en remplacant remplagant Gn par par Q et en en considérant considerant l'homomorphisme Fhomomorphisme IF —* r->GL 2 (Z)

provenant l'action sur provenant de Faction sur le le module de Tate de de la la courbe. courbe. L'application L'application Gal(Q(i)ab/Q) —+ Aut(g)

ainsi définie definie est est injective. infective.


113

3.3. Applications. 3.3. Applications. Dans Dans ce qui précède precede nous avons avons étudié etudie une une famille famille de de dessins dessins d'enfants d'enfants galoisiens. Dans la suite nous allons nous intéresser galoisiens. interesser aux aux quotients quotientsde deceux-ci ceux-ci et en particulier aux aux dessins dessins de genre 1 obtenus par ce ce procédé. procede. (propre) en genre 1 ayant n2 Dn est le seul dessin dessin galoisien galoisien (propre) n2 faces faces et somsommets de valence valence 4. Ceci revient a dire dire que que le le groupe groupe Aut(G n ) opère opere transivif = 1,1, o(f) o(f) — = tivement tivement sur l'ensemble l'ensemble des presentations Gn =< v, v,l,f 1, f \ vlf 2o(i) = o(v) = 4>. 2o(l) o(v) 4 >. Soit E(4, 4) l'ensemble 25(4, 2, 2,4) l'ensemble des des classes classes d'équivalence d'equivalence de paires paires de de Belyi Belyi(pré(prepropres) /3) ou oh l'ordre l'ordre des des zeros zeros et et de poles de /3 divise 44 (ce sont tous propres) (V, (P, /?) ft divise les dessins dessins de de type type [4,2,4]... [4,2,4]... cf. cf. l'introduction). 1'introduction). 1 {('D,/3) Posons dememe mêmeEE'(4,2,4) Posons de (4, 2,4) = {(£>, /?) EG E(4,2,4) £7(4, 2,4) |VV aagenre genre 1}. Pour toute 2,4), il existe unun entier une suite toute paire paire(D, (X>,/3) /3)E GE(4, 25(4, 2,4), il existe entierpositif positifn et n et une suite de revêtements revetements C -*£>Vv A P1 C j

telle que /3 /?(/>£> = fn- V V est est alors alors le le quotient quotient de de CC par par un unsous-groupe sous-groupe de de

10.En Aut(/ n ) 10 . En particulier, particulier, vu vuque queles les éléments elements de de ce ce dernier dernier son son tous tous définis definis oon n obtient en La a sur C sur Kn C Q(i)ob' °bî ^ ^

Proposition. 2, 4) peut Proposition.Tout Toutdessin dessindedeE(4, 25(4,2,4) pent être etredefini defini sur sur une une extension extension abélienne abelienne de Q(i). Q(i).

Passons 2,4). Ceux-ci Passons maintenant aux aux éléments elements de de E1 25X(4, (4,2,4). Ceux-ci correspondent correspondent a des C. On (4,2,4) des courbes elliptiques isogènes isogenes a C. Onpeut peutalors alorsdécomposer decomposerE1 25*(4,2,4) 1 1 11 en "strates" E'(4, 2, = {(V, = 4n}'1. Décrire /3) E E'(4, 2,4) deg/3 £' (4,2,4) n {(P,/3) G £' (4,2,4) | deg/3 = An} . Decrirelesles X éléments de 25 E'(4,2,4) elements de (4,2,4)revient revientaadeterminer determinertous tousles lessous-réseaux sous-reseaux d'indice d'indice n de de A A = 7L Z 0 iZ iL aa homothéties homotheties près. pres. C'est le le problème probleme des des homothéties homotheties qui rend les calculs compliques compliqués et et fastidieux fastidieux (determiner le le nombre nombre de de soussousréseaux d'indice n est relativement reseaux d'indice relativement facile). facile). En En travaillant travaillant avec avec les les groupes groupes Gn on peut peut simplifier simplifier cette cette démarche. demarche. Voici Voici lalaméthode methodegénérale: generale: V est est le le quotient de de C par par un un sous-groupe sous-groupe d'automorphismes H H , alors H = O^HO =< a >. Si H =< Si H crq H° 01H0 9{Q) En posant (x, y) E E Fp, posant Q Q— = (x,y) # = £T H=H"

x2 + y 2 = 0 mod(p)

