Introduction to the Theory of Algebraic Numbers and Functions

INTRODUCTION TO THE THEORY OF ALGEBRAIC NUMBERS AND FUNCTIONS PURE A N D APPLIED MAT H EMAT IC S A Series of Monogra...

Author: Martin Eichler

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INTRODUCTION TO THE THEORY OF

ALGEBRAIC NUMBERS AND FUNCTIONS

PURE A N D APPLIED MAT H EMAT IC S A Series of Monographs and Textbooks

Edited by PAUL A. SMITHand SAMUEL EILENBERG Columbia University, New York 1: ARNOLD SOMMERFELD. Partial Differential Equations in Physics. 1949 (Lectures on Theoretical Physics, Volume V I ) 2: REINHOLD BAER.Linear Algebra and Projective Geometry. 1952 3 : HERBERT BUSEMANN A N D PAULKELLY.Projective Geometry and Projective Metrics. 1953 4: STEFAN BERCMAN A N D M. SCHIFFER. Kernel Functions and Elliptic Differential Equations in Mathematical Physics. 1953 5 : RALPHPHILIP BOAS,JR. Entire Functions. 1%54 BUSEMANN. The Geometry of Geodesics. 1955 6: HERBERT 7 : CLAUDE CHEVALLEY. Fundamental Concepts of Algebra. 1956 8: SZE-TSEN H u . Homotopy Theory. 1959 9 : A. M. OSTROWSKI. Solution of Equations and Systems of Equations. Second Edition. 1966 Foundations of Modern Analysis. 1960 10: J. DIEUDONNC. Curvature and Homology. 1962 11 : S. I. GOLDBERC. 12 : SICURDUR HELCASON. Differential Geometry and Symmetric Spaces. 1962 Introduction to the Theory of Integration. 1963 13 : T. H. HILDEBRANDT. ABHYANKAR. Local Analytic Geometry. 1964 14: SHREERAM Geometry of Manifolds. 1964 L. BIS OP AND RICHARDJ. CRITTENDEN. 15 : RICHARD 16: STEVEN A. GAAL Point Set Topology. 1964 17 : BARRYMITCHELL.Theory of Categories. 1965 18: ANTHONYP. MORSE.A Theory of Sets. 1965 19: GUSTAVE CHOQUET.Topology. 1966 20 : Z. I. BOREVICH A N D I. R. SHAFAREVICH. Number Theory. 1966 21 : Josh LUIS MASSERA A N D J U A N JORCE SCH~FFER. Linear Differential Equations and Function Spaces. 1966 22 : RICHARD D. SCHAFER. An Introduction to Nonassociative Algebras. 1966 Introduction to the Theory of Algebraic Numbers and 23: MARTIN EICHLER. Functions. 1966 ABHYANKAR. Resolution of Singularities of Embedded Algebraic 24: SHREERAM Surfaces. 1966

T

I n preparation: FRANWIS TREVES. Topological Vector Spaces, Distributions, and Kernels. OYSWINORE The Four Color Problem. PETER D. LAXand RALPHS. PHILLIPS. Scattering Theory.

INTRODUCTION TO T H E THEORY O F

ALGEBRAIC NUMBERS AND FUNCTIONS Martin Eichler DEPARTMENT OF MATHEMATICS UNIVERSITY OF BASEL BWEL, SWITZERLAND

Translated by George Striker &TTINGEN,

GERMANY

1966

@ ACADEMIC PRESS

New York and London

First published in the German language under the title Einfiihrung in die Theorie der algerbraischen Zahlen und Funktionen and copyrighted in 1963 by Birkhiiuser Verlag, Basel, Switzerland

COPYRIGHT 0

1966, BY ACADEMIC PRESS INC.

ALL RIGHTS RESERVED. NO PARTS OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRllTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS INC. 111 Fifth Avenue, New York, New York 10003

United Kingdon Edition published by ACADEMIC PRESS INC.(LONDON) LTD. Berkeley Square House, London W.l.

L m w w OF CONGRESSCAT-

CARDNUMBER: 65-26396

PRINTED IN THE UNlTED STATES OF AMERICA

Preface to the En&h Edition Several errors and ambiguities which had been indicated in an Addenda et Errata sheet to the German edition have been eliminated. Three important changes were made: (a) Chapter III,§l has been shortened considerabiy by rearranging the material. (b) Section 4 of Chapter III,§3 has been rewritten as sketched in the Addenda et Errata. (c) Fields of genus g 4 1 had been implicitly excluded from representations of correspondences by hth degree differentials in the case h > 1. This assumption, in Chapter V,§2and in the preparations in Chapter 111,$5, has been avoided. The author extends his most sincere thanks to Academic Press for publishing an English translation of this book. Indeed, the attempt to keep up with the times while not adhering to the representation of the subject matter which has today attained a virtual monopoly certainly does place certain demands upon the reader. He also thanks the editors of the Series in Pure and Applied Mathematics, Professors Samuel Eilenberg and Paul Smith, for choosing this book for their esteemed series. He particularly thanks Mr. George Striker for his clear translation into a pleasant English as well as his agreement to special wishes during the translation work. M. EICHLER

Basel, Switzerland May 1966

V

This Page Intentionally Left Blank

Preface to the German Edition The purpose of this book is twofold. First it serves to introduce the general notions, the concepts, and the methods which underlie the theories of algebraic numbers and algebraic functions, primarily in one variable. A certain mastery of algebra and function theory, as might be gained from a modest one-volume text on each, is presupposed. It seems most important to me to illuminate this subject from various points of view, projecting a multidimensional image. In choosing the topics and methods to be included in this book it became particularly clear to me that mathematics obeys the law of the biological theory of evolution: all individuals follow the general development of the species. It may well be that this law holds for other sciences as well, but the immense formative force of mathematics on the human mind assures particular adherence to this law. After some basic ideas are presented in the first chapter, the development of the actual subject is commenced in the second chapter, with the divisibility theory of algebraic number and function fields. At this point no distinction is necessary between these two types of fields, for within the bounds of divisibility theory they obey the same laws. In an appendix to the second chapter the theory of algebraic number fields is pursued only a small step beyond this interesting point. From that point on our sole interest will be the study of algebraic fields of functions. In the third chapter the foundations of the classical theory of algebraic function fields are laid down, using the Puiseux power series. The reader can thus omit Chapters I and I1 if he so desires. We find, however, that the power series themselves are not indispensable, and that this classical approach eventually merges with another method even applicable to fields of arbitrary characteristic ,provided the field of constants is algebraically closed. The second object of this book is to introduce the theory of elliptic modular functions, with its wonderful and deep applications in analytic number theory. This theory has provided an unquenchable effervescent spring of new discoveries for more than one hundred years. Some of these discoveries have crystallized in the theory of correspondences of general algebraic function fields, which is dealt with in Chapter V. Of course, brevity of a book such as this one is possible only at the cost of omitting many interesting topics. It might have been particularly attractive to compile the knowledge of theta and zeta functions of algebraic number vii

viii

PREFACE TO THE GERMAN EDITION

fields available today. This would have led to a reunion of the algebraic number and function theories at a higher plane. Those familiar with the subject might still be interested in the following remarks. In occasional lecture courses in the past I had always used the methods of valuation theory. During the preparation of this book it became clear to me that the older development of divisibility theory, that of L. Kronecker is by far the most elegant. It leads directly to principal ideal rings, the divisors of which correspond to the ideals. It was first presented in this form by H. Weber, and seems all the more attractive today, a time when the basic properties of principal ideal rings are taken up in elementary algebra. It is not surprising that Kronecker did not realize this possibility himself, for the notion of a principal ideal ring would have seemed too abstract in his day. Instead, he proved the lemma that the contents of the product of two polynomials is equal to the product of their contents. It was the difficulty of this proof that lead to the general adherence to E. Noether’s development of ideal theory. The valuation-theoretical foundations of divisibility theory, supported chiefly by H. Hasse, also avoid this lemma. But these methods each have their own disadvantages. It would be wrong, though, to interpret this to mean that the entire theory should be developed along Kronecker’s lines. On the contrary, it is R.Dedekind’s notion of ideals which permits the valuable methods of linear algebra to be introduced. Similarly, the local considerations introduced by Hasse are quite indispensable. I have avoided the term ualuation,operating instead with the corresponding valuation rings. In particular, I have avoided using the perfect closures of fields with respect to valuations, for these not only are unpleasant from an esthetic point of view but, moreover, represent an unnecessary detour. The references at the end of later sections of this book are by no means intended to be complete. It is a pleasure to acknowledge the stimulation provided by Dr. H. Kappus who read and edited the drafts. His thesis also contributed decisively to the final form of Sections 111,s and V,2. The support of Mr. A. Heeb who read the proofs to the German edition was most valuable. In particular, I want to express my appreciation to the publisher who made this book possible and gave it an exemplary appearance.

Basel, Switzerland March 1963

M. EICHLER

Contents PREFACE TO THE ENGLISHEDITION PREFACE TO THE GERMAN EDITION

V

vii

Introduction 1. The Subject 2. The Method

Table of Several Abbreviations and Symbols

1 2 3

Chapter I: Linear Algebra $1. Modules in Principal Ideal Domains 1. 2. 3. 4.* 5.+

Finite Modules The Theorem of Elementary Divisors Dud Spaces and Complementary Modules Noetherian Rings A Further Basis Theorem

1. Minkowski's Point Lattice Theorem 2.* Siegel's Proof 3. Generalization to Function Fields

14 16 18

20

93. Linear Divisors Basic Concepts Norm and Degree of a Linear Divisor The Dimension of a Linear Divisor The Riemann-Roch Theorem and the Minkowski Linear Form Theorem

$4. Traces, Norms,and Discriminants 1, 2. 3. 4.

5 7 10 11 13

14

92. Systems of Linear Inequalities

1. 2. 3. 4..

5

Representations by Matrices The Transitivity Formulas The Discriminant Separable and Inseparable Extensions

* The sections marked with an asterisk might be omitted in the first reading. ix

20 23 23 26

27 27 28 29

30

CONTENTS

X

Appendix to Chapter I: The Theta Function !jl? The Symplectic Group The Basic Properties Symplectic Geometry The Hyperbolic Plane and Hyperbolic Space The Symplectic Modular Group 5. The Fundamental Domain 6. The Theta Function 7. Proof of the Reciprocity Formula

1. 2. 3. 4.

$2.* Theta Functions for Quadratic Forms 1. Simple Gaussian Sums 2. The Quadratic Reciprocity Law and the Sign of Gaussian Sums 3. The Theta Function of a Definite Quadratic Form

32 32 34 35 36 39 41 43

44 44 46 48

Chapter II: Ideals and Divisors

$1. Ideals 1. Integral Dependence 2. The Finiteness of the Principal Order 3. Kronecker Divisors 4. Ideals 5. Proof of the Principal Theorem 6.* Extension of Divisibility Theory

$2. Local Rings 1. 2. 3. 4.

Basic concepts Local Rings in Algebraic Extensions Local Rings in Algebraic Number and Function Fields The Component Decomposition of Ideals

$3. Ideals in Different Fields; the Norm 1. Extension of an Ideal 2. TheNorm 3. The Prime Ideals

§4. The Complement, Different, and Discriminant

45.

53 53 54 56 58 60 61

63 63 64 66 67

70 70 70 72

73

1. The Complement 2. Different and Discriminant 3. The Dedekind Discriminant Theorem

73 75 77

Divisors

79

1. The Rational Functim Field 2.* Projective Invariance

79 81

CONTENTS

3. 4. 5. 6. 7.

Divisors in Algebraic Number and Function Fields The Behavior of Divisors under Field Extensions The Prime Divisors Divisors and Linear Divisors The Linear Degree

$6.* Decomposition of Prime Ideals in Galois Extensions 1. The Decomposition Group and Inertia Group 2. The Ramification Groups

3. The Discriminant

xi 82 84 86 88 89

90 91 94 96

Appendix to Chapter II:* Topics from the Theory of Algebraic Number Fields $1.

The Finiteness Theorems 1. 2. 3. 4.

The Finiteness of the Ideal Class Number The Discriminant The Dirichlet Unit Theorem The Regulator

$2. Quadratic Number Fields and Cyclotomic Fields 1. Quadratic Number Fields 2. Special Cyclotomic Fields

98 98 100 101 104

104 104 106

Chapter 111: Algebraic Functions and Differentials $1. Power Series Expansions of Algebraic Functions 1. The Field of Power Series 2. Divisibility, Rearranging of Power Series 3. Inversion of a Power Series 4. Algebraic Functions; Regular Places 5. Continuation; Critical Places 6. Puiseux’s Theorem

$2. Algebraic Function Fields Divisors in Rational Function Fields Divisors in Algebraic Function Fields Decomposition of Rational Divisors The Principal Orders Divisors and Linear Divisors 6. The Invariance of the Concept of Divisors 7.* Extension to More General Constant Fields

1. 2. 3. 4. 5.

$3. The Riemann-Roch Theorem 1. Dimension of a Divisor Class 2. The Riemann-Roch Theorem

110 110 112 113 115 116 118

120 120 121 124 125 128 131 131

132 132 133

xii

CONTENTS

RCfCrCnCCS

135 135 139 140 141 142

@ Differentials I.

143

3. Questions of Invariance 4. Extension of the Constant Field 5. The Fields of Genus 0 6. Likroth's Theorem 7. Further Proofs and Generalizations of the Riemann-Roch Theorem

1. 2. 3. 4. 5. 6. 7. 8.*

Differential Quotients The Differential Calculus with Characteristicp The Concept of the Differential Continuation; Separable and Inseparable Prime Divisors Cartier's Operator Residues of Differentials The Residue Theorem The Differential Class

$5. Differentials and Principal Part Systems 1. 2. 3. 4. 5.* 6.

Differentials of Higher Degrees Principal Part Systems The Scalar Product The Relationship to Integral Calculus TheDiagonal The Analog of the Green Function Notes

!$6. Reduction of a Function Field with Respect to a Prime Ideal of the Constant Field 1. 2. 3. 4. 5. 6.

The Irreducibility Theorem Regular Prime Ideals Behavior of Ideals under Residue Formation Behavior of Divisors under Residue Formation Continuation; Behavior of Differentials under Residue Formation Behavior of the Field under Residue Formation and Extension Notes References

143 144 147 149 150 151 153 156

158 158 159 160 163 165 167 171

171 171 174 177 178 181 183 183 184

Chapter IV. Algebraic Functions over the Complex Number Field $1. Riemann Surfaces 1. The Riemann Surface of an Algebraic Function 2. The Riemann Surface as Complex Manifold 3. The Riemann Surface as Topologid Manifold

185 185 186 188

$2. Fields of Elliptic Functions

190

1. Introduction 2. The Addition Theorem 3. Automorphisms

190 191 193

CONTENTS

...

Xlll

196 197

The Integral of the First Kind The Addition Theorem and the Abel Theorem The Weierstrass Normal Form Elementary Elliptic Functions Notes References

204

53. The Group of Divisor Classes of Degree 0

204

4. 5. 6. 7.

1. 2. 3. 4. 5.

The Riemann Period Matrix A Hermitian Metric for Differentials of the First Kind Abelian Integrals of the Third Kind Abel’s Theorem The Jacobian Variety

Notes References

&I Modular . Functions The Modular Surface Covering Spaces of the Modular Surface Congruena Subgroups Modular Forms The Field of Modular Functions Modular Forms and Differentials 7. Fourier Expansions of Eisenstein Series 8. Theta Functions References

1. 2. 3. 4. 5. 6.

rn

202 203

204 206 208 209 211 213 214

214 214 215 217 219 221 223 225 229 232

Chapter V: Correspondences between Fields of Algebraic Functions $1. The Correspondences Basic Concepts Multiplication of Correspondences Properties of the Product Correspondences of a Field with Itself Effect of Correspondences on Divisors Prime Corresqondences Inseparable Extensions a. The Frobenius Automorphism 9. Correspondences of a Field of Automorphic Functions with Itself 1. 2. 3. 4. 5. 6. 7.

233 233 236 239 241 242 245 246 249 250

$2. Representations of Correspondences in the Space of Differentials 252 1. 2. 3. 4. 5.

Definitions The Classid Case Continuation ;Representations of Rosati-Adjoint Correspondences TheTrace Evaluation of the Trace Formula Notes

252 256 25 8 259 263 265

xiv

CONTENTS

$3. Modular Functions 1. 2. 3. 4. 5. 6. 7.

The Modular Correspondences Products of Modular Comspondences Representations of Modular Correspondences by Differentials The Petersson Metric Fourier Expansions of Modular Forms Ramanujan’s Conjecture Results for Modular Forms of Odd Dimensions; Notes References

$4. Castelnuovo’s Inequality 1. 2. 3. 4. 5. 6. 7. 8. 9.’

Introduction Reduction to the Classical Case Extension of the Notion of Correspondence The Fixed Points of a Correspondence The Connection with $2 TheTracc Second Proof of the Principal Theorem: Preparations Second Proof of the Principal Theorem: Conclusion Remarks Concerning the Ring of Correspondence Classes Notes References

$5. Applications in Number Theory 1. 2. 3. 4. 5. 6. 7.

The Zeta Function of a Field of Functions The Functional Equation Extension of the Field of Constants Riemann’s Conjecture Modular Functions The Eigenvalues of Modular Correspondences Modular Functions of the Principal Character Notes References

$6. Elliptic Function Fields 1. The Ring of Correspondence Classes 2. Complex Multiplication

AUTHORINDEX Sumcr INDEX

266 266 269 271 212 275 211 219, 281

28 1 28 1 282 284 286 289 291 293 295 297 298 299

299 299 301 303 305 307 311 313 314 315

315 316 318

321 322

INTRODUCTION TO THE THEORY OF

ALGEBRAIC NUMBERS AND FUNCTIONS


Introduction 1. THESUBJECT

A finite algebraic number field K is a finite extension of the rational number field k = Q ; a finite algebraic function field K is a finite extension of the rational function field k = ko(xl, ..., x,) in the rn variables x i over some field of constants ko.We will, however, simply talk of number and function fields, thus taking for granted the prefix “finite algebraic,” but sometimes distinguishing between “ algebraic” and “ rational.” In a function field, all elements of K which satisfy an algebraic equation with coefficients in ko form a subfield k,,called the exact constant field. K can always ‘be considered as a function field over the exact constant field, and in general this is convenient. There is a remarkable similarity between algebraic number and function fields, which at the stdrt permits a common treatment of both, and is first manifested in the formal analogy between the integral domains i = Z of rational integers and i = k,[x] of rational functions. Both are Euclidean domains. In such domains the theorem of unique prime factorization holds. This property is also shared by the polynomial ring i = ko[x,, ..., x,,,] in view of the theorem: r f an integral domain 0 has unique prime factorization, then so does the ring D [ x ] . In this book only marginal treatment will be given function fields with rn > 1 variables, as long as the arguments do not require a distinction for different numbers of variables. The special problems and difficulties of these fields will not be considered. The few points of a general theory that can be included, however, will finally prove to be useful for the theory of functions of one variable. Among the first problems is the development of a theory of divisibility for algebraic number and function fields. This is the subject matter of Chapter 11. With the extension of k to K one arrives at an integral domain 3 in K which can, in the corresponding sense, be considered an extension of i. In 3,however, the theorem of unique prime factorization no longer holds. The simplest counterexample is found in the quadratic number field K = Q(,/-6), where 6 = -(,/-6)’ = 2 . 3 are two factorizations of an integer into integral irreducible factors. The uniqueness of prime factorization is restored by the introduction of ideal elements. Chapter I gives several theorems of linear algebra, which arose in the course 1

2

INTRODUCTION

of the continual reformulation of the theories of algebraic numbers and functions. They are attractive enough to stand apart from their applications. Their study can be postponed, however, until they are needed.

2. THE?METHOD To explain an important method which will be among those used, we consider the example of the rational function field C(x) of one variable over the field C of complex numbers. We will often refer to this example simply as “classical function theory.” One usually first expands a function in a power series (Laurent series) at some point or place (we use the term “place” immediately, anticipating the usual algebraic terminology), and asks for the first nonzero coefficient. The places x = 5 are interpreted as being in the complex number plane, which is closed in the number sphere by introduction of the place symbolized x = 00. The expansion of the function at x = 00 is simply the expansion in a power series in x-’. This preparatory local theory of functions then leads to the global theory, which makes statements about the behavior of a function in an entire domain-for the present case, in the entire number sphere. An example is given by the theorem: A function holomorphic everywhere except for poles is a rational function. The power series in x - ( which contain no negative powers form an integral domain. The rational functions with power series of this property also form an integral domain, which we always designate i,. These functions can be characterized without reference to power series; in the decomposition

to first degree factors the term x - 5 appears with a positive power v ( r ) , if at all. The rational functions which have no negative powers when developed in powers of x-’ form an integral domain iw. They are characterized by their degrees (Grad = G) being nonpositive. The integral domain i = C[x] is clearly the intersection

i=

n

i,.

i#W

This equation shows that a function holomorphic at every finite place is a polynomial. The method of algebraic function theory consists, first of all, of finding not only a corresponding extension 3 of i, but also an extension 3, of i , , (There will turn out to be more than one.) Functions are then examined with

INTRODUCTION

3

respect to their relation to 3,; the results lead to the “local” properties. The goal is, of course, to find “global” properties of functions, i.e., such that hold for all 3,.The separation into a local and a global theory is the natural graduation from the easier to the more difficult. It is perhaps amazing that this distinction between local and global properties, which stems from classical function theory of one variable, can be carried over profitably to algebraic numbers and functions in several variables. From what has been said it becomes plausible that in the detailed investigation one can use power series expansions, but that they are not indispensible. Certain advantages in the study of functions over an abstract field ko are gained by remaining free of power series; we do so in Chapter 11. On the other hand the method of power series is so standard, from classical function theory, that it would be too biased not to base some of our representation on it. Thus we reformulate function theory in Chapter I11 with its help.

Table of Several Abbreviations and Symbols (a) GENERAL greatest common divisor, degree (Grad) of a polynomial or rational function f(x), degree of a divisor a (II,§5,4), norm with respect to a field extension K / k , trace with respect to K / k , pseudotrace with respect to K / k (1,§4,4), pseudocomplement or complement of a module a with respect to K / k (II,N,l) pseudodifferent or different with respect to K / k (11,§4,2), ring of residue classes of o modulo an ideal a, ring of residue classes of a modulo an a E a.

(b) MATRICES E = (811) At

A*

= (At)-l

unit matrix, transposed matrix of the matrix A, contragredient to a matrix A. (In Chapter V the hermitian adjoint denoted A*.)

is also

(c) FIELDS Q field of rational numbers, R

C ko k

=Q

or

ko(x), kx, Kx

field of real numbers, field of complex numbers, any field, in particular the exact constant field of a function field K , depending upon whether one wishes to consider algebraic numbers or functions. In Chapter 11, however, we often designate the quotient field of an integral domain o by k, the multiplicative groups of the fields, k , K.

4

INTRODUCTION

(d) INTEGRALDOMAINS Z t = 2 or

ko[x],

3 0

0 iP

domain of rational integers, depending upon whether algebraic numbers or functions are to be considered, principal order of a finite extension K/kwith respect to t (II,$l,l), integral domain, such that it is a principal ideal domain and has quotient field k. principal order of K with respect to o (II,gl,l), integral domain of all integral elements of k = Q or ko(x) at a placep.

CHAPTER I

Linear Algebra 51. Modules in Principal Ideal Domains

1. FINITE MODULES We first consider principal ideal domains, that is, integral domains o in which every ideal a E o is a principal ideal a = oa. Given two elements a, b E 0 , there always exists a greatest common divisor (g.c.d.) d, defined by the fact that every common divisor of a and b divides d. The theorem of the greatest common divisor states that there exist two elements u, v E 0 , such that au bv = d. In principal ideal domains the theorem of unique decomposition into prime elements holds: if a = epT1 ... p: = uqf1 ... 9 )

+

are two decompositions of a into prime elements p p , qd and units e, u, then r = s and suitable indexing gives flp = ap,qp = uppp,with units up. The residue classes of a principal ideal domain with respect to a prime element form a field, as we know from the theorem of residue classfields with respect to prime elements. These three theorems will be assumed known from elementary algebra. In many cases it is difficult to verify the assumption that not only finitely generated o-ideals, but even infinitely generated ones are principal ideals. A different definition becomes handier: A principal ideal domain is one in which every finitely generated ideal is principal, and each of whose elements can be decomposed into prime elements in at least one manner. If o is a principal ideal domain in this sense, we can first prove the uniqueness of prime decomposition and then the basis theorem, the finiteness criterion, and the ascending chain theorem, as is done below. The last step shows that infinitely generated ideals can be finitely generated, assuring us that they, too, are principal ideals. NO& let o be a principal ideal domain and k its quotient field. It is known that a linear vector space of dimension n over k is isomorphic to the space k" of vectors a = (al, ..., a,) with coeRcients in k. An o-module a in k" is an additive group of vectors of k" such that whenever it contains the 5

6

I. LINEAR ALGEBRA

vector a of k” it also contains all aa with a E 0 . The maximal number of vectors in a linearly indepenedent with respect to o (or equivalently, with respect to k ) is called the rank or dimension of a. An o-module a is calledjinire if there exists a finite number of vectors p,, ..., B, E a such that every vector a E a has at least one representation as a sum

with ai E 0. The vectors pi are called a system ofgenerators of a. A one-dimensional o-module a with elements in the quotient field k is called an o-ideal of k. If a E o we call a an integral 0-ideal. Up t o now we had used the term ideal to designate an integral o-ideal in this sense, and this is often done. In this book we will distinguish the more general o-ideal from the integral o-ideal.

Basis Theorem. I f a is ajinite o-module of rank r, then there exist r vectors ai in a such that every a E a can be represented in a unique manner as a =Caiai with a, E 0 . The ai are called a basis of a. Proof. First, let Pi = ( b i l , ..., bin)( i = 1, ..., m) be a system of generators of a, and let bij = cij/h, with cij, h E 0 . Such a “common denominator” h must exist, as k is the quotient field of 0 . Due to the existence of the reprcsentation (I), the components of any vector a = (a,, ..., a,,) E a satisfy haj E 0 . Thus the components of all the vectors in a have the common denominator h. The first components a, of all a E a form an o-ideal a, (and it is no loss of generality to assume that they are not all 0). For, if a = (a,, ...) and a’ = (a,’, ...)aretwovectors,anda,a’ E 0,thenaa a’a’ = (aa, + a’a,‘, ...) €a, and thus aa, a‘al’ E a,. We remember that ha, c 0 , and that therefore ha, is an integral o-ideal, and thus by our hypothesis a principal ideal, ha, = clo with c, E 0 . We have thus shown that a contains a vector a, whose first component is a, = cJh, while the first components of all a E a are of the form ac,/h with a E 0 . This proves the contention for the case r = 1. By assuming it correct for modules of rank r - 1, we can continue as follows. For every vector a E a consider the difference a’ = a - aal whose first component is 0. These a‘ form a module of dimension r 1, which has a basis a,, ..., a, by inductive hypothesis. Thus, a,, ..., a, form a basis of a. rt

+

+

-

Up to now we have not used prime decomposition. We will need it t o prove the following:

t The symbol 7 signifies “end of proof.”

01. MODULES IN PRINCIPAL IDEAL DOMAINS

7

Criterion for Finiteness. A submodule b of a finite o-module a is finite. Let ai be a system of generators of a. Any P E b may be written as P = C b p i with bi E 0 . With a sequence P, p‘, p“, ... E b we associate a

ProoJ

sequence of o-ideals ob,, (ob,, ob,’), (ob,, ob,’, ob’;),..., each of which is contained in all the following.All are principal ideals,generated by theelements b , , = b,, b , , , b13,..., each divisible by all the following, all b,, E 0 . As an element of o can only have a finite number of divisors, this sequence must break off, so that the ideals stop increasing after a finite number of steps. Thus, there exists a 8, whose first component b, divides the first components of all the other P. Now, choosing suitable b E o , the first components of P - bp, are 0, and the proof can be completed as above by induction on the rank of a. 7 The criterion leads immediately to the following :

Ascending Chain Theorem for o-Modules. An ascending chain a, c a2 c of submodules of afinite module ends after a finite number of steps.

...

Proof. From hypothesis, the union a of all the a, is a submodule of a finite module, and therefore itself a finite module with, say, generators a i . Each of these ai is contained in some a,,, and then in all those following it. Thus, there exists an index n such that a, contains all the a i , which means that a, = a, that is, the chain ceases to ascend after a,. 7

2. THETHEOREM OF ELEMENTARY DIVISORS

If a, b are finite o-modules and b E a, then there exists a basis u, of a and elements t i E o such that Pi = t,ai is a basis of b. Furthermore, any nonzero t i divides t i + l . Several remarks will precede the proof. First, let u i , Pi be any bases of a, b. As b c a, a system of equations

holds. If ai’, Pi’ are other bases we further have

The matrices (uij),(rlj) are inverses to the matrices (uij),(uij). Now, matrices with coefficients in o having inverses whose coefficients also lie in o are called unimodular matrices; they form a group.

I.

8

LINEAR ALGEBRA

Equations (2) and (3) yield

PI' = C a t q ' 1

with (a;,) = (ui,)(ai,)(uJ

'.

The theorem of elementary divisors asserts the existence of unimodular matrices (u,,), (u,,) such that

is in diagonal form, and t, divides tz and so on. Thus, we can state our theorem as a proposition concerning matrices. The ti will concur, up to multiplication by units, with the elementary dioisors commonly defined for the matrices (ai,) and (a)). The hth elementary divisor (h = 1, 2, ...) is defined as the quotient Dh/Dh-l, where D, is the g.c.d. of all subdeterminants of order h, and Do = 1. It is clear by (4) that this is so for (aij). We must assure ourselves that (ai,) and (a;,) have the same elementary divisors. Along with the theorem we prove the following:

Corollary, The group of unimodular matrices with n > 2 columns is generated by the special matricesformedfrom the n-column unit matrix by replacing the elements at the places pp, pv, vp, vv by the elements a, b, c, d of a twocolumn unimodular matrix. For example, a b : d 1

1

(with zeros in the empty spaces). For D = Z it suflces to let

el,

Left multiplication of a matrix A with row vectors ..., c, by V,,,(: I;) is equivalent to replacing the pth and vth rows by a(,, + be, and cell + d r y , respectively. Multiplication from the right does the same to the columns. We will now demonstrate the invariance of the determinant divisor D, of A under multiplication by V,& :) from the left. Certainly for those subdeterminants not containing any elements of the rows p or v , or

$1.