^

inod(4), alors alors H H ne peut pas être En particulier, si p = 3 mod(4), etre distingue dans Gn et on obtient donc dessins non isomorphes. done ^ ^ dessins isomorphes. mod(4), alors alors on obtient obtient deux deux sous-groupes sous-groupes distingués distingues distincts distincts Si p = 1 mod(4), 1 4, p est est constitué dessins dont ^ ^ non-galoisiens et 2 (2,4,4) constitue de de ^y^ dessins et E' £' (2, galoisiens. galoisiens on on obtient obtient un automorgaloisiens. De plus, pour les deux dessins galoisiens phisme d'ordre 4 qui fixe un point. Les phisme Les courbes courbes correspondantes correspondantes sont sont donc done isomorphes c, et isomorphes aa C, et les lesrevêtements revetements obtenus obtenus correspondent correspondent aades desisogénies isogenies non triviales (autres que . .cf. remarque du 3.2) .. En que la la multiplication multiplication par par n. n...cf. résumé, resume, on obtient la

Proposition. Proposition.Soit Soitpppremier; premier;alors alors — E"2 " '4 '4) ') —

I

^

sil Pp 33 mod(4) ~ si p 1 mod(4)

particulier, sip sip = 3 mod(4), En particulier, mod(4); alors alorsles lesdessins dessinsadmettent admettentun unmodèle modeledefini defini stir une extension de Q(i), Q(i), de sur extension abélienne abelienne de de degré degre inférieur inferieur on ou égal egal aa ^yk Si p = 1 mod(4), alors alors on obtient obtient 2 dessins dessins réguliers reguliers et tons tous les les autres autres peuvent être stir une extension abélienne Q(i) de degré etre définis definis sur abelienne de Q(i) degre au plus ^-^. Exemple. Nous Nous allons allons reprendre reprendre les les exemples exemplespresentes présentésdans dans [SS]: [SS]: On s'intéresse ici au an cas cas pp = 3, 5. Avant s'interesse ici 3,5. Avant toute chose, chose, nons nous allons allons utiliser le théorème que la courbe est définie snr Q seulement theoreme 1 p.3 qui affirme affirme que definie sur seulement dans dans deux cas: JJ == 1728 et deux cas: et JJ = = 287496 287496 qui correspondent correspondent respectivement respectivement a r == i et et rr = 2i. En particulier, particulier, pour p / 2 2,, les les dessins dessins non galoisiens galoisiens de 1 E'(2, 4, 4),, ne sont pas définis sur Q. E' (2,4,4) p pas definis Pour p = 33 on obtient deux dessins Pour dessins qni qui sont donc done conjugués. conjugues. Pour pp = 55 les les dessins dessins sont sont 4; 4; deux deux sont sont galoisiens et les les deux deux autres sont Pour galoisiens et sont conjugués et définis sur une extension quadratiqne conjugues et definis sur extension quadrat ique de Q (pour (pour un un exemple exemple "graphique" voir voir [SS] [SS] p.6). Ces résultats sont tout en Ces resultats sont en accord accord avec avec cenx ceux de [SS], [SS], tout en utilisamt utilisamt des des méthodes différentes. methodes differentes.


115

3.4. 3.4. Conclusion. Conclusion. Remarques. Remarques. Les dessins reguliers régulierspropres propresenengenre genre11(et (et on on le le deduit déduit par le Les dessins le calcul calcul de la caractéristique caracteristique d'Euler-Poincare) d'Euler-Poincaré)sont sontdu du type type [4,2,4] [4,2,4] ou ou [3,2,6]. [3,2,6]. Le premier cas cas aa été premier ete étudié etudie a fond fond dans dans les les paragraphes paragraphes précédents. precedents. Pour Pour le le deuxième cas, on on petit peut suivre exactement la deuxieme cas, la mêine meme démarche demarche en en remplaçant remplagant la courbe courbe C C par celle celle d'invariant d'invariant modulaire modulaire J == 0.0. Les Les résultats resultats obtenus obtenus sont sont alors une simple simple transcription de de ceux ceux présentés presentes ici, ici, avec avec des modifications modifications moindres ou des des corps corps mis mis en en jeu). jeu). On moindres (dans la definition definition des groupes groupes C,-, Gn ou On 3 une réalisation realisation du du groupe groupe Gal(Q(p)ab/Q) Gal(Q(p)a6/Q) ou pP3 = 1. obtient en particulier une