9

MODULES IN PRINCIPAL IDEAL DOMAINS

containing both, there is no change in the g.c.d. Let el, ..., eh-,be a set of indices different from each other and from p and v. Using self-explanatory symbols, the subdeterminants of VPv(: f;)A = A’ arising fro& elements of rows el, ..., eh-,,p are

,...,Q h - 1 . P

= ’(CQ19

*.*7

CQh-1,

= aDUlw...Ph-l,P

aCP

+

bCV)

’

+ bDQl.”’,Qh-l.Y

,,

and similarly for those from rows el, ..., @ h - v . Thus, the hth determinant divisor of A‘ is divisible by that of A. But the converse is also true, for the matrix inverse to VPv(: f;)is also unimodular. The same argument holds for multiplication from the right, by replacing the rows by columns. Proof of the Elementary Divisor Theorem. We prove the second form of our assertion, remarked upon above, by induction on the number of elements of (aij).The assertion is trivial for one-row matrices. In’the general case, a suitable permutation of rows and columns yields all # 0; this permutation is accomplished by left and right multiplication with certain Vi,(y The matrix (aij)is then multiplied from the left and right by suitable unimodular Vli(: :) in such a way that the first coefficient a; of the product matrix becomes a proper divisor of the first coefficient a, of the former. This process breaks off when this is no longer possible, and it is the case after a finite number of such multiplications, since a,, has but a finite number of divisors. We maintain that the final a;, is even a divisor of all the aij. Our indirect proof of this is simplified by again writing ail for a;,. Say a,, does not divide azl, and set a ; , = (a,,, aZ1).There then exist a, b E o such that = aali bazi,

A).

,

,

+

where (a, 6 ) = 1. This means that there exist c, d E o with ad - bc = 1. The matrix V12(: f;)is thus unimodular, and applying it as left multiplier for (aij)replaces a, by the proper divisor a; contrary to our assumption. The same method shows that a, divides all the ail and, using right multiplication, all the a l i . This shown, the matrix (ai,) can be put into a first normal form in which ali = a i l = 0 for i # 1, using repeated left and right multiplication by suitable V,i(: y ) and V ,i(h i), respectively. We finally show that all the a,, are divisible by a,,. If, say, a i j with i, j # 1 were not divisible by a,,, we could set VIj(A :) (aij)= (a;j)and would have a;, = a,,, but ail = a,, is not divisible by a;,, which was shown to be impossible. All further transformations using left and right unimodular multiplication can be carried out with matrices whose first row and column are 1,0, ..., 0.

,

,

,,

I.

10

LINEAR ALGEBRA

The first row and column remain unchanged, while the rest of the matrix is transformed as desired. Use of the inductive hypothesis thus permits the matrix to be brought into the diagonal form asserted, while a,, remains as divisor of all the elements occurring. The theorem is proved. 7 The same argument applied, in particular, to a unimodular matrix (aij) yields the corollary to the elementary divisor theorem.

3. DUALSPACESAND COMPLEMENTARY MODULFS In addition to k" we consider another vector space over k of the same dimension n, and denote it k"*. We agree to write scalar multiplication of vectors a* E k"* by elements a E k in inverse order: a*a. We further assume that to every pair of vectors a E k", a* E k"* a product with the following properties is defined: (1) aa* E k ; (2) (alal azaz)B* = ala18*

+

+

+ a2a2B*,

+

a(B1*bl p2*bZ)= ap1*bl a/3,*b2; ( 3 ) if a E k" is such that aa* = 0 for all a* E k"*, then a = 0; if a* E k"* is such that aa* = 0 for all a E k", then a* = 0. Let pi, pi* be bases of k" and k"*. Multiplication of a = cai/3, by jl* = Cpi*bi can be written a/* =

a,Bijbj,

Bij = flipj*.

The third postulate assures the nonsingularity of the matrix ( B i j ) .Conversely, starting with such a matrix, one can define a product of vectors a = (ai) by fl* = ( b J ; it will satisfy the above requirements. Under these assumptions one calls the two vector spaces k" and k"* duu1.t With every o-module a E k" we associate its complementary o-module a* E k"* as follows: a* consists of all a* E k"* such that ~ * E O

for all a E a ;

(5)

a* is also called the complement of o. The following is evident:

a,

c a,

implies a,* 2 a,*.

Let a be a finite o-module of rank n and pi, /Ii*, a, bases of k", k"*, and a; further let ai = C mijPj * I

t By the first part of the second postulate one can consider kn* to be the module of all linear functions on k n with values in k. The rest of the postulates must then be proved.

4 1.

MODULES IN PRINCIPAL IDEAL DOMAINS

We formally define elements a,* =

11

C fij*mfi i

in k"* with indeterminate coefficients m;j and require apj* = hi, =

[1

for i = j for i # j

to hold; this can be written in the matrix form (mij)(Pifij*)(m$= (mij)(Bij)(m$)= (hi,).

For a of rank n there is a unique solution (m:) for this equation. The a,* are what is called the complementary basis to the ai;they are a basis of the complement a*. For clearly 1mz.l # 0 and thus every a* E a* may be written as a* = c a i * a i with a, E k. As a,a* = a, we have a,a* E o for all i, or equivalently aa* E o for all a € a as long as all a , E 0 . This gives the contention. The symmetry of the complementary basis gives a** = a,

(7)

so that we know complementation is an involution. 4*. NOETHERIAN RINGS Certain applications of module theory require only a finite system of generators of a module in place of a finite basis. The term noetherian ring denotes a commutative ring o with the property that every integral o-ideal a has a finite system of generators. Manipulation within noetherian rings is, essentially, similar to that within principal ideal rings. Of course the assumptions and results are weaker. Two theorems will give examples of noetherian rings which in general are not principal ideal rings. If o is a noetherian ring and a an 0-ideal, then the residue class ring o/a is also noetherian. Proof. Residue class formation o -,B = o/a maps an o-ideal b onto an &ideal 6. Conversely, every &ideal 6 corresponds to the set containing all elements of all residue classes of 6, an o-ideal b. Thus, a finite system of generators of b yields the same for 6. 7 If o is a noetherian ring, so is o[xl, ..., x,]. Proof. Of course we need only consider the case m = 1, for the variables may be adjoined one after the other. Let a be an o[x]-ideal. The leading

I.

12

LINEAR ALGEBRA

coefficients c, of all polynomials a(x) = cox" + c,x"-' + ..-+ c,, of a, of some fixed (formal) degree n, including 0, give an o-ideal c,. For if a'(x) = co'X +..- is another polynomial in a, and d some element of 0 , then a(x) - a'(x) and h ( x ) are elements of a and have nth coefficients c, - c,' and dc, , respectively. The same argument shows that the leading coefficients of all polynomials a(x) E a, along with 0, form an o-ideal c. For, now setting a'(x) = co'x"' + with, say, n' n, we find that a(x) - x"-"'a'(x)has leadingcoefficientc, - c,'. Let c,,, c,,, ... be a finite system of generators of c, and a,(x) = colx"*+ a,@) = co2x"z ... be polynomials in a with these generators as leading coefficients, Say that n is the highest degree occurring among them. We start forming the desired system of generators of a by choosing the polynomials b,(x) = a,(x)x"-"',certainly contained in a. Now, if d(x) is a polynomial in a of some degree m >= n, our construction of the b,(x) permits us to find e, E o such that d'(x) = d(x) - x"-"Ce,bi(x) is of degree m' < m, and continuation of the process yields a polynomial of degree less than n. We need now only extend our b,(x) by a finite system of polynomials b(x), with leading coefficients that generate each of the ideals c,, m = 0, 1, ..., n - 1. This gives a finite system of generators of a. 7 In noetherian rings we have the following: + n e e ,

Ascending Chain Theorem for 0-Ideals. Eoery ascending chain a, t a, c ... of 0-ideals breaks off aferjinitely muny steps. The proof is identical to that for o-modules of $l,l. Let o be a noetherian domuin (integral domain and noetherian ring), and let k be the quotient field. We then have the following: Let of be some basis of k", and write the vectors a = z a p , as component vectors (a,).An o-module a in k" is finite if and only if there exists a nonzero h E o such that for all a E a we have hal E o for every i. (We think of h as a common denominator for the a, .) As in $l,l, this leads to the following: Finiteness Criterion. A submodule of a jinite module is jinite. Proof. As a may be replaced by ha, it is no loss of generality to assume h = 1. For n = 1 the contention is identical to that of the ascending chain theorem. Assume the theorem proved for n - 1, and consider the o-ideal formed by the first vector components a,. Choose somefinite system of vectors a,, a,, ... E a whose first components generate this ideal. For any a E a our construction permits us to choose b, E o such that z' = a - b,a, - b,a, - -.has zero for its first component. But the inductive hypothesis assures the finite generation of the ideal, in a, of vectors with vanishing first components. The assertion is proved. 1

81. MODULES IN

5*.

PRINCIPAL IDEAL DOMAINS

13

A FURTHERBASISTHEORFM

As before, let o be a principal ideal domain with quotient field k . The ring o[x] of polynomials in one variable x with coefficients in o is no longer a principal ideal domain unless o = k . A finite o[x]-module a of rank n, corztained in a linear space L over k(x), has a basis consisting of n elements if and only if it satisfies the following condition: i f a and b are mutually prime elements of o[x] and a E L such that au E a, ba E a, then a E a. The necessity of the condition is obvious. To prove the sufficiency, consider the principal ideal domain of = k [ x ] and the 0'-module a' of all finite sums c a i a i with a , E o', ai E a. Now, a' is a finite 0'-module of rank n and thus, by §l,l, it has a basis a,'. Since k is the quotient field of 0 , there exists an a E o such that aa,' E a. But this is also a basis of a', so that we lose no generality in assuming that ai' E a. If the ai' are not yet a basis of a, then there exists a vector ,

in a, with thef, not all divisible by p . There is even such an a for which p is a prime element in 0 . We consider the polynomials f i with respect to the module p, that is as polynomials over the residue class field o/p. As such they have a g.c.d., say d . Then, fi = dg, + pu,, with polynomials ui and g , , the latter of which are mutually prime modulo p. Substitution into (8) yields a =d

1 P

gpi'

+ c up,'

= da'

+ c uia{

The vectors du' and pa' are in a, and since d and p are mutually prime, we see that u' = l/p giai' is in a, by our assumption. This shows that we may assume, henceforth, that thefi in (8) are mutually prime modp. We can therefore find polynomials g , , u in o[x] such that

c

We mdst study the solution of the diophantine equation (9) in detail. We again calculate within the polynomial ring (o/p)[x]= o[x]/p. Transform the one-row matrix composed of thefi by subtracting multiples of one element from another repeatedly (Euclidean algorithm) until the row I , 0, 0, ... remains. These operations correspond to the right multiplication by a unimodular matrix (g,,) = G in (o/p)[x]: (fi,

...,f")G = (190, .*.,0).

14

I.

LINEAR ALGEBRA

And the first column gl, ..., g,, solves (9) as a congruence mod p. By the corollary in $1,2, G is a product of the special matrices

l m 0 1

1 0 1

. '

1

,

etc.

We then construct a matrix H with coefficients in o [ x ] by substituting, in G, the 0 and 1 of k for the 0 and 1 of o/p, and any representative of its residue class in o [ x ] for a nonzero element of (o/p)[x].The matrix H is unimodular in o [ x ] , and its residue class modulo p is G = (gij). Designate the elements of the first row of H as gi; they satisfy (9). The equation a,' = hij$ can now be used to introduce a new basis a': for the module spanned by the a,'. With (9) we can write the elements (8) in the form

1

withf,'

E o [ x ] . The

vector

also lies in a. The elements a", a;, ..., c t i form a basis of a larger module contained in a. As we presupposed a to be a finite module, a finite number of these steps must suffice to arrive at a basis of a. Exercise (nontrivial). Let x and y be independent variables, p a prime element in 0, and pl, p2 two linearly independent vectors. The o[x,y]module a generated by the vectors pl, p2 and I/p(xpl + y p 2 ) satisfies the conditions of the above theorem for o[x, y ] instead of o [ x ] . But it has no basis consisting of only two elements. 52. Systems of Linear Inequalities 1. MINKOWSKI'S POINTLATTICE THEOREM As always let R" be the n-dimensional (affine) vector space over the field R of real numbers. 2" will signify the module or, using geometric language, the lattice of vectors a = (al, ..., a,) with components a, E Z.

Minkowski's Point Lattice Theorem. Let '3 be an open convex region in

$2.

15

SYSTEMS OF LINEAR INEQUALITIES

R" having the origin as centerpoint (i.e., if the vector u E% then also - a E %), which has the volume V in the sense of Riemann.? Then including u = 0, there are at least 2-" V vectors in Z" n 93. Proof. Let N be a natural number and Z," the lattice of all vectors a = (ai) such that Nui E Z; this point lattice consists of cubes of edge length N-'. We take a small fixed positive constant E and form two new regions similar to % but shrunk in the ratios 1 : 4 and 1 : (4 E ) and denote them 4% and (4+ E)%. The numbers bN(4)and bN(++ E ) of cubes of the lattice Z," lying completely within these two regions, respectively, multiplied by the volume N - " of each cube yield, essentially, the volumes of these regions:

+

N-"b,(+) = (f)"v - )IN

N-"bN($

+

E)

= (4

+ E)"V -

VN',

where q N , )I,' approach zero as N increases. These numbers of cubes are essentially the same as the numbers of points of the lattice Z," in the regions, for one can associate each cube with that corner point which has the smallest coordinates. Counting thus is correct, except in the boundary zone. The number of points a, in Z," A 4% can certainly be approximated, for sufficiently large N, as follows:

5 UN 6 b N ( i +

bN(4)

Thus, for any given small positive 2-"V - ~1

and a sufficiently large N we have

s N-"uN s 2-"V +

EI.

On the other hand, Z," and 2" are additive groups, the latter being a subgroup of the former with index N". The a, points in Z," n 4% are distributed among the N" residue classes of Z," mod Z", and thus clearly some residue class must contain at least N-"aNpoints: denote them ai ,i = I , 2, ... . As % is symmetric, -ul E +%, and as '3 is convex, even +(ai - ul) E *% for all i. Thus ai - a1 E 93. Since all ai lie in the same residue class mod Z", we have also ui - u1 E Z". The number of these vectors is 2N-"a, 2 2-"V for every > 0, which proves the theorem. 7 We shall need the following application: Given the system i

n

I

of inequalities, where the mij are complex, and the c j positive real numbers such that if the m i j , of a column are not real, the complex conjugate column mij, = E i j , occurs, with cj2 = c j , . Further, assume ( 4 2 ) " n j c j > Idet(mii)l

t The last assumption always holds, in fact, for convex regions. We need not take the trouble, however, to prove this.

16

I.

LINEAR ALGEBRA

(absolute value of the determinant), with 2r2 being the number of nonreal columns of (mi,). Then (1) has a nontrivial solution in rational integers xi. Proof: An open convex symmetric region is given by (1) in the space R" with coordinates x i . Let the nonreal columns of (m,) be mij2 = mi,, - im;,. mi,, = mi,, + im;, Consider the real n-column matrix formed by the real mi, and mi,,, in$, ; the absolute value of its determinant is 2-'21det(mij)(. A linear transformation with this matrix puts the inequalities (I) in the forms 9

lx,'l < c, and xj:

+ xj; < cj,.

n,

This defines a region of volume 2"-2'2n'2 c,. Thus, the region defined by (1) has the volume V = 2"(n/2)'2 c,ldet(m,,)l-l.

n I

By our hypothesis we have 2-"V > 1. By the lattice point theorem there is more than one solution to (l), and thus at least one nontrivial solution. 7 2*. SIEGEL'S PROOF

A second proof of Minkowski's theorem, which is in fact a strengthening of the part essential in application, is due to Siegelt: Let % be a convex region centered at the origin, which contains no further lattice points. Then

where V is the volume of %, and the sum on the right is taken over all lattice points p # 0 E 2". The consequence is, then, that such a region cannot exist with V z 2". For the proof one defines a function q(() in R" by setting q(t)= I when) 0 otherwise. ever the point t with coordinates xi lies within %, and ( ~ ( 5= Further, kt summed over all p E Z". Now, f ( ( ) is periodic in all variables with period 1 and can thus be expanded in the (convergent, at least in the mean) Fourier series

Uber Gitterpunkte in konvexen Korpern und ein damit zusammenhiingendes Extremalproblem, Acta Math. 66, 301-323 (1935).

$2.

17

SYSTEMS OF LINEAR INEQUALITIES

The coefficients are

cP =

/

1

/ f ( < ) e - 2 x i c Pd x 1

dx,,

0

(3)

and we have the relation of completeness

/

0

/f(t)'

dx,

dx, = C P IC,,~',

(4)

which, properly analyzed, will yield the theorem. First let g ( t ) be some integrable function with period 1 in all x i . Then,

Let %, be the intersection of the cube 0 5 xi 5 2 with the region R translated are disjoint for p # v, for if t were in both, then t + 2p by -2p. !XPand and t + 2v would be in 3. By the symmetry and convexity of R we would have +(t + 2 p - t - 2v) = p - v € 3 ,contrary to hypothesis. The integrals to the right in ( 5 ) are actually taken over the regions 9IP with the integrand g ( + t ) . The substitution t + 5 - 2 p is applied in the integrals, leaving g ( K ) invariant because of the assumed periodicity. The translated regions RP form, together, the region R,and one has

This formula is applied on the one hand to (3), cP = 2 - n / - ~ . / e - x i ~d"x ,

w

... dx,,,

in particular co = 2-"V,

while on the other

For p = 0 the integrand to the right has the constant value 1, while for fi # 0 it is = 0. For, cp(t 2 p ) # 0 and t E R would, with the symmetry and convexity of %, imply that )(< + 2 p - t) = p ER,while R contains no such lattice point. Thus,

+

This result, inserted into (4), gives (2).

18

I.

LINEAR ALGEBRA

3. GENERALIZATION TO FUNCTION FIELDS The theorem of 52,l has its counterpart in function theory, which strengthens the resemblance, mentioned in the introduction, between numbers and functions. Let k = ko(x)be the field of rational functions in one indeterminate over the constant field ko , and li the field of formal power series in x - l : m

f(x)=

1

avo#O,

vo $ 0

v=vo

with coefficients in ko. The negative of first exponent power series is called the degree G(f(x)).We have

+ d x ) ) s max(G(f(x)), G(g(x))), G ( f ( 4 g ( 4 ) = G ( f ( x ) )+ G ( g ( 4 ) .

G(f(4

-yo

of such a

(6)

This concept of degree corresponds to the usual one for rational functions, when expanded in powers of x-'. In this connection the following fields and integral domains correspond to each other: Q, R, 2 and k = ko(x),&, ko[x]. In place of the absolute value in R we use the degree of a function in E. The problem analogous to (1) and its solution are as follows. Let (mij)be an n-rowed nonsingular matrix with coeficients in E and (m;) be its contragradient. Further, let there be given n rationul integers d j . The solutions of

in vectors xi ,xi* with components in ko[x]each form a linear space of finite dimension over ko . The equation

c d, - G(lw,I) n

1-1*=

j= 1

+n

(9)

holds between the dimensions I and I* of these spaces. For each pair of solutions x i , xi* of (7) and (8) we have n

c xixi*

i=l

= 0.

(10)

This theorem supplies information concerning the number of solutions to (7) similar to that which the theorem in §2,1 supplies for (I), as (9) gives an approximation from below because 1* 2 0. The link between the problems

$2. SYSTEMS OF LINEAR INEQUALITIES

19

(7) and (8) and the exact formula (9) are, on the other hand, characteristic of function theory and have no direct analogies in number theory. Formula (10) is, incidentally, a simple consequence of (7) and (8). For, as the matrices ( m i j )and (mz)are assumed contragredient, we have

while with (6) through (8) the degrees satisfy

G

1

C xix? 5 max(dl - 2 - d,)

( i

=

-2.

Nevertheless, the term on the left in (10) is a polynomial which cannot have negative degree and must therefore be 0. In proving (9) we first assume the m i j all to lie in k = k,(x). This proof, which specially covers the case of terminating power series for the m i j ,will be given in the paragraph after next, where concurrently a new interpretation of the theorem will evolve.? At this point we will only derive the general case from this special one, although we will make no use of it below. To this end we set

c xi * i

* -- yl *,

mij

that is, xi* = 1 mrjyl* 1

and, instead of (8), consider the equivalent system

G(y:) 6 -2

- dl,

C mijyi*E kOCxl.

(11)

1

The theorem is assumed to hold for terminating power series, and the general mij are approximated by their partial sums rn;;):

mil = mi;) + r$), the remainders r$) starting with the power x-' a t the earliest. The equation

(mf;Y)*)= ((m!J))')-' is introduced. The matrix identity M - ' to (mil) and (m{J))yields

- N-'

= M-'(N

- M)N-'

applied

(mc)= (mi;)*)- ( m i J ) * ) ( r { ~ ) ) f (=m(mi;)*) ~) + (r$J)*), with rfJ)* as an abbreviating symbol. The powers of x-' with which the sequences r$)* start increase unbounded as v increases; thus the mi;)* also approximate the mij*.

t Another proof is given in M.EICHLER, Ein Satz uber Linearformen in Polynombereichen, Arch. Math. 10,81-84 (1959).

I.

20

LINEAR ALGEBRA

Let I('), I(')* be the number of linearly independent solutions xi('), XI")* of (7) and (11) with in place of (mij). From some certain v onwards the degree G(Im$'I) ceases to change. Further, from ml;)yy)* = xi')* we have the uniform boundedness of the degrees of the polynomials xi")*, the same argument holding for the x!"). Hence the degrees of the power series c x / ' ' t $ ) , cx{")*rj;)* remain under a certain bound, which goes to - co as v increases. As a consequence the xi'), xi')* are solutions of (7) and (1 1) for some sufficiently large v and then I(') = I, I(')* = I*. As I('), I(')* satisfy Eq. (9) the theorem is proved. 7

(MI;)

c

$3. Linear Divisors

1. BASICCONCEPTS We now come to a first and very important application of the methodical principle sketched in the introduction. First, let k = k,(x) be the field of rational functions in one variable over some constant field ko . Every rational function f ( x ) E k,(x) can be decomposed in a unique manner into prime polynomials p = p ( x ) : A x ) = Y JJ PV? Y E k, * P

The p which occur here are, in case k, is algebraically closed, all of the form p = x - with E k, , while in general, prime polymonials of higher degrees occur. In place of the i, mentioned in the introduction we have here the integral domains i, of those rational functions f ( x ) in whose prime decomposition p occurs with a power v ( p ) 2 0. In addition to the i, we have the integral domain i, of thosef(x) with degrees G ( f ( x ) )5 0. The i, can be defined not only for k,(x), but even in general for all such fields k in which a unique prime decomposition is possible. These include, in particular, the fields k = Q and k = k,(x,, ..., x,,,) in m > 1 variables. It is easy to see that the i, are always principal ideal domains. In the last case one introduces i, as the integral domain of functions having a nonpositive degree (the degree of the quotient of polynomials being the difference between the degree of the numerator and the degree of the denominator). One could also introduce several integral domains i, i, which would then consist of those functions having nonpositive degrees as functions of some xi. The proof that even here i, or i, are principal ideal domains may be left as an exercise for the reader, as these fields will play no part in what follows. He must convince himself that every function can be expressed in the form f ( x i ) = e(xi)f, where e(xi) is a unit of i, or imi , respectively, and y is any function of degree one. Although we are concerned, in the end, only with the field k = k,(x), we

0; when v(ai) 6 0 they are satisfied by all polynomials ai = q(x) of degree - v(ai). Hence there exist

linearly independent multiples with respect to ko . The basis ai* complementary to ai is a basis of a,* = a,', and the basis xv(ai)ai* complementary to x-'(")ai is a basis of am*= x'a,'. Hence a multiple a' of a is represented doubly as a' =

ai*ai* =

C bi*~v(a')-2ai*,

a,* E i,

bi* E i,

.

Using the same argument, among these there are

that are linearly independent. Equations (16)-( 18) yield the asserted equation (12). 7 4*. THE RIEMANN-ROCH THEOREM AND THE MINKOWSKI LINEAR FORM

THEOREM Let ai be an arbitrary basis of a, and ami a basis of a,. As the basis wi we use oi = tli. Let (19) ai= mijam J *

1

As above, where we proved (16), this yields G(a) = G(Imij1).

(20)

The multiples a = 1 xiai of a can now be determined by the conditions x i E i and Cxirnij~i,. The latter condition can, due to the meaning of i,, be formulated as G(C ximil) 5 dj = 0. Thus /(a) turns out to be the number of linearly independent solutions of the special system of inequalities (7) in $2. Also /(a') can be similarly interpreted. Let ai* be the basis complementary to a i , and thus a basis of the module a,* = a,', and aZi the basis complementary to a,[, i.e., a basis of x%,'. Then, using the argument of §3,1 (proof of (7)), similarly to (19) ai* =

C m$aZj,

(21)

where (mz.) is the contragredient matrix to (qj). Here the multiples of a' are a' = C xi*ai* with x i E i, xi*m$ E x-'i,. As above, the latter conditions can be replaced by G(C xi*m$) 5 - 2. Thus, I(a') is the number of solutions of §2,(8), and the Riemann-Roch theorem coincides with the linear form theorem of §2,3.

$4. TRACES,

NORMS, AND DISCRIMINANTS

27

Conversely, let ( m i j ) be an arbitrary nonsingular matrix. Choose n arbitrary linearly independent vectors ami in k" and define the ai by (19). Call the i,-modules and i-modules, generated by a , and a i , a, and a,, respectively. Here the latter is the intersection (2) of the i,-modules a, with the same basis a i . For the complementary bases (21) holds, and the number of solution, I, I* of $2.3 (with dj = 0), coincides with ,(a) and l(a'). It is seen, incidentally, that the assumption dj = 0 in $2,3 does not inhibit the generality, for one can replace mij by mijx-dJ. $4. Traces, Norms, and Discriminants 1. REPRESENTATIONS BY MATRICES

Let K be an extension of the field k of finite degree [K:k] = n. Whenever we wish so accentuate this fact, we write K/k (k over K). Confusion with residue class formation is impossible. Using a basis ol,..., onof K/k one arrives at a representation of K by n-row matrices in k. For every a E K one has amj = C o i a i j ,

ail E k .

i=l

Abbreviating the one-row matrices with elements oi and ami as o and ao, and setting ( a i j )= A, this system of equations is written as am = o A .

The matrix A corresponding to a is the zero matrix if and only if a = 0. For another element /3 E K with /3w = oB, (a

t B)o

tB)

holds (where the symbol indicates addition or multiplication). The mapping a + A is thus an isomorphic mapping of K onto (or a representation by) a ring of n-row matrices in k. With a change of basis w --+ o'= wC, C being a nonsingular matrix over k, it is well known that A goes into A' = C-'AC. Letting E denote the unit matrix and x a variable, the determinant IxE - A1 = X" - s(A)x"-'

+ - + (- l)"n(A) *.*

is known as the characteristic polynomial of A or also of a. It does not change when the basis changes. The coefficients n

s(a) = s(A) =

aii, i=1

n(a) = n(A) = laill

I.

28

LINEAR ALGEBRA

are called the trace and norm of a. It is easy to verify the sum and product formulas s(a + B) = s ( 4 + s(B), n(aS) = n(a)n(B). (1)

If a E k the corresponding matrix is A = aE, and thus s(a) = na,

n(a) = a",

for a E k.

s(aa) = as(a)

(2)

In particular let the minimal polynomial x" - alx"-l +

n

- ... f a,, = n ( x - ai)

(al = a)

I=1

of a E K be of degree n. Then 1, a, ..., an-' form a basis of K/k, and the corresponding matrix has the form

which implies n

n

s(a) = a , = C a I , I=1

n(a) = a, = n a I . I=1

2. THETRANSITIVITY FORMULAS

Let an intermediate field L be given: k c L c K. One can take traces and norms both with respect to L and to k, the difference being clearly expressed in an obvious terminology. We will prove the transitivityformulas

sK/k(a)= SL/k(sK/L(a)),

nK/k(a) = nL/k(nR/L(a)).

(3)

Let wl, ..., onand ql, ..., q,,, be bases of KIL and Llk. Then, o l q i , ..., wnql; ...; q q , , ..., WJ,,, is a basis of K/k. We again express these bases as one-column matrices o,q and o x q, where o x q consists of the products wiqj in lexicographical order. Now, first

a o = oA'

(4)

holds with an n-row matrix A' over L, and then, for every coefficient a I j of A', one has aijq = V 4 j (5) with an m-row matrix A:, over k. Both equations can be combined to

a ( o x q) = (0 x ?)A,

#4. TRACES, NORMS, AND DISCRIMINANTS

where A is the mn-row matrix A = (A!,

A’’,

:::

29

Ajn)

...

4

n

Its trace is

c s(Ai). n

SK/k(a)

= s(A) =

i=1

By

(9,using the sum formula (l), this comes out to

The sum on the right, C aii,is equal to sKIL(a), according to (4). Thus the first of the formulas (3) is proved. By interchanging columns and adding multiples of one column to another one can, as is well known, put the matrix A ‘ = (aij) into triangular form with zeros above the principal diagonal. This leaves the determinant invariant: A 1

Now, the m-row matrices A$ are representations of the aij . Transformations of the matrix A = ( A i j ) corresponding to those of A’ = (aij) bring it into the form

s; I B=

with /Iift = ftBi

*

B:

and then

3. THEDISCRIMINANT With any n elements wi E K one associates the determinant

No,)= No)= IS(0iwj)l

I.

30

LINEAR ALGEBRA

and calls it their discriminant. Under the change of basis o o’= oC with a matrix C over k we have, for the matrix of this determinant,

(s(otlq9)= c’(S(oio,))c,

so that D(w’) = D ( w ) ~ C ~ ’ (u’ = wC).

Thus it is clear that D(o) = 0 whenever the wi are linearly dependent, although the converse does not always hold. Now let the configuration of #4,2 prevail. We want to prove the transitivity formula DK/k(W V) = nL/k(DK/L(o))DL/k(q)cK’L’. (7) To avoid confusion we use subscrpits to denote the extension which is meant. Equations (2) and (3) yield q) = IsK,k(wi~i‘w,‘lj’)l = ISL/k(?i’SK/L(wioj)qlj’)l, where we are dealing with an mn-row determinant whose columns and rows are each given by double subscripts ii‘ and j j ’ , respectively. Into this formula by the basis, i.e., we substitute the representation of sKIL(oio,) DK/k(W

~ ~ / d ~ i=~@ij* j ) q

For any fixed pair of subscripts i, j we have the equation (SL/k(‘ti’SK/L(o~,)q,~))

= (SL/k(qi’qj’))sij = D s i j

with the m-row matrices D and Si,. These matrices, ordered into an n-row x q). Thus, now scheme, yield the matrix of DKIlr(o q) = ID/”The S, are representations of the elements ail = SKlL(oiw,)of L. Elementary transformations of the n-row matrix (aij) over L and the corresponding transformations of the mn-row matrix (S,,)over k lead, as in §4,3, to DK/k(a

lSijl = n L / k ( l a i j l )

= nL/k(DK/L(m)).

Thus (7) is proven. 7 4. SEPARABLE AND INSEPARABLE EXTENSIONS

If K/k is separable there exists, as is known, an (primitive) element 9 that generates K/k. Let 9 = 9,, ..., 9, be the elements conjugate to 9. Then 1 ... 1 1 g1 ... 9”- 1

D(1,9,

a * * ,

gn-l)

=

91

91-1

... 9, ... . ...

9;-1

. . . *

1 9,

... ...

...

1

9;-1

w. TRACES, NORMS, AND DISCRIMINANTS

31

As the value of the Vandermonde determinant on the right is known, we now have D(1,9,..., v - 1 ) = (9,- 9,)?

n

i 1 .)