§4 Remerciements. §4 Une grande grande partie partie des des resultats, résultats, et en Une en particulier particulier pour pour cc ce qui qui regarde regarde les les formes modulaires, provient provient de de ma ma these these de de "Laurea". Un formes modulaires, Un premier premier remerremerciement va Arbarello qui premier m'a adressé adresse vers vers la théorie theorie ciement va done donc aa E. Arbarello qui en premier des dessins dessins d'enfants. d'enfants. Le developpement développement ulterieur ultérieur n'aurait jamais été ete possipossiconseils et Schneps et de Pierre Lochak Lochak que que ble sans les conseils et le soutien soutien de Leila Schneps je remercie remercie entre fourni les preprints de F. Pakovicth Pakovicth et et entre autre pour m'avoir fourni D.Singerman et R.I.Syddall. Un Un remerciement remerciement particulier est adressé adresse aa de D.Singerman et R.I.Syddall. l'lstituto Nazionale Nazionale Di Alta Matematica "F. "F. Seven" Severi" (INDAM) (INDAM) qui a rendu rendu l'Istituto possible mon sejour Paris. possible séjour a Paris.

References [C] [C] [GJ [G]

[GI] [GI] [I] [I]

K. Chandrasekharan, Elliptic Elliptic functions, functions, Springer-Verlag, Springer-Verlag, 1980. 1980. A. Grothendieck, Grothendieck, Esquisse Esquisse d'un programme, programme, Geometric Geometric Galois Galois AcActions, volume I. M. Girardi, G. Israel, Israel, Teoria Teoriadei dei campi, campi,Feltrinelli, Feltrinelli, 1976. 1976. of Gal(Q/Q), Gal(Q/Q), in Y.Ihara, On the embedding of in The The Grothendieck Grothendieck theory theory ants, L. L. Schneps, Schneps, ed., ed., London Mathematical Society of dessins d'enf d'enfants, Society Lecture Note Series 200, Cambridge Cambridge University University Press, Press, 1994. 1994.

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N. Koblitz, Introduction to elliptic elliptic curves curves and and modular modularforms, forms, GTM GTM 97, 1993. 1993.

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J. Lehner, Lehner, Discontinuous Discontinuous groups and automorphic automorphic functions, functions, AmeriAmerican Mathematical Mathematical Society, Society, 1964. 1964. planaires et et points d'ordre fini F. Pakovitch, Pakovitch, Arbres Arbres planaires fini sur les les jacobi-

{P} [P]

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Leonardo Zapponi

ennes des courbes courbes hyperelliptiques, hyperelliptiques,Prepublication Prépublication de de l'institut l'institut Fouennes des Fourier, Grenoble, Grenoble, 1996. 1996. L. Schneps, Schneps, Dessins Dessins d'enfants d'enfants on the the Riemann sphere, in The Grothen[S] [S] dieck Theory Theory of of dessins dessins d'enfants, d 'enfants,L. L. Schneps, Schneps, ed. ed. London dieck London MatheMathematical Lecture Note Note Series, Series, 200, 200, Cambridge Cambridge Univ. Univ. Press, matical Society Society Lecture 1994. [Si] [Si] J. Silverman, Silverman, Advanced Advanced topics in the theory theory of of elliptic elliptic c'urves, curves, GTM 151, Springer-Verlag, 1994. [Sh] [Sh] G. Shabat, Plane Plane trees, trees, in in The TheGrothendieck Grothendieck Theory Theory of dessins d'enfants, L. Schneps, fants, Schneps, ed., London London Mathematical Society Society Lecture Lecture Note Series, Series, 200, Cambridge Cambridge Univ. Press, Press, 1994. 1994. [SS] D.Singerman, D.Singerman, R.I.Syddall, R.I.Syddall, Belyi Belyi uniformisation uniformisation of elliptic elliptic curves, curves, [SS] preprint, 1996. 1996. [SV] [SV] G. Shabat Shabat and andV. V.Voevodsky, Voevodsky, Drawing Drawing curves over number fields, in Grothendieck Festschrift Festschrift3,3, Birkãuser, Birkauser, 1990. 1990. The Grothendieck [Z] L. Zapponi, di Strebel [Z] Zapponi, Grafi, Grafi, differeuziali differenziali di Strebel e curve, curve, Tesi Tesi di di Laurea, Laurea, università universita degli studi di di Roma Roma "La "LaSapienza", Sapienza", 1995. 1995.