5. THEFUNDAMENTAL DOMAIN

Consider the case n = 1, in particular. The modular group by the three elements u - , = ( - 0l

-10)).

J = ( -1O 0I),

r'

is generated

7 - q ; ;).

Under the corresponding fractional linear substitutions of the complex upper half plane, U-, is the identical mapping. By a fundamental domain iJ of r' in sj', a point set with the following two properties is meant: ( I ) to every T E sj' there exists exactly one point ~7E 5

40

APPENDIX TO CHAPTER I. THE THETA FUNCTION

and one M Er' with M(z) = a; (2) in every neighborhood of every Q E there are inner points of 5. Figure 1 depicts a fundamental domain of r' in fi', the so-called modular triangle. Here the left half of the boundary, up to and including the point a = i, is to be included in 8, but not the other boundary points. Proof. Using T, T - ', and J, further " equivalent " triangles are attached to 8 on its three sides. Further, the substitutions TJ, (TJ)', JT, and (JT)' represent FIG. 1 rotations of angles 2x13 and 4x13 about the points 112 i&2 and - 112 + i,/i/2 of 5.The images 8, T - ' g , JTS, J T T ' S , (JT)'S, (JT')T-'S fill out theentireneighborhoodof theleft corner. The neighborhood of the right comer can be covered similarly. The images ME for all M = Ta1JTa2 Tancombine smoothly and without gaps to a subregion fi' of fi'. For if M,S are the neighbors of 8 and M is arbitrary in r', then MM,S are the neighbors of Mg. We must now show that fi' = fi'. Were fi' c fi', then fi' would have a boundary point zl. Let z be an inner point of fi' near zl. There exists an M = Y 1 J T a 2 such that a = M(z) lies in 5.As M is a hyperbolic motion, the hyperbolic distance between the points c1 = M(z,) and a is the same as between the points zl and z. This would imply the existence of a point Q of 8 and a neighboring point a1 belonging neither to 8 nor to a neighboring region. But this is impossible. The consequence is that every point of 8' can be transformed to a point of 8 by some M = Ta1JTa2 . The Mg with M = Ta1JTa2 cover fi simply. Indeed each Mg is surrounded by the three neighbors MT-'iJ, MTS, MJS which, along with Mg, cover the tleighborhood of MS simply. Since fi is simply connected the covering can also be but simple in the large. As a consequence, no point of 8 can be transformed to another point of 5 by such an M.It is only possible that a point of 5 remains fixed under such a transformation, and this is the case for the points i(- 1 and i under the substitutions JT, (JT)', and J. This implies that 8 is a fundamental domain for the group generated by these substitutions. The last section showed that this group coincides with the entire modular group r'. The proof is complete. 7 The discussion also yields the fact that those substitutions M,which transform the fundamental domain into a neighboring domain generate the group r'. This clearly holds true for all groups which have fundamental domains. On the other hand, a system of generators of a group need, by no means, correspond to a fundamental domain in this sense. -1

-+ 0 +

1

+

9 . -

+ n)

41

$1. THE SYMPLECTIC GROUP

The existence of a fundamental domain for r"is proved by Siege1.t It is difficult to describe because of the behavior of the subgroup of the U, . There is, however, even an arithmetically distinguished system of only four generators for r".$ 6. THETHETA FUNCTION

We introduce a new notation before defining this function. T = ( t i j )will always denote an n-row symmetric matrix with complex coefficients whose imaginary part is positive definite in the sense of the beginning of this section. Also, m, x, and y will be one-column matrices with the entries m, , xi, and y i (i = 1, ..., n), respectively. The mi are always rational integers, the x i and y i arbitrary complex numbers. We use the abbreviation n

T[m] = m'Tm

n

zijmimj,

=

x'y = y'x =

xiyi .

i= 1

i,j= 1

The function announced in the title is 9(T, x , y ) =

1exp(niT[m - y ] + 2nim'x - nix'y),

(1 1)

m

where the summation is over all integral m. The series, along with all its partial derivatives with respect to all variables, converges absolutely and uniformly in every compact domain in which the imaginary part of T is positive definite. In the next section we shall prove the reciprocity formula 9(T, X, y ) = I - iTl-1/29(-T-1, y, -x).

(12)

The square root in (12) is to be taken positive when T is purely imaginary. In general the sign is found by analytic continuation of the function liTI-'I2 in Sj". No ambiguity can arise, as Sj" was shown to be simply connected. A simple calculation directly from the definition yields the functional equation for a unimodular V 3(T, X, y ) = 9(VTV', V X , V * y ) ,

V* = (I")-'.

(13)

Finally, for an integral symmetric matrix S,

W, x, y ) = exp[ - niS(S)'y]S(T

+ S, x + S y - 5(S), y )

(14)

t Einfiihrung in die Theorie der Modulfunktionen ersten Grades, Math. Ann. 116,617-657 (1939); also Symplectic geometry, Amer. J. Math. 65, 1-86 (1943). $ L. K. Hua und I. Reiner, On the generators of the symplectic modular group, Trans. Amer. Math. SOC. 65, 415-426 (1949); also H. Klingen, uber die Erzeugenden gewisser Modulgruppen, Nachr. Akad. Wiss. Gottingen Math.-Phys. KI. Ha, 173-185 (1956).

42


holds, where ( ( S ) is the one-column matrix with entries j s i i (S = (sij)). To prove (14) we must, because of (1 I), verify

fT[m- y]

+ ( m - +y)'x = J(T + S)[m - y l + ( m - jy)'(x + S y - ( ( S ) ) - j((S)'y mod 1.

A quick calculation reduces this to

jm'Sm - ((S)'m = 0 mod 1. This congruence arises from fm'Sm = f

CI mlzsll= f C misii= &S)'m mod 1. i

The functional equations (12) to (14) can be combined to one relation: 9(T, X, y ) = ICT

+ DI-'"

x exp[ni(A, - &JMz)]S(M(T),

Ax

+ By - fs,',

CX

+ 0~- $qM1)(15)

with A, = 0; here t M , q M , and lM= (5MqM) are integral row vectors and z is the column vector (:). In the cases M = J and M = U,, is even the zero vector, while for M = T,, q,,, is also zero and tM = 2((S) lies in the

(,,,

residue class mod 2 of the vector tTs introduced in §1,4. The functional equation (15) holds for all M E r". Here A,,, is a rational q,,,, and = ((MqM) are integral number with denominator dividing 4. row vectors, representing the residue classes introduced under these notations in §1,4; when is changed for another representative, A,,, must be changed accordingly.

(,,,,

cM

cM

Proof. One first verifies that the right side of (15) goes into itself under an arbitrary variation of CM mod 2 and a suitably chosen variation of A,, . It suffices to do this for the unit matrix M. As r" is generated by J, U,, and Ts and (15) holds for these elements, one must verify the following three equations (where MIMz = M):

(CiMz(T)

+ DJ(CzT + 02)= CT + D,

(16)

~ M , J M+ z ~CM, JMiMzz = CM JMz, CMzMlt

+ [ M I= CM.

(17)

Equation (16) is easy to check. The second equation follows from (17). Equation (17) is identical to (9); however, first view (17) as a recursion formula for (, as a vector with integral coefficients, and then remember that CM in (15) can be arbitrarily changed up to mod 2. Finally, one can also compute I , by a corresponding recursion formula; it is uninteresting in this connection though. 7

81. T H E SYMPLECTIC GROUP

In the case M equation

E

0"one can set

9(T, X, y ) = x(M)ICT

tM = q M = CM

+ DI-'l29(M(T),

AX

43

=0

in (1 5 ) arriving at the

+ By, CX + Dy),

(18)

where x ( M ) is an eighth root of unity. The determination of x(M) is difficult, because it requires the determination of the square root ICT D1-1/2, not only for M = (," but also for all other matrices which occur while M is traced back to the generators of the group 0". The function 9(T) = 9(T,0,0) as any funciton with certain regularity properties satisfying the functional equation

x)

+

9(T) = x(M)(CT + DI-"29(M(T)),

M E 0"

is called a symplectic modular form of the dimension -4 with the multiplier system x(M). The solutions of these functional equations with the general factor ICT + DI-' on the right are modular forms of dimension -k. For k = 0 one speaks of symplectic modular functions. Those modular functions of the entire group r"with multiplier system ~ ( 2 = 1, which are meromorphic analytic functions for all finite values of the variables, form a finite algebraic function field in fn(n 1) variables, if n > 1. t In the case n = 1 an additional assumption on the behavior of the functions as T = t approaches the real axis is needed; see Chapter IV, $4. For n > 1 these modular functions do not fall within the scope of this book.

x)

+

7. PROOFOF THE RECIPROCITY FORMULA The proof of (12) is carried out under the additional assumption that T is purely imaginary. As an analytic function in the complex variables t i , , x i , y , is being dealt with, the functional equation continues to hold under analytic continuation. In this manner the sign of the square root in (12) is simultaneously fixed. The function 9(T, x, y)exp(-niy'x) = =

exp(inT[m - y] - 2ni(y - m)'x)

~

m

1exp(inT[y - m - T-'x] m

- niT-'[x])

(19)

is periodic of period I in all the y i , and can therefore be expanded in the Fourier series 9(T, x, y) exp( - niy'x) = c,. e~p(2nim'~y) (20) m'

with coefficients

t C . L. Siege], ober die Abhangikeit von Modulfunktionen n-ten Grades, Nachr. Akad. Wiss. Gottingen Math.-Phys. Kl., 257-272 (1960).

44 c,.


=/.:*STexp(niT[y - m - T-'x]

- niT-'[x]

- 2nim"y) d y , ... dy,.

(21) The proof consists of the evaluation of this integral. First, summation and integration may be interchanged. Then, the substitution y - m - T-'x = y' T-'rn' is carried out in each summand. The argument of the exponential function in the integrand then becomes

+

niT[y'] - niT-'[m'

+ x] - 2nim'm',

and the last summand can be dropped off. Now summation and integration can be combined, so that one finally integrates over the entire y'-space. Setting f+ao

J;;

f

Y=J

exp(niT[y']) dy,'

dy,',

we have, by (20),

9(T, x, y) = y

exp(-niT-'[m' m'

= y 9( -T-',

+ x] + 2nim"y + nix'y)

y, -x).

The positive definite quadratic form -iT[y'] becomes a sum of squares under a linear substitution with determinant I -iTI -I/'. This implies y = l-jTl-1/2S...Sexp[-n(y,2

+ + y:)]

dy,

dy,

completing the proof. 7 §2*. Theta Functions for Quadratic Forms 1. SIMPLE GAUSSIAN SUMS

In the following we use the abbreviating notation e(x) = elnix.

(1)

By a simple Gaussian sum we mean the expression G(a, b) =

1

rmod b

e(ar'/b)

(2)

where a and b are relatively prime rational integers, and r runs through a complete residue system mod b. For odd b G(a, b) = ( 4 b ) G(1, b), (3)

52.

THETA FUNCTIONS FOR QUADRATIC FORMS

45

where (a/b)is the Legendre symbol and G(1,b)= k d ( - l ) ( b - 1 ) ' 2

b,

(4)

More precisely,

for b for b

G(1, b) = {i$:

1 mod 4, 3 - 1 mod 4. 3

(5)

The proof starts with a reduction. Let b = b,b2 with relatively prime b,, b , , so that integers cl, cz exist with

bit, + b 2 ~ 2= 1. Set r = b,clr

+ b,c,r

= blrl

+ b,r,

If rl and r2 run through a complete residue system mod 6 , and b,, respectively, then r runs through a complete residue system mod b, and conversely. Thus, G(a, b) =

C

e(a(bici + b2~2Xbiri+ b2r2),/b) = G(aci, b2) G(ac2, bi).

rmodb

Assuming (3) to be proved for both factors on the right, one has G(a, b) = (a/bl)(a/b,)(cl/b,)(c,/bl)

G(1, bl)G(1, b2).

By (6) this gives G(a, b ) = (a/b)(bi/b,)(b,/bi)G(1, b i ) G(1, b2).

Further, if one assumes (5) to hold for both Gaussian sums on the right, then (5) is verified for the left side using the quadratic reciprocity law. Hence one need but prove (3) and (5) for powers of primes. Let p be an odd prime and b = ph, h > 1. Set

r = r,

+ ph-lr,

and let r l , rz traverse a residue system modp"-' and p, respectively. If rl f 0 modp, then 2r1r2 runs through a complete residue system m o d p along with r, so that one has

C

~ ( a p, h ) = p

aOmod p rI mod ph - 1 rl

e(ap-hr12)+

C

+ Omod P rl mod

e(ap-"r12) C e(ap-'r2).

Pi

r2 mod p

ph-1

The second term is clearly 0. This leaves

G(u, p") = p G(a, p"-,). If (3) through (5) hold for the Gaussian sums on the right, then they also hold for the left side. It thus suffices to prove the equations asserted for a prime b = p . If c is

46


any quadratic residue mod p , then clearly G(ac, p ) = G(a, p ) . If, however, c is not a quadratic residue, one has C(a, p )

+ G(ac, p ) = 2

r mod p

e(r/p) = 0.

Thus for a c relatively prime to p , GW9 P) = (ClP) G(ad-4

holds, and this implies (3). Because of (3)

As p is odd, when rl, r2 run through complete residue systems independently so do rl + r2 and r, - r2 . Hence (-1/p)

W ,p)'

e(p-'rlr2)

=

=p.

r1,rz mod P

By inserting the value of the Legendre symbol on the left from the first complementation theorem to the quadratic reciprocity law, one has (4). It is more difficult to determine the sign of (4); this will be done in the next section. In anticipation we can remark that our calculations, up to now, remain valid when in place of (1) we set e(x) = eZnidxwith an arbitrary number d relatively prime to b. This can be interpreted as a rearrangement of the bth roots of unity in the sums, and the algebraic equations dealt with thus far are invariant under automorphisms of the field of roots of unity. The problem of determining the sign in (4) does not appear until one bases the convention ( I ) on more than purely algebraic information. SUMS 2. THEQUADRATIC RECIPROCITY LAWAND THE SIGN OF GAUSSIAN Let a and b be relatively prime odd numbers and 7 a complex variable with positive imaginary component. Taking n = 1, x = y = 0 in (I I ) of $1 * gives the function +m

3(z) =

+

1

exp(nirrn2).

(7)

m=-ca

We will substitute z = 2a/b i l and let 1 approach 0. The index of summation is written m = r bm,, and r is permitted to traverse a complete residue system mod b, while m, takes on all integral values. This yields

+

+m

exp(-db2rn12 - 2nlbrm1)

exp(-nlr2) rmodb

ml=-m

=

1 e ( i r2)exp(-dr2) rmod b

9(ilb2, ilbr, 0).

$2.

THETA FUNCTIONS FOR QUADRATIC FORMS

41

The reciprocity formula (12) of §I* is applied to the theta function on the right, and then I is allowed to approach 0. The result is

The same limit is then computed in another manner. Formula (12) of § I * is applied immediately to (7) :

(

92-+iI

) (I - i 2 =

;)-ll29(-

%+

iIl),

1

(9)

The square root on the right is to be taken positve for a = 0 and general values found by analytic continuation with 2a/b + iI remaining in the upper half-plane. This has the consequence

i)

- 112

l i m ( i - i2 I-0

= exp[ni sign(ab)/4]

The second factor of (9) is changed in form, as was (7), in particular by setting m = r + 2am1:

exp(-4nI,a2m12 - 4n1,arrn~);

x ml

(8) is again applied to the theta function on the right. Noting the connection between I and I, in (9) one finds

Finally the sum on the right can be operated upon as in the beginning of this section by setting 4C1 aC2 = 1

+

with integral c,, c2, This yields

C

r m o d 2 a e("

4a r 2 ) = r mod C 2e ( - 2 r 2 ) G ( - b c l , a ) .

Now, c, is a quadratic residue mod a and ac2 = 1 mod 4. Thus,

C r mod 2a

e(2

r2) =

C r mod 2

e -( a b7

r2) G(-b, a).

48


Combining (8) through (1 1) we have exp[+ni sign(ab)]

C

rmd2

.(4"r 2 ) G ( - b , a).

The sum still remaining on the right is easy to compute, and this leads to the reciprocity law for Gaussian sums: G(u, b) = ~lb/aI'"G(-b, a )

with E

= el

- +nil for a b = 1 mod 4, exp[jtni sign(ub)], c1 = exp[ exp[ *nil for ab = - 1 mod 4.

(13)

By specializing a = 1, b > 0, it is seen that G( -b, a) = 1, so that as a first application we have the sign determination ( 5 ) for Gaussian sums. Secondly one has a proof of the quadratic reciprocity law by putting two different odd primes in (12) as a and b. It was shown in the first section that formula (3) can be used. This leads to (a/b) = (- b/a) exP[b.ni( 1 - (va - 1) =(-b/a)

exp[)ni(i - va

+ (vb - 1) - v&)]

+ - Vab)], vb

where va, v b , v&, take the values 1 or - 1, depending upon whether a, b, and ub, respectively, are congruent 1 or - 1 mod 4. This last formula becomes the reciprocity law if one applies the first complementation theorem. The generalized form of the reciprocity law for arbitrary integers a, b is a simple consequence of the law for primes. 3. THETHETA FUNCTION FOR A DEFINITE QUADRATIC FORM A definite quadratic form in n = 2k variables mi,

uij)

is given. Let the matrix F = have rational integral coefficients, and let thehi be even. For integral values of the m i , F[m]can take on only even values. The smallest natural number N,such that the matrix F' = NF-' also has integral coefficients and, moreover, even diagonal coefficients, is called the level of F. Because NF-' is integral, so is INF-'I = N2klFI-' so that IF1 is divisor of a power of N. It is easy to verify that (- 1)'IFI is always congruent 0 or 1 mod 4. Now let z be a complex variable with positive imaginary component. Then, T = zF is a matrix of the sort on which our considerations in §1*,6 were based. The theta function for F arises from the following specialization

52. THETA FUNCTIONS FOR QUADRATIC FORMS

49

of (1 1) of the last section:

S(T, x, y ) = C exp(nirF[m - y ]

+ 2nim'x - aix'y).

(14)

m

For every matrix funcrional equation ax

i) E I"

with the property that c z 0 mod N, the

+ bFy, cF-'x + dy = (sign

d)*(-)

S(T, x, y),

(15)

with the generalized Legendre symbol? on the right holds; the symobl IF] here is the determinant. The substitutions (: f;) with c = 0 mod N obviously form a subgroup of finite index in the modular group r'. Pro05

Using the assumption on (:

i) it is easy to prove that

A B M = ( c D)=(:;-'

);t

is an element of the subgroup OZkc rZk described in $1*,4. Thus, by §1*,(18) and

Eq. (15) holds as maintained, with some eighth root of unity x(: f;) depending upon that matrix on the right side. It only remains to prove that

An easy consequence of the fact that (15) holds with of S(T, x , y ) on the right is the equation

x(t f;)

as a factor

for two transformations belonging to our group. Multiplication of a, b, c, d with - 1 leaves the theta function on the left of (15) invariant because of §1*,(13) with V = - E, so that both sides are multiplied by (Thus it is permissible, from now on, to assume d > 0 in the proof of (1 5 ) or (17).

t The generalized Legendre symbol a =1

(a/b) is also defined for even b, when b > 0 and mod 4. Then, (a/b) = (b/lal)holds.

50


In determining x(: :) one may assume x = y = 0. The procedure is similar to that of 92,2. By the reciprocity formula (12) of §1*, we have, for z = il,l real, Iim lk9(iA,0,0)= IF(-^/' lim 9*(iA-', o,O), 1-0

Ad0

where 9* is the theta series which is to be formed analogously with F-' in place of F. The limit on the right is reduced to the summand with m = 0. Thus Iim Ik 9 ( U , O,O) = IF[-'/'. (19) 1-0

The same limit is now determined for the left side of (15). Writing (aiI + b)/(cil + d ) = b/d

+ ill,

A1 = l/d(ciA + d )

(20)

+

and decomposing the summation index of the theta series to m = r dml so that r traverses a residue system of integral vectors mod d and m1 runs through all integral vectors, the exponent of the general summand is ni(b/d

+ ill)F[r + d m , ] = ni(b/d F[r] - nA1 F[r] + nibd F[ml] + 2nibJ Fml - d 1 d 2 F[m,]

- 2n11dm1'Fr.

The third and fourth summands on the right are integral multiples of 2ni and can be omitted. The last two summands lead to the theta function

9 ( i l l d Z F ,idl,Fr, 0) = IZ;kd-2klFI-1/29((i/llld2)F-1, 0, idIIFr). Taking (20) into account, this gives (cill

+ d)-k = IF(-'/'

exp(nibd-'F[r]), rmodd

on the left-hand side of (15). The same limit is formed for the right-hand side as well, while (19) is observed. Thus we find that (15) is true with the constant factor

Eventually we have to transform (21) into (17). For this we first reduce the task to the case where d is odd. The proof of (I 7) is first reduced to the case that d is odd. Let d, therefore, be even. Then c is odd and therefore so is N . But if IF1 is a divisor of a power of N , then IF1 also must be odd. Now, it can be read directly from (21) that :) = 1 f o r d = 1. Set

xc

$2. THETA FUNCTIONS FOR QUADRATIC FORMS

51

with integral s. These are three substitutions of the sort permitted. Take, in particular, s = t(FI with t odd, then d' = d ct(F1= 1 mod 2, and by the quadratic reciprocity law (set footnote on page 49),

+

xc:

xc 3.

Because of (18) and the fact that x('iNS i) = 1 we have ::) = Thus it suffices to prove (17) with a', b', c', d instead of a, b, c, d. As d' is odd, we may, without losing generality, assume d to be odd. For the case of an odd d, the proof of (17) is as follows: The quadratic form F[r] can, in the residue class ring mod d, be transformed to 2k

Zk

F[r] = 2 C fvsVzmod d ,

s,

v= 1

= 2 c,,,r,, mod d, p= 1

(22)

where the determinant lcvpl is relatively prime to d. As r traverses a residue system mod d, so does the vector s = (s,). Thus (21) becomes

x(ca db )

C exp(2nibd-lC fVs:) smodd

= d-k

Zk

=d - k n v=l

d

C exp(2nibd-'fVs,2). s,=l

By (2) through (4)and the first complementation theorem this is equivalent to

n

But because of (22), 2zk f , E IF( I C , ~ ) ~ mod d, and we have finished. 7 Finally we give another important property of the function 9. There is an integral power seriesfo with qo = exp( - 2 n i / ~ .I/z).

G(r, 0, 0) = (- l/z)kfo(qo),

(23)

For a reducedfraction s = alc # 0, determine rational b, d, so that ad - bc = 1. Then there is a power series f, with G ( t , 0,O)=

(-cr: a)>s(qs),

q, = exp[-2ni/N.(dz

- b)(-cr

+ a)-']. (24)

The first series expansion is gotten directly from the reciprocity formula 41 *,(12) under the consideration that NF-'[m] always represents even integers. To get the second expansion, set T = z' + s and dz - b = - -d-CT

+a

c

1 c(cz

- a)

= - -d- c

As above, in the sum over m for 9(rF, 0, 0), set n = r

1 c*z"

(25)

+ cm, with r traversing

52

APPENDIX TO CHAPTER I. THE THETA FUNCTlON

a residue system mod c and m1 all integral vectors. Then $(7,O,

0)=

C

rmod c

exp(ni(z’ + s ) ~ [ r ]$(c2r‘ ) F , Z ’ C F ~ ,0)

1

x

F-’, 0, -7’cFr

C exp(-ni/N.l/c2t’.NF-’[ml] + 2ni/c*ml‘r). mi

Up to the factor (-ic2T’)-klFI-1’2,the right side is an integral power series in exp ( - 2 n i / N . l/c2T’);using (25) this can also be written as an integral power series in qs. 7 The function $(z, 0,O)can be written as W

S(Z, 0, 0)=

c(p, F ) eZn‘Nr, p=O

where &,F) is the number of representations of p in the form p = +F[rn]. As is easily observed, this number and, in fact, the function $(T, 0,O) are invariant under unimodular transformation of the variables of the quadratic form Flm].They are class invariants. The functional equation (15) and the expansions (23) and (24) are the basis of the analytic number theory of quadratic forms. They demonstrate that quotients of functions $(T, 0,O)of forms F with the same determinant and level are, in principle, algebraic functions of one variable. The algebraic function fields thus generated are among the most remarkable examples of our theory since they serve, to a great measure, in the illustration of the more advanced concepts. In the course of history they have often inspired the formation of extensive general concepts. We will make a particular study of these functions in IV,W and V,§3.

C H A P T E R I1

Ideals and Divisors $1. Ideals

1. INTEGRAL DEPENDENCE Let an arbitrary field k, and a noetherian or, more specifically, a principal ideal domain o with quotient field k, be given. Further, let K/kbe a finite extension. An a E K is called integrally dependent upon o if all its powers, 1, a, a2, ..., belong to a finite o-module. u a n d only i f a satisfies an equation a"

+ c p " - l + + C" = 0 * * a

(1)

in k with coefficients in 0 , is a integrally dependent upon 0. Proof. If (I) holds with ci EO, then all powers of o! belong to the o-module formed by 1, a, ..., a"-' . Conversely, assume all powers of a belong to a finite o-module. By the finiteness criterion of I,§l,l or 1,§1,4, respectively, they themselves generate a finite o-module, a. Let al, ..., a, be a system of generators of a and ai = C atja' with a i j E 0 . Let n - 1 be the highest power of a appearing here. Then a is generated by 1, a, ..., a"-' , and we can represent a" = -c, - ... with ci ED. Hence, a satisfies an equation f ( x ) = 0 with coefficients in 0 . 7 If a satisjes an equation (1) with ci E 0 , it also satisfies an irreducible equation of this kind. Indeed, due to the theorem of Gauss, if the left hand side of (1) is factorized into polynomials in k[o!]with leading coefficients 1 these have coefficients in 0 . From the theorem just proven it is seen that the trace trk(a),k(a)= - c1 of an elemeht a integrally dependent upon o is in 0 . Applying the transitivity formula 1,§4,(3) to L = k(a) yields: If a E K is integrally dependent upon o then the trace trK/k(a) E 0. The set of elements of a Jinite extension field Klk integrally dependent upon o form an integral domain 0,with K as its quotient field. It is called the principal order of K with respect to 0.

Proof. If a, jl are integrally dependent upon o then the o-modules generated by all powers of a and /3, respectively, have finite generating systems, ai ,pi. 53

54

11. IDEALS AND

DIVISORS

The powers of a f p and crp are contained in the o-module generated by all a t , pi and all products a i p j . This is again a finite module. Thus, the elements cr integrally dependent upon o form an intergal domain 0 c K. Now let a be arbitrary in K and (1) be the normalized irreducible equation in k which a satisfies. As k is the quotient field of o there exists an u EO, such that all the a'c,, i = 1, ..., n, are in 0 . Now, /3 = aa satisfies the irreducible equation p" + ac1p"-l + ..-+ d c , = 0. Thus, /3 is integrally dependent upon 0 , and a = pa-' is the quotient of two elements of 0. 7 In the same vein we still show that the linear rank of D with respect to k is equal to [K:k]. In fact, starting with [K:k] linearly independent elements ai E K,there exists an u E o so that all the aat ED. If0, and 0,denote the principal orders of K and an intermediary field L with respect to 0 , then OKis also the principal order of K with respect to DL. For, if the a" are in a finite o-module, then because of (3, 2 o they certainly lie in a finite 0,-module. Exercise. If 0,is a finite o-module, then the principal order of K with respect to DLis the same as the principal order of K with respect to 0 .

2. THE FINITENESS OF THE PRINCIPAL ORDER

If Q is a principal ideal ring, then the principal order D with respect to o of a finite separable extension Klk is afinite o-module of rank [ K :k].The theorem holds even if o is only a Noetherian ring, as can be seen from the proof. Proof. Let mi be a field basis of Klk. We have already seen that an a E O Exists such that all awl €0. Thus there is no loss of generality in assuming the w i E O . Now, let 5 = ximi so that their traces are in 0. The disbe some element of 0.The ( w j €0, criminant D(w,) = Is,/k(wiwj)l is # 0. Hence the xican be determined uniquely from the linear system of equations

c

sK/k(<wj)

=

1

XiSK/k(Wiwj).

i

Therefore, they are all contained in the module oD(w,)-'. We thus see that

0 E D(w,)-'(oo,, ..., 0 0 , ) ; i.e., D is contained in a finite o-module. By the finiteness criterion of I , § l , l , 0 itself is a finite o-module. 7 This theorem does not hold unconditionally for an inseparable extension K/k.We will prove here only as much as we need for our applications. Let the field k be of the form k = ko(x1, ..., xm),

ko

= koo(w1,

Wr; Wr+1,

Wr+s).

55

$1. IDEALS

where koo is a perfect field and the wl, . . ., w, are independent indeterminates while w,,~, ... are algebraically dependent upon the first w p . Further, let o = ko[xl,

..., x,].

Then, the principal order 9 of an arbitrary finite extension Klk with respect to o is a j n i t e o-module of rank [ K :k ] . Proof. For m > 1, o is no longer a principal ideal ring, but it is a noetherian ring and we must apply the previous theorem with that hypothesis. Let K, be the largest subfield of K separable over k. K is then contained in the field K2 = K1P-', created by adjoining all p'th roots of elements of Kl , wherep is the characteristic and e is a sufficiently large number. The principal order O1of Kl is a finite o-module. Letting wi denote a system of generators of D,,we maintain that the elements with 0 e j , pi < p' form a system of generators of the principal order D2of K , as an o-module. In fact, let ct E O2. The p'th power of a lies in XI1, and can thus be represented as ape=

c Ri(wl,...; xl, ...)w i , i

where the R i are rational functions in the wg and polynomials in the x, with coefficients in k,, . They can be written as Ri(Wl, ...; X I , ...) = [N(wI',

c

...)I-' s,,,...

,,,,; i ( W g ,

... ; XI',

...)w y

... x;' ...

where N and the S,, ,.,,;, ,...; are polynomials in all variables. The p'th roots of the coefficients of these polynomials are again contained in k,,. They are used to form new polynomials, which are plausibly denoted Np-', etc. Thus,

holds, which verifies our assertion. By the finiteness criterion of 1,§1,4, 0, being a submodule of the finite module 0,, is itself finite as an o-module. 7 From the last theorem it is easy to derive another, which we shall also need. Let k be as before, and p(xi) a prime polynomial in k,[xl, ..., x,,] . The integral domain of all rational functions in the xi whose denominators are not divisible by p(xi) is denoted op . Then, the principal order 0,of a finite extension Klk with respect to 0, is a finite 0,-module. Proof. Let ct E 0, satisfy the irreducible equation (1) with ci E 0,. A common denominator a of the ci is relatively prime to p(xi), and thus a unit

56

11.