Laboratoire de Mathématiques Mathematiques Faculté Faculte des des Sciences Sciences 25030 Besancon CEDEX 25030 Besangon zapponi©dmi.ens.fr zapponi©math.univ-fcomte.fr [email protected] [email protected]

Part II. II.The The Inverse Inverse Galois Galois Problem Problem

The regular inverse inverse Galois Galois problem over over large large fields fields Pierre Dèbes Debes and Bruno Bruno Deschamps Deschamps

The symplectic The symplectic braid group and Galois Galois realizations realizations Karl Strambach and Helmut Helmut Völklein Volklein

Applying modular towers towers to to the inverse Applying modular inverse Galois Galois problem Michael Yaacov Kopeliovich Kopeliovich Michael Fried and Yaacov

The Regular Inverse Inverse Galois Galois Problem over over Large Large Fields Fields Pierre Dèbes Debes and and Bruno Bruno Deschamps Deschamps

§1. Introduction. recent progress progress on on the the Inverse Inverse Galois Galois Problem. Problem. Most Most of of itit There has been recent consists of of results results on on the the absolute absolute Galois Galois group group G(K(T)) G(K(T)) when K isis aafield consists field various good goodarithmetic arithmetic properties. properties. This paper is a survey survey of of that that recent recent with various progress. questions that progress. Our Our goal goal isis also alsototo try try to to unify unify the the results results and and the the questions that have have arisen arisen in the last last few few years. years. each finite finite group group the Historically Inverse Galois is: is each Historically the the Inverse Galois Problem Problem (IGP) is: Galois group group G(E/Q) G(E/Q) of Galois of an extension extension of Q? The The modern modern approach approach consists consists in studying rather the the Regular Regular Inverse Inverse Galois Galois Problem (RIGP): (RIGP): isis each each finite group withE/Q E/Qa a group the the Galois Galoisgroup groupG{E/Q(T)) G(E/Q(T)) of of aa Galois Galois extension E/Q(T) E/Q(T) with regular Galois extension regularextension? extension?(As (Asusual, usual,we wejust justsay sayininthe thesequel sequelregular regular Galois extension E/Q(T)). Regularmeans means that that EEflOQ Q, or, or, equivalently, equivalently, G{E/Q(T)) E/Q(T)). Regular G(E/Q(T)) == Q == Q, Here are three reasons G(EQ/Q(T)). Here reasons why why considering considering this problem problem is is more more natural: (1) A positive positive answer answer to to the the RIGP implies a positive answer answer to the IGP. This This classically follows followsfrom fromHilbert's Hubert's irreducibility irreducibility theorem. classically

(2) Regular Galois extensions extensions E/Q(T) E/Q(T) correspond correspond totoGalois Galoiscovers coversf/: X : X—* —> p11 that are along with with their their automorphisms. automorphisms. Thus the RIGP P are defined defined over over Q along RIGP essentially consists consists in in studying studying the the action of G(Q) G(Q) on covers of the the projective essentially covers of projective that the line. This fits the general feeling feeling that the action action of of G(Q) G(Q) on on geometric geometric objects objects expected to reflect much of of its its structure. is expected reflect much

(3) for an an arbitrary (3) The RIGP RIGP can can be be formulated formulated more more generally generally for arbitrary field K (possibly of Given any field K, each each (possibly of positive positivecharacteristic) characteristic)inin place place of of Q. Q. Given group Galois group K(T): group might might well well be be the Galois group of of a regular Galois extension extension of of K(T): at least, no no counter-example counter-example isis known. RIGP known. In In contrast contrast with with the IGP, the RIGP might not not depend depend on the base field K K but rest might rest on on some some universal universal property of K(T) of the field K(T) of rational rational functions. functions. Because of the regularity condition, solving Because of solving the RIGP RIGP (in (in an anaffirmative affirmative way) over over a given field automatically solves way) given field solves it over over each each overfield. overfield. Therefore Therefore to solve solve the the RIGP RIGP over each prime prime field field Q. Q. The The RIGP has been it is sufficient sufficient to over each been solved solved over each each algebraically algebraically closed closed field: field: the real problem problem is the the descent descent from from Q to Q. Roughly speaking there are then two two directions directions of of work. Group Group theorists fix fix aa group group and and try try to realize theorists realize it over over Q(T) regularly regularly (i.e., (i.e., as the the Galois group group of of aa regular regular Galois Galois extension extensionofofQ(T)). Q(T)). Thanks to the Galois the so-called so-called rigidity criterion criterion and its developments, developments, there have been been since since the the late late 70' many many