IDEALS A N D DMSORS

in 0,. The element /I = aa is in the principal order 0 with respect to o = ko[xl, ...I, as the same argument used in §1,1 shows. By the previous theorem there exists a finite system of generators w i for D.Therefore a = (l/a) biw, = clwi with bi E o or c, E op. Hence the w i also form a system of generators for 0, with respect to o,,. 7

3. KRONECKER DIVISORS We make the assumption that o is a principal ideal domain, and stay with it until $1,5. An indeterminate x is now adjoined to k. A polynomial in x is called primitive when the greatest common divisor of its coefficients is 1. By Gauss’s theorem the product of primitive polynomials is primitive. o(x) will denote the integral domain of quotients of polynomials with coefficients in 0, the denominator being primitive. O(X) is again a principal ideal domain. Proof. Every flx) E o(x) can be represented in the form flx) = ag(x), where a is the g.c.d. of the numerator offlx), and g(x) is a quotient of primitive polynomials. Let f’(x) be another such function, f ’ ( x ) = a’g‘(x), and let d be the g.c.d. of a and a’. With suitable u, u‘ of o we have au a‘u‘ = d, or

+

I(x)ug’(x) +f’(x)u’s(x) = dg(x)g’(x),

and the right side is a divisor of both f l x ) and f ‘ ( x ) . It follows that every finitely generated ideal is principal. Prime decomposition of elements follows from the same property in 0 . 7 Similarly we adjoin infinitely many indeterminates xl, x 2 , ... to k and form theintegral domain o(x) = 0(x1)(x2) .... It again is a principal ideal domain. In addition, let K/k be a finite extension, D the principal order of K with respect to 0 , and 0(x) the principal order of K(x) with respect to o(x). D(x) consists ofthe quotients a = Pa-’ with fl E D[x] andprimitioe a E o[x]. Proof. Let an a E D(x) satisfy the irreducible equation (1) in k(x). The ci are elements of o(x) and thus have a primitive polynomial a as their common denominator. Then, /I = aa is a polynomial in the x with coefficients in K. We contend that the coefficients are even in 0.Assume first that /I= Po plxl /12x12 depends on one variable only. Inserting x1 = 0 we see that /Io depends integrally on 0 , so Po E 0. Then xl-’(fl - /Io) = /I1 B2x1 is in O(x), so that /Il E 0 , etc. If fl depends on n variables we can consider /I as a polynomial in the first variable the coefficients of which are polynomials in the others and thus proceed by induction on n. Hence, every a E D(x) can be written as

+ + + +

+

a = /I/a,

/3 E D[x],

a E o[x]

$1. IDEALS

57

with primitive a. Conversely, such a are contained in D(x). 7 D(x) is a principal ideal domain.

Proof: Let a E D(x) be a finite D(x)-ideal and ai a system of generators of a as an o(x)-module. We take just as many indeterminates xi not occurring in the ai ,#forma = aixi,and maintain a = aD(x). Let p be an arbitrary element of a. For some indeterminate ( consider the quotient

It is a polynomial in t, in fact the characteristic polynomial for y in the sense of 1,§4,1. For, with a field basis mi of K/kand elements 6 , of the unit matrix,

holds, and thus nK/k(r

- 7)

=

IE( - < c i j > l *

If we also represent 8 in the system of generators of a as B =

1a i b i ,then

Both denominator and numerator here are polynomials, their only difference being the variables x i and u i , and these are independent of each other as of the variables occurring in the a,. Dividing by the g.c.d. of the coefficients yields a rational function, whose numerator polynomial has coefficients in o(x, 5 ) and whose denominator polynomial is primitive. As a consequence all coefficients off(() are in o(x). The two facts, that f(() is the characteristic polynomial of y and that its coefficients are in o(x), the highest being 1, show that y E D(x). Hence /j' = ay with y E D(x), so that a c a D(x). The opposite inclusion is obvious. The possibility of prime decomposition arises by norm formation with respect to k. The norm of elements of D(x)lie in o(x). Now, if an a(x)E D(x) were decomposable without limit, the possibility of prime decomposition in o(x) would assure that, with a finite number of exceptions, all the norms are units. But then the factors themselves would be units. T The theorem just proven permits us, because of the uniqueness of prime factor decomposition in principal ideal domains, to draw the consequence: The factor group R, = K(x) /(E(x) of the multiplicative group K(x)" of K(x) with respect to the group of units (E(x) of D(x) is the direct product of the cyclic subgroups generated by the

58

11. IDEALS AND DIVISORS

prime elements of D(x). R, contains the factor group

Sj, = K " / @ ,

@ = K n @(x),

where 6 is the unit group of 0.We call the elements of 52, Kronecker divisors of O(x) with respect to 0 , the elements of 8,(Kronecker) principal divisors with respect to 0 . Later another concept of divisor will be introduced and placed more in the foreground. Kronecker divisors play a small part in the current literature and we too will only apply them twice. But then they prove themselves superior to other conceptual aids. The first application will be the construction of ideal theory, to which we now turn. We conclude with a remark which is not necessary in what follows. After a single variable z is adjoined, O(z)is already a principal ideal domain. Proof. We maintain that, if

is a polynomial in the variables xl, ...,x, , one can find exponents ni = such that, setting P(z) = C ai ,,...znc, the quotient a(x)p(z)-' is a unit in D(xl, ..., x,, z). This statement yields the proposition. For, as we showed above, the g.c.d. of a finite number of a,(x) E O(x) can be written as a polynomial a(x). But, by our construction, it is also equal to p(z). If the a,(x) depend only upon a single variable x their g.c.d. is also equal to p(x), as /I(z)P(x)-' is, of course, a unit. Choose n , in such a manner that the polynomials a(x) and p(z) have the same coefficients. Also, choose d(x) = d(x,+', ...) so that it represents the g.c.d. of all the a,,,... . Then, a(x) d(x)-' is a primitive polynomial in xi, ..., x, with coefficients in O(x,+,, ...). Its norm with respect to k is a unit in o(x,, ..., x,,,+~,...). For, the norm is a product of elements, conjugate with respect to k, all of which are primitive in this case. By Gauss's theorem, their product is also primitive. As a consequence, a(x) d(x)-' is a unit in O(x,, ..., x,+', ...). The same argument shows that p(z)d(x)-' isaunit in D(z, x,+,, ...). The quotient a(x)/I(z)-' is therefore a unit in D(z, xi, ..., x,), which was to be proved. 7 4. IDEALS

D will from here on in be the principal order of K with respect to 0 . An D-ideal (simply called an ideal when no confusion can result) is defined as a finite o-module a of elements of K not consisting only of the 0, and such that for every pair o E D and a E a the product o a E a. This last condition can be written symbolically as

Da = a.

(2)

$1. IDEALS

59

This concept of ideal is somewhat more general than that of abstract algebra, where a _C 0 is required. If a, b are two ideals, then the finite sums c a i S i with arbitrary a i e a , p i E b also form an ideal called the product of a and b and written ab. If ai , pi are systems of generators of a and b, then aipj is a system of generators of ab. In particular, (2) represents a product, so that it becomes plausible to call Q the unit ideal, as ideals remain unchanged when multiplied by it. By 51,2, Q is a finite o-module, and thus an ideal. If a, b and c are three ideals, the associative law

(ab)c = a(bc) holds. Special ideals, those of the form Qa with a # 0, are called principal ideals. The short notation Dct = (a) is used. Clearly

This can be genralized to a(a) = ad. Because of (3) the principal ideals form a group, the unit element of which is Q = (1). This group is a homomorphic image of the multiplicative group K of K. The units of Q, i.e., those elements E E Q whose inverse is contained in 0, are mapped onto the unit ideal under this homomorphism. An ideal a is said to be diivisible by an ideal b if a s b

(4)

holds. If a is divisible by D it is called an integral ideal. Multiplying (4) by the inverse ideal b-' (which we shall soon prove to exist) shows that the divisibility of a by b is equivalent to the fact that ab-' is integral, as is familiar from numbers. An element a is also said to be divisible by a if a E a. If a and b are arbitrary ideals, the o-module consisting of their union is an ideal written (a, b). It is a divisor of a and b in the sense of (4,and is called the g.c.d. of a and b. In apparent contradiction to its name, the g.c.d. is the smallest ideal containing both a and 6. The g.c.d. of several ideals is defined accordingly. For principal ideals a = (a), b = (P), etc., one writes (a, 8, ...) for short. As is familiar, an integral ideal p is called a prime ideal if its residue class ring O/p contains no divisors of 0. Our hypothesis being that o is a principal ideal domain, the residue class ring 0 / p is even a j e l d . Proof. All elements in o divisible by p form an o-ideal. By hypothesis it is a principal ideal ( p ) . The fact that p is a prime ideal implies that p is a prime element (i.e., not decomposable into proper factors). By I,§l,l, the residue classes of a principal ideal domain with respect to a prime element form a field.

11.

60

IDEALS AND DIVISORS

+

+

c, E 0 mod p with c iE o Now let a E 0,a p , and ar + clar-l + be the congruence of least degree that a satisfies. Certainly c, f 0 mod p , for otherwise we would have a(a'-' + + c , - ~ = ) 0 mod p and, because of our assumption, the second factor would be S O mod p. Since the residue class ring o/p is a field, there exists a c E o with cc, = 1 mod p. But then

-ac(a'-l

+ + c,-~) = 1 mod p,

and the assertion is proved. 7 A prime ideal p is divisible only by itself and 0. Proof. Let a lie between p and D.The residue classes of the elements in D form an ideal in 0 / p , But in this field there can be only 51 zero ideal and unit ideal, corresponding to whether a = p or a = 0. 7

Principal Theorem, If 0 is the principal order of K with respect to a principal ideal domain o of k, then the 0-ideals ,form a group. Furthermore, all ideals can be written uniquely as power products of prime ideals. The principal ideals form a subgroup of the group of all ideals. The cosets of this subgroup are called classes of ideals, and they also form a group, the factor group. Ideals of the same coset, or simply class, are called equivalent: a-b

if b = a a ,

aEK.

5. PRWF OF THE PRINCIPALTHEOREM

As in §1,3 we adjoin infinitely many indeterminates x . Then every element a(x) E 0 ( x ) can be written as quotient a,(x)a,(x)-' of polynomials, the denominator ad(x) being a primitive polynomial in o[x]. The D-ideal generated by the coefficients of the numerator a,(x) is called the content of a(x). It will

turn out that the content does not change when the quotient suffers cancellation or extension with primitive polynomials in o[x]. Conversely, it is possible to associate an element a(x) = aixi with a given ideal a having the system of generators a,; its content is a. To every ideal a there exists an inverse ideal a-': aa-' = 0.

Thus the ideals form a group. Proof. Introducing another set of indeterminates x' consider the a(x) as rational functions with coefficients in the principal ideal domain 0 ( x ' ) . As such, they (in particular the polynomials in x ) have contents which are Kronecker divisors, i.e., cosets of the form a,(x')@(x'),where CZ(x') is the group of units of D ( x ' ) . By Gauss's theorem the content of a product is equal to the product of the contents.

$1. IDEALS

61

Let an ideal a be given, and let ai be a system of generators of a as a o-module. Weforma(x) = CaixiandtheinverseP(x) = a(x)-l = b,(x)b,(x)-'. Then a(x)bn(x) = b d ( X ) , b d ( X ) E O[xl with primitive bd(x).Now we consider the contents of these polynomials as Kronecker divisors with respect to the variables x'. Because bd(x)is primitive the contents a,(x') and b,,(x') of a(x) and b,(x) are inverse to each other, and the products of the coefficients of a(x) and b,(x) are in D(x'). As they are also in K , they must be contained in D. Furthermore, it becomes clear that the coefficients of bd(x)can be written as sums of products of coefficients of a(x) and b,(x). Since o is a principal ideal domain and the coefficients of bd(x) have g.c.d. I , it is possible to represent 1 as the sum of products of coefficients of a(x) and of b,(x) and of elements of 0. This finally implies that every element of D can be so written. Denoting the ideal generated by coefficients of b,,(x) with a-', the result attained is aa-' = 0, which is what was to be proven. 7 The point should not be missed that the assumptions of o being a principal ideal domain was necessary. An example in §1,6 will demonstrate this clearly. The group of ideals is isomorphic to the group of Kronecker divisors, i.e., the factor group K(x)"/E(x) of the multiplicative group of K(x) with respect to the group of units of D(x). Proof. Comparison of coefficients shows that the rational functions a(x), p(x), y(x) with a(x)P(x) = y(x) have contents a, 6, c satisfying ab E c. Using p(x) = a(x)-'y(x) leads to a-'c E h, and multiplying this with a yields c G ah. This proves the assertion. 7 The earlier assertion, that the contents of a(x) remain invariant under cancellation and extension, also follows directly. The unique prime decomposition of Kronecker divisors yields, finally : Any ideal can be written uniquely as the product of prime ideals.

6*. EXTENSION OF DIVISIBILITY THEORY To conclude this paragraph we show that the concept of Kroneckerdivisors leads further than the concept of ideals. We return to this possibility in Chapter V. If o is a noetherian domain in which eirery element can be represented uniquely as the product of' prime elements, then O(Z) is a principal ideal domain. Proof. Given taking of the

As in $1,3, we first adjoin an infinite number of indeterminates, xi. ai(x)E o(x), their g.c.d. can be represented as a(x) = a i ( x ) x i , variables x i not occurring in the ai(x). For, cancelling by the g.c.d. denominators in cci(x)a(x)-' leaves elements with g.c.d. 1 in o(x)

62


remaining. A second step now shows that adjunction of a single variable suffices. This is done, e.g., by associating the polynomials a(x) =

C ail,...

xlil..-

where

mi = mi,,.,,= i l

and a(z) = 2 ail,.,.zmi

+ n,i2+

n,n2i3

+ ...,

the n p being the degrees of the a(x) in the x p , The feasibility of prime decomposition of a polynomial is a consequence of the fact that o[z] is also a noetherian ring (cf. 1,§1,4). The prime decomposition of an element of o(z), i.e., of a quotient of polynomials of o[z]with primitive denominator, comes from the decomposition of the numerator. 'I The principal order of K(z) with respect to o(z) is a principal ideal domain. The proof is word for word that of §1,3. Feasibility of prime decomposition is, as in §1,3, demonstrated by the formation of norms with respect to k . Again as in §1,3, the group R, = K(z)"/@(z)of Kronecker divisors with respect to o and its subgroup 9, = R /@ of principal divisors can be formed. Ideal theory does not lead this far. Take the example k = ko(Y1, Y21,

K = k(Jy,y,),

0

= k,CY,, Y21.

A basis of D with respect to o is 1, Jy,y2. Now, let a, b be ideals with bases y,, &yZ and y , , J y Z with respect to 0 . Then, ab can be generated by y l J y z , y 2 , / y z , y1y2 with respect to 0 , but has no two-element basis. There can be no inverse to a. Such an ideal would, at the most, contain elements of b(,/yz)-', but ab ( J y X ) - ' c 0.The calculation with Kronecker divisors is the following: a(x) = Y l

+ Jy,Y, x ,

with the unit E ( X ) = (yl follows from the norm

.

P(x> = Y 2

+ y2)x + &(I

%,k(E(X))

=(Y,

+JYlY2

+ x').

x,

a(x)P(x> = Jy,Y,

The fact that

E(X)

4 X )

is a unit

+ Y 2 I 2 X 2 - Y,Y2(1 + X 2 l 2 ,

which is a primitive polynomial in x. The theory of divisibility based on divisors is due to Kronecker. He, however, uses them only in the proof that the contents of the product of two polynomials is equal to the product of their contents. This yields, in principle, the theorem that the ideals form a group, and from there it is not far to the unique prime decomposition of ideals. As was seen here, divisors attain prime decomposition quicker, and the ideals would be superfluous. The concept of ideals turns out to be indispensible for other reasons, however. Nevertheless, multiplication of ideals could often be omitted from the theory, which would be a simplification.

$2.