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Pierre Dèbes Debes and Bruno Bruno Deschamps Deschamps

results in in that direction groups in in particular. particular. We refer results direction for for simple simple groups refer the reader to the works of Matzat Matzat ([Mat], [MatMal)and and his hisstudents students for for that that aspect aspect of of works of ([Mat], [MatMa]) question (see (see also geometers try to to the question also [Del]). [Del]). On On the the other other hand, arithmetic geometers realize regularly all finite groups given algebraic extension groups over over K[T) K(T) with K aa given possible) of of Q. Q. There has been more more recently recently some some progress progress in in (as small as possible) that direction. direction. We Wewill will focus focus on on this this second second aspect aspect of of the the problem, problem,which which was was developed Of course developed in in particular particular by by Fried, Fried, Harbater Harbater and and Pop. Of course this this distinction is somewhat somewhat artificial artificial for for in in practice practice there there isis aa strong strong interaction interaction between between the group arithmetic. group theory theory and and the arithmetic. Basically most most of of the the recent can be summarized by saying saying that Basically recent progress progress can summarized by the is solved solved if base field "large". By By large large we we actually actually mean mean the RIGP is if the the base field is is "large". the following following precise preciseproperty, property,which whichwas wasintroduced introducedby byF.F.Pop: Pop: each each smooth smooth geometrically geometrically irreducible irreducible curve curve defined definedover overKK has has infinitely infinitely many many if-rational K-rational points We will study property more more preprepoints provided provided there there is is at at least one. We study this property cisely §3. Haran Haran and Jarden Jarden call call itit "ample" "ample" inin[HaJa], [HaJa],which which expresses expresses cisely in §3. a certain tendency tendency of of these fields fields to develop abundantly points to develop abundantly through through the points of AMPLE can of aa variety. variety. We We note note that that AMPLE can also also be be understood understood as as aa property property of of "Automatique will use "Automatique Multiplication Multiplication des des Points Points Lisses Lisses Existants". Existants". We will use this terminology.

The statement does not not account The statement above above however however does account for for aa second second series series of of results of Similarly, the the general general RIGP RIGP does results of the same same spirit. spirit. Similarly, does not account account for a second conjectures of of the the area. Those results results and for second series series of classical classical conjectures area. Those conjectures give, give,ororpredict, predict, under under certain certain conditions, conditions,the the exact exact structure structure of of conjectures some Historically, the problem of of this this second second some absolute absolute Galois Galois groups. groups. Historically, the starting problem circle is is the conjecture circle conjecture of Shafarevich: Shafarevich: the absolute absolute Galois Galois group group G(QP") G(Qa6) is a free profinite profinite group group (on countably many free many generators). generators). These two two circles circles of ofthe the area area are are closely closelyrelated. related. They are both concerned These concerned with the structure of groups. The methods use the same with of absolute Galois groups. same kind of arguments, arguments, namely some patching and gluing techniques for analytic covers of covers and some some specialization specialization arguments arguments for for ample ample fields. fields. Still was our our feeling feeling Still it was that the the exact exact connection connection had had never never been made completely completely clear. clear. The goal of of this this paper is

(1) to state state aaconjecture conjecture that thatunifies unifies all all classical classical questions. questions. This This conjecture conjecture is Conjecture. We We state itit in in §1 §1and and show show the the connection connection with with all all is the Main Conjecture. other conjectures. conjectures. (2) to state aa theorem results of of the the area. area. This theorem that that summarizes summarizes most recent recent results This theorem Main Theorem. merely asserts asserts that the the Main Main Conjecture Conjecture theorem is is the the Main Theorem. It merely is true true if the base field KK is ample. ample. This contains the above above statement statement that is base field contains the the RIGP is is solved solved if K is ample. ample. The The Main Main Theorem Theorem is is precisely precisely stated in in K is §2 where special cases. cases. A §2 where we we also also show showhow howtotodeduce deducemost most recent recent results results as special general was given by F. Pop Pop [Po4]. [Po4]. general proof proof of of the Main Theorem was

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(3) to explain Theorem. For (3) explain the main arguments arguments of the proof proof of Main Main Theorem. For simsimplicity, we will will restrict restrict to the solution plicity, we solution of of the the RIGP RIGPover overlarge largefields. fields.

The necessary backgroundon onthe the arithmetic arithmetic of of fields fieldsand and the the theory theory of of necessary background covers can be found in the Fried-Jarden book [FrJa] and in the recent books covers can be found in the Fried-Jarden book [FrJa] and recent books [MatMa] by by Matzat-Malle and [MatMa] and [Vol [Vo] by Völklein. Volklein. We wish to to thank thank M. M. Fried, Fried, D. D. Haran, Haran, D. D. Harbater, Harbater, M. Jarden, F. We wish F. Pop, Pop, H. Völklein for very very helpful helpful comments comments and many Volklein for many valuable valuable suggestions. suggestions.

§2. Conjectures. §2. 2.1. 2.1. Classical Classical conjectures. conjectures. 2.1.1 2.1.1 First Firstcircle. circle.Recall Recallfrom fromthe theintroduction introductionthese thesevarious variousconjectures conjectures around the Inverse Inverse Galois Problem. Problem. The The Galois Galois group group of of aa Galois Galois extension extension Elk E/k isis denoted denoted by by G(E/k). G(E/k). Given Given aafield field K, K, we we denote denote by by K5 Ks (resp. (resp. by by K) K) a separable of KK and by G(K) separable (resp. algebraic) algebraic) closure closure of G(K) the theabsolute absolute Galois Galois group G(K) = G(K5/K) group G(K) G{KS/K) of K. K. every field field KK and for every Conjecture. (RIGP/K(T)) (RIGP/K(T)) — For every every finite finite group

G, there regular Galois extension EG/K(T) such =— there exists exists aa regular extension EQ/K(T) such that G(EG/K(T)) G{EQ/K(T)) G. G.

For every Conjecture. (IGP/K (IGP/#hub.) each finite finite every hilbertian hilbertian field fieldKK and and for each Conjecture. hub) — For G, there there exists exists aa Galois Galois extension extension EQJK such that that G(EG/K) G(EQ/K) = C. G. Or, Or, group G, EG/K such

equivalently, each quotient of of the the absolute absoluteGalois Galoisgroup group equivalently, each finite finitegroup groupGC is is a quotient G(K) of K. G(K)ofK.

Conjecture. Conjecture. (IGP) (IGP)

C, there — For each finite group G, there exists exists aa Galois Galoisextenextension E EG/Q G(EG/Q) = C. G. G/Q such that G(EG/Q) We have: have:

IGP R I G P / A ( T ) = > IGP/K I G P / ^hub. RIGP/R(T) hllb. = > IGP

Proof. it follows fromthe thehilbertian hilbertianproperty propertythat, that,ififEEr/K(T) isis aa Proof. Indeed, Indeed, it follows from T/K(T) regular Galois extension extension of of Galois Galois group group G, G, then then there exists exists some some specializaspecialization t Ee K K of of T such such that that the thespecialized specialized extension Et/K is a Galois extension of Galois Galois group group G. G. Whence Whence RIGP/ RIGP/K(T) of IGT* / KhUb.. hub.-From From Hilbert's irreHilbert's irreK ( T ) => IGP/K ducibility ducibility theorem, Q is a hilbertian field. field. So SoIGP/K I G P / ^hUb. IGP. 0• hiibm =ÎGP.

2.1.2 2.1.2 Second Secondcircle. circle.Conjectures Conjecturesofofthis thissecond secondcircle circle are are conjectures conjectures about embedding an embedding embedding problem problem for group rFisisaa embedding problems. problems. Recall Recall that that an for aa group diagram of group homomorphisms homomorphisms

Pierre Dèbes Debes and Bruno Bruno Deschamps Deschamps

122 122

F

r

1

H

)

1

where the horizontal horizontal sequence sequenceisisexact exactand and the the map map /f :: FF —>HHisissurjective. where the surjective. A proper solution is a surjective surjective group group homomorphism homomorphism gg ::FF —* —> G G such such that that Without the condition such aa map map g is said said to be aa ag f. Without condition "g "g surjective", surjective", such ag = f. weak solution. The The embedding embedding problem problemisissaid saidtoto be be finite finite ifif GC isis finite. finite. It is weak solution. said to be split H has aa group-theoretic said split if aa ::G G —* —» H group-theoretic section. section. profinite group F is said to be projective each finite finite embedding embedding problem A profinite protective if each for for F is weakly weakly solvable. solvable. Free profinite profinite groups are some examples of projective groups. groups. Finally Finally recall recall that, that, given given aaweakly weakly solvable solvable embedding embedding problem problem for for aa group procedure that that generally generally allows allows to reduce to group F, there exists a standard procedure a split situation. situation. More Moreprecisely, precisely, this this procedure procedure (e.g. (e.g. [Po4;§l B) 2)]) 2)]) uses uses aa weak weak solution solution to to construct construct a split embedding embedding problem problem for for FF such such that that existence of of aa proper proper solution for for it implies implies existence existence of ofaaproper proper solution solution for for the the original one. occasions. For simplicity, simplicity, we one. We will will use use this reduction in several occasions. we call call itit the weak—* weak-^split split reduction. reduction. Conjecture. Conjecture.(Fried-Völklein (Fried-Volklein[FrVo2]) [FrVo2])— —Let Let K K be beaahilbertian hilbertiancountable countable field field such that G(K) G(K) is (e.g. cohdim(K) cohdim(K) < 1), then such that is projective projective (e.g. then G(K) G(K) isispro-free. pro-free.

Conjecture. Conjecture.(Shafarevich) (Shafarevich) — G(Qa6) is pro-free. pro-free. These conjectures conjectures are more or less These less classical. classical. We will denote them respecrespectively by by FrVo FrVo and SHA. The tively The latter latterisisactually actuallyaaspecial special case case of of the the former. former. Indeed, from from a result of Indeed, of Kuyk, Kuyk, Qab Q°6 is hilbertian, see see [FrJa,Th.15.6]; [FrJa,Th.l5.6]; and Q^6 1: this follows fromthe the fact fact that any is of cohomological cohomological dimension < 1: follows from any dividivision algebra algebra over sion over every every number number field has aa cyclotomic cyclotomic splitting splitting field field [CaFr]. [CaFr]. In turn, FrVo In FrVoisisa aconsequence consequenceofofthe thetwo twofollowing following equivalent equivalent conjectures conjectures about embedding embedding problems. problems.

Conjecture. (Split Conjecture. (Split EP/K(T)) E P / # ( T ) ) — Let K be be an an arbitrary arbitraryfield. field. Then each problemfor for G(K(T)) G(K(T)) has split embedding embedding problem hasaaproper propersolution. solution. K be Conjecture. (Split (SplitEP/K E P / #hub.) be aa hilbertian hilbertian field. field. Then Then each hilbm) — Let K problemfor for G(K) G(K) has split embedding embedding problem has aaproper propersolution. solution. We We call these these conjectures conjectures the the Split SplitEmbedding Embedding Problem Problem conjecture conjecture over over

Regular Inverse Regular Inverse Galois Problem

123 123

K(T) (resp. K(T) (resp. over over hilbertian hilbertianfields). fields). We Wehave: have: FrVo = > SHA Split EP/K(T) EP/*(T) «=> Split EP/K EP/tf hub. SHA hilb. = > FrVo

from the the fact fact that ProofofofSplit Split EP/tf EP/K(T) Proof E P / K M 6 .. («=) follows from (T ) Split EP/K for every every field field K, K, the field K(T) for K(T) isishilbertian hilbertian[FrJa;Th.12.1O1. [FrJa;Th.l2.10]. Given aa split split embedding embedding problem problemfor forG(K) G(K) with with K K hilbertian, (=>): Given hilbertian, conconsider the the embedding problem for for G(K(T)) G(K(T)) obtained sider embedding problem obtained by by composition composition with the the map G(K(T)) G(K). The The Split EP/K(T) map G(K(T)) —* -» G(K). EP / K(T) conjecture conjecture provides provides aa proper proper solution. solution. Then, Then, as in the proof proof of R RIGP/K(T) I G P / K ( T ) => IGP/K hub. hub.above, above,use usethethe hilbertian property property to specialize this proper proper solution solution to to aa proper of hilbertian specialize this proper solution solution of embedding problem problem for for G(K). G(K). D• the original original embedding Proof of EP/ Khub. hub. =>FrVo. FrVo.We Wewill willshow showthat thatany anygiven givenembedding embedding of Split EP/K U problem for for G(K) G(K) has problem has a proper proper solution. solution. Conclusion Conclusion "G(K) G(K) pro-free" pro-free" will will follow then from Iwasawa's theorem recalled below. Since G(K) is assumed follow then from Iwasawa's theorem recalled below. Since G(K) is assumed to be projective, projective, the given given embedding embedding problem weak solution. From problem has has aa weak solution. From the weak—*split weak—>split reduction, embedding problem problem isis reduction,one onemay mayassume assumethat that the embedding split. Therefore, from from the Split EP/K EP / Khi/b. conjecture, this split embedding embedding split. Therefore, the Split hiib. conjecture, problem solution. • problem has has a proper solution. D

(Iwasawa [Iw],[FrJa;Cor.24.2])Le* [Iw],[FrJa;Cor.24.2J)LetKK be be a countable Theorem T h e o r e m 2.1. 2 . 1 . (Iwasawa countable field, field, G(K) isis pro-free problemfor forG(K) G(K) has a G(K) pro-free ifif and and only only ifif each each finite finite embedding embedding problem proper solution. solution. 2.1.3 Conclusion. Conclusion.Finally Finallywe we have have Split Split EP/K EP/K hi/b. G P / ^ hhu/b.• ub.- Indeed Indeed hub. => IIGP/K realizing aa group group G G over over KK amounts amounts to solving realizing solving the split embedding embedding problem problem for G(K) ininwhich is 1isl—)•£?—»C?--»1-»1. —* G —÷ G —÷ 1 —* 1. The following for whichthe theexact exactsequence sequence following diagram summarizes §2.1. §2.1.

IGP/K

RIGP/K(T)

IGP

hi/b.

1.t

Split EP/K(T) EP/K(T)

Split EP/K EP/K hi/b. Mb.

=>

FrVo

=>

SHA SHA

Remark Conjectures of of the the second circle seem seem to to be stronger R e m a r k 2.2. 2.2. Conjectures second circle stronger since since they give give results on the structure structure of of absolute absolute Galois Galois groups groups whereas whereas those those of of the first circle the circle only only deal deal with the finite finite quotients quotients of of absolute absolute Galois Galois groups. groups.

But there than Split EP/K(T) But there is in in fact fact no no other other obvious obvious implication implication than EP/K(T) => both these these sets sets of there is no IGP/K(T) between between both of conjectures. conjectures. For example example there no IGP/K(T)

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Pierre Dèbes Debes and Bruno Deschamps Deschamps

obvious implication between betweenFrVo FrVo and and IGP IGP 1,x, nor there there is is between between Split Split obvious implication EP/K(T) to state EP/tf(T) and RIGP/K(T). R I G P / V ( T ) - The Themotivation motivationofofthe thefollowing following section section is to more general general conjecture unifies both The a more conjecture that that unifies both these these circles circlesofofthe the area. area. The difference between EP/^(T) and RIGP//^) will difference between the the two two conjectures conjectures Split Split EP/K(T) RIGP/K(T) will then become become clear clear (see (see Remark Remark 2.4). 2.4).

2.2 2.2 Main Conjecture. Conjecture. 2.2.1 Statement Suppose given Statementofofthe theMain MainConjecture. Conjecture. Suppose givena acommutative commutative diagram of of group group homomorphisms homomorphisms G(.K5a(T)) (T)) G(K

G(K)

G(.K(T))

I/

/

9/

/

7/

the sequences sequences in which which the arrows are lined up are are exact, exact, '—* •means meansthat thatthe thehomomorphism homomorphismininquestion questionisisinjective, injective, -- where where —* —» means means that that the thehomomorphism homomorphism in in question question isissurjective. surjective. -- where where

-- where

Such Such a diagram diagram isiscalled called an anembedding embeddingproblem problemofofexact exactsequences sequencesover over K(T). solution isis aa triple (9,g, maps as as in the K(T). A A proper solution (g,g,j)'y) of of surjective surjective maps the diagram above, above, such the enlarged enlarged diagram diagram commutes. commutes. If If the the condition condition diagram such that the "surjective" "surjective" is removed, removed, the triple triple (9, (g,g, g,'y) 7) is is said said to to be beaaweak weaksolution. solution.

Main Conjecture. — Let K be embedding bean anarbitrary arbitraryfield. field.Then Thenfor foreach each embedding EPhas hasa aweak weaksolution, solution,then then problem EP EP of exact sequences sequences over K(T), K(T), ififEP EP has has aaproper propersolution. solution.

Remark Remark2.3. 2.3.(a)(a)Weak Weaksolutions solutions(9,9,7) (#,#,7)actually actuallycorrespond correspondininaaone-one one-one way with the maps g: g :G(K(T)) G(K(T))—* —>CGsuch suchthat that ag==f.f. Indeed, Indeed,given giveng,g,take take for for g9 the restriction restriction of g to G(K8(T)). G(KS(T)). The The containment containment g(G(K5(T))) g(G{Ks(T))) C CG holds because the the two groups in in the the diagram diagram are equal (to holds because two down-right down-right groups (to F). F). There There exists exists then aa unique unique map map 7 that makes makes the diagram diagram commute. commute. The The triple (g, (

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