LOCAL RINGS

63

Another possibility of extending the theory of divisibility is the theory of quasi-equal ideals of van der Waerden and Artin.? The assumptions made are weaker yet, but with them it is easy to show that if o is an integral domain in which the theory of quasi-equal ideals holds then O ( Z ) is a principal ideal domain. Thus, both extensions of the theory of divisibility turn out to have the same effect. Without doubt, the Kronecker divisors are easier to handle than quasi-equal ideals in concrete problems. $2. Local Rings 1. BASICCONCEPTS

In arithmetic, the local considerations indicated in the introduction play the same part as in analysis, where, for example, a curve is considered in relation to its tangent. Since the basic operations and concepts of arithmetic do not have the same visual meaning in the immediate sense of analysis and geometry, the meaning of local considerations only becomes apparent step by step as the theory is built up. Our first task is to fix in a conceptual way the integral domain i, constructed in the introduction. An integral domain o is called a special local ring$ if it has the following properties: (1) the quotient field k of o is not equal to 0 ; (2) if a E k then either ~ E orO a-l E O . If k is a function field, we will also require: (3) o contains the exact field of constants. A special local ring contains a single maximal prime ideal p; it consists of all a E o such that a-l 6 0 . Proof. Let ~ E O a-' , $ 0 . Then, if y ~ so o is ay, but (ay)-'#o. Thus, a E p, y E o implies ay E p . Further, let a l , a2 E p . Suitable indexing gives ala;' E 0 . Then, a1 - a2 = (a,at - 1)a2 E p , proving p to be an o-ideal. Those a E o not in p have, by definition, the property a - l E 0 . Hence, the residue class ring is a field, and p is therefore prime. We emphasize: the residue class ring of a special local ring with respect to its prime ideal is afield. Now let q be some maximal prime o-ideal. If K E ~ then , I C - ~ $ 0 . Thus, q E p, and as q is maximal, q = p . 7 A Jinite ideal a for a special local ring 0 , that is, the union of afinite number of principal ideals 01,is a principal ideal O M . Proof. (a) 5

For any two nonzero elements a, p ~ k we , define an ordering E 0 . Then we have: (1) either (a)S (p) or (p) 5 (a), with

(fi) if MP-'

t B. L. van der Waerden, Algebra, 4th ed., Vol. 11, $131. Berlin-Gottingen-Heidelberg, 1959. 1A local ring as such is defined as an integral domain having a single maximal prime ideal.

64


both certainly holding if a = 8,we write (a) = (8);(2) if (a) 5 (8)and (8)I( y ) then (a) 6 ( y ) ; (3) if (a) 5 (8) and y # 0 then (ay) S (BY); and, finally, (4) 0 consists of all a with (a) S (1) and a = 0. A finite ideal a has a largest element. Assume this to be false, and there to exist an infinite series a, with (a,) < i.e., (a,) 5 (a,+1) but not (a,). For no subscript n, then, can we write a, as (a,+ 1) a,, = ylal

+ + 9..

yn-lan-l,

7,

E 0,

for this equation would imply

+ + 7.-

1 = yl(al/aA

I(@,tlan).

By hypothesis a,/a, = ( ~ , / a , +m--a,,-l/a,, ~ are contained in 0, but not their inverse elements. Thus, the a v / a n E p ,yet 1 cannot be in p. We have thus shown that a cannot be a finite ideal, which is a contradiction. For a maximal a E a we have (8) 5 (a) for all B E a, which means that 8 E oa. Thus, a = oa, as maintained. 7 The discussion also implies : The finite o-ideals form an ordered group under the definition oa 5 08 if(a) 5 (B). Exercise. The triangle inequality o(a + 8) 5 max (oa, OD) holds; i.e., the mapping a -,oa is a valuation of k. A special local ring o will be called discrete if its prime ideal p is finite and if to every a E k there exists a power of p such that uph E 0 . Taking the least power h, one has ap" = 0 . In fact, let p = on. Were ap" # 0, then ap" G on and an-'ph = a$-' E 0 , and h would not be minimal. Thus, for these discrete local rings all finite ideals are powers of the prime ideal. Further (maximality) : An integral domain or properly containing a discrete local ring o and having the same quotient field k coincides with k. Proof. Let K E o', K # 0 . Then, a = K - ~is in p. As o is discrete, there exists to every a E k an exponent h, such that an" = E 0 . Clearly one may choose h 2 0. Then, every a can be represented a = This, however, implies that adjoining K = n-' to o yields the field k. 7 Example. Let the field k have the property that every element can be written essentially as a power product of prime elements, a = ep?' ...p:, with a unit e. These units and the prime elements belong to an integral domain o in k. Distinguishing some prime p , all a in which p occurs with nonnegative powers form a discrete local ring o p . The i, mentioned in the introduction are of this variety. 2. LOCALRINGSIN ALGEBRAIC EXTENSIONS

Let a discrete local ring o with quotientjeld k be given, K be a finite extension of k, and 0 the principal order of K with respect to 0. Then, 0 is a principal ideal domain with only finitely many prime ideals.

$2. LOCAL RINGS

65

These prime ideals have the form p, = Dn,, where n, comes from a prime decomposition p = En;' ... n,e. (1) of a prime element p of 0. ( p is determined up to a unit factor by p = op.) E is a unit of 0. The n, are called the prime elements for the p,. Proof. As o is a principal ideal ring the ideal theory holds for 0. Let q be a prime 0-ideal, then o n q is a prime o-ideal; thus it must be p, so that q is a divisor of 0 p . Let 0 p = pi' ... pp

be the prime decomposition in 0. All ideals are power products of the p, . We now maintain: there exists a nl E p1 such that nIp;l is relatively prime to p1 -..ps (s = 1, 2, ..., r). This assertion holds for s = 1, as pf c pl. Further, plp2 c p1 implies the existence of a nl' E p1 not divisible by p 2 . Therefore, either n1 or n1 + ni2 is divisible by pl' exactly once and not a t all by p 2 , proving the assertion for s = 2. We proceed by induction, denoting an element of pi as if it is such that ni;j,...p,:l is relatively prime to p i p j ... . The inductive hypothesis gives us x ~ ; ~ , . . . , We ~ - ~ have . finished if this element is not divisible by ps. But if it is, then 7 h ; 2(...( s

- n1;2,....s-

1

2 + 7h;sAl;s ..-Zs-l;s

is an element with the property asserted. In particular, takings = r and forming the analogous elements n 2 , n3 , ... n,, one has p, = Dn,, proving the theorem. 7 Those a E K for which a fixed p , does not occur with a negative power in It contains the prime decomposition of'Da = (a) form a discrete local ring 0,. 0 , and

0 = O1 n ... n Dr holds. Conversely, these are all the special local rings 8 containing

0.

Proof. The first statements are obvious (see the example at the end of $1). Keeping the notation from above, let 23 be such a ring. Any a E 0 satisfies an equation u" + clan-' + + c, = 0 , ci E 0 . If a were not contained in 8 we could write a = -cl - c2a-' - ...- a - n + l , the right side of which would be in 8 . Thus, a E 8 and 0 c 8 . Let q be the prime ideal of 8 and set 0 n q = r; it is an integral 0-ideal. In order to show that it cannot be divisible by two different prime 0-ideals, we assume the contrary, that it is divisible by pl, ..., p,. Then, n1 is not in r, norinq.Thus,z;I Eq;similarlyn;', ... ~q.Altogether,a= ( n r l n s - l ) h E q for any natural h. Choosing a sufficiently large h, by hypothesis a-' E t , and thus in q. This is impossible.

66

11. IDEALS AND DIVlSORS

The possibility remaining is D n q = p , h, with suitable indexing. Certainly h # 0; otherwise 1 E q. h > 1 drops out, as then n1 # q, but n;' E q is just as impossible. By elimination, then, we have D n q = pl. Proceeding, n z ,..., R, E 0 c B.They are not in pl, thus cannot belong to ... nYhrE B,if a E D for arbitrary h, . These are exactly the elements that form the ring D1.7 In conclusion we draw attention to the fact that these theorems can be proved without resort to the ideal theory of $1, which is, in fact, accomplished by the use of valuation theory in several texts. This leads to another foundation of arithmetic meeting with ours in 45. It borrows its basic concepts from classical function theory and is thus quite intuitive. On the negative side, certain difficulties are encountered when it comes to inseparable field extensions, as we saw in 41,2, and these can only be overcome by limitation of the generality. It can, in fact, be shown that the important equation43 ,(14) is incorrect without such limitations. Our method is preferable for these reasons, as well as because it is shorter and makes fewer assumptions. The results of this section, incidentally, can be achieved without reference either to $1 or to valuation theory, by simply deriving them from the properties of 0 , as long as the residue class field IE = o / p contains infinitely many elements. We will do this in III,42 for the field k = k,(x), k, algebraically closed, thus choosing the path otherwise arrived at by valuation theory. For k with only finitely many elements the procedure of 41 can be shortened considerably. It is a worthwhile exercise (to be contemplated, say, after reading Chapters I1 and III,42) to carry this out. q, and are units of %. Therefore, all elements an;hz

3. LOCALRINGSIN ALGEBRAIC NUMBER AND FUNCTION FIELDS Let k = Q or k = ko(x), i being as in the introduction. Let p denote a prime element and i, be the integral domain of all a E k , in whose prime decomposition p does not occur with a negative power. In the function case, let i, be the integral domain of functions with nonpositive degree, which we can now write in the form i, = ib. , with x' = x - ' , i' = k,[x'], and p' being the prime element x'. The i, (including i,) are all the special local rings of k. Proof. A local ring o always contains 1, and thus - 1, & 2, ... . In the number case, therefore, o 2 i. In the function case let us assume first that x E 0. The constant field is always contained in a local ring, hence again, i E 0 . Not all a E i can be units of 0 , for then o would contain k, the quotient field of i. If some a is a nonunit of 0 , then one of its prime factorsp must be

$2.

67

LOCALRINGS

a nonunit. Thus, the prime ideal p contains a t least one prime element p , but it also contains only one. For, if it contained two, say p and q, i, would also contain 1, because of the representation p u qv = 1. As all primes of i except for p are now units in 0 , this ring contains the ring i,. But i, is a discrete local ring, so that due to the maximality property we have o = i,. We have yet to discuss the possibility that k is a function field but x # 0. But then x’ = x - l E 0 , and the same argument holds: o = i;. with some prime polynomial p’ = x ’ ~+ + c,,. Now, if p‘ # x‘ we would have p‘ = x - 9 , where p = c,,x” + 1. It is easy to see that this would lead to i’,,, = i, , and then x E 0. Hence we have p’ = x’. 7 By now applying the last theorem of $2,2, a complete view of all the special local rings of an algebraic number or function field has been obtained. This result is especially remarkable for functions in view of the following consideration. An algebraic function field K is always given as a finite extension of a rational function field k = ko(x) over the exact constant field ko. The pair K , k o ( x ) is called a model of K. Given the function field K, a model is determined by an element x. Clearly, any x 4 ko can be used. In our construction of all the special local rings of K, we used a particular model, but the local rings are actually invariant of the model. We shall return to this fact in $5. We here extend the concept of a place of a field by associating a place with every discrete local ring. The remark of 1,$3,1 holds here in that our definition is essentially only a manner of speaking to be used, for example, in the sense: an element is integral at a place if it is contained in the corresponding local ring. We extend the terminology by saying a place of a finite extension K / k lies over a place of k if the local ring of K contains that of k. This expression takes on geometric meaning in the case of classical function theory, but this will not appear until IV,$l.

+

+

4. THECOMPONENT DECOMPOSITION OF IDEALS We now consider some methodical preparations for $03 and 4. Let k and o satisfy the assumptions of $1, and L be an intermediate field between k and the finite extension K, the cases where L coincides with K or k not being excluded. The ideals of K are not only o-modules, but also 0,-modules, where 0, is the principal order of L with respect to 0 . In order to study the correspondence between the ideals of K and L, the ideals of K must be taken as 0,-modules. The fact that 0, is not a principal ideal domain is a hindrance here, but it is overcome by the use of local considerations. To avoid the complication of subscripts we write k and o for L and D,, and we will use the result of $1, that all o-ideals and all D-ideals form a group in which unique prime decomposition always holds.

11.

68

IDEALS AND DIVISORS

First we form the local rings 0,; for all prime ideals p of k. These consist of those a E k which are not divisible by a negative power of p. Clearly

We list further results as (a),

... .

(a) Every a E k lies in almost all o, (i.e., in all 0, with at most finitely many exceptions). For a # 0 the same holds for a-', so that any nonzero a E k is a unit of 0, for almost all p. For an 0-ideal a we now introduce the p-components a, = ao,.

(b) 0,= Do, is the principal order of K with respect to

0,.

Proof. Denote this principal order by D, , and assume a E 0,. Let tl satisfy the irreducible equation an clan-' c,, = 0 in k, the ci lying in op by assumption. The common denominator b of the ideals (ci) is relatively prime to p . As in 52,2, choose a b E b not divisible by bp, and in addition a C E o such that bc 5 1 mod p. Then, al = abc satisfies the equation aln c,bca;-' c,,(bc)"= 0. Thus, a1 E 0.By construction (bc)-' E o p , and thus a = a,(bc)-' E Do,. T

+

+ +

+ +

+

(c) Every a E K is contained in almost all 0,.

+ +

Proof. Once more let a" c,, = 0. By (a) almost all c i , and thus, by (b), a lies in almost all 0,.7

0,

contain all the

(d) Ifa is an 0-ideal, then a, = 0,almost always.

Proof. Let a', ..., a,,, be a system of generators of a as an 0-module. As the ai and a;' lie in almost all D,, they are almost always units of 0,. For these O,,a, = 0,. 7 (e) Two ideals, a, b satisfy

'.

(ab), = apbp, (a- '), = (a,)- = a; The proof is an immediate consequence of the definitions. (f) Every 0-ideal is the intersection of its p-components.

Proof. First, two ideals a, b satisfy

b, E

a P

n ab,.

(3)

?

Applying (3) to a-', ab in place of a, b yields, with (e),

n b, = n a;'a,b, P

P

=

n a-'a,b, P

3 a-'

n a,b, P

= a-'

nP ab,.

42. LOCAL

69

RINGS

Multiplication by a and comparison with (3) yields a

n 6, = n ab, = 0 apbp. P

P

P

Now, set 6 = 0 to get the result a=

n a,.

(4) 7

P

(g) Let there be given an arbitrary D,-ideal a, to every p, with the provision that almost always a, = 0,. Then, the intersection (4) is an 0-ideal. Proof. The equation a 0 = 0 is obvious, and what must be shown is that a is finite in the sense required. Let a, E a2 E E a be a sequence of finite 0-ideals leading to a. By assumption a E a, , and therefore ao, E ap holds for the p-components of a. By hypothesis in conjunction with (d) we have, for almost all p,

c a,, = 0, for all i. Because of anop c a, the ascending chain theorem 0,= a,,

= a,o, E ao,

and then a . = a, 'P can be applied to the modules a,o, . For each of the finitely many exceptional p there exists an index n, such that a,o, = an+lop= ... . The maximal of these n applies to all, and for this index is a, = a,,, = = a. 7 (h) Let a be generated as in (g). Then, the p-components of a are ao, = a,. Proof. Distinguish p and note that, by §2,2, a, = 0,a, is a principal ideal. By (c), a, lies in almost all the other a, , Say a, # a,, for a finite number of p i # p . Because pplp2 ... c p l p 2 ... there exists an a E k which is divisible by all the pi but not by p. With this a we have a, = Dpu,ahwith any h and, with suitably large h, apahE a,,. Thus, we can set a, E ap, for all p' # p. But, with this choice we have ao, = D P a pcompleting , the proof. T (i) Let q be a prime D-ideal. Then, p = q n o is a prime o-ideal, and we have the residue class isomorphism

=

D/q ~ , / q ,* Prooj The residue classes of o mod p form a subring in the residue class field O/q. As such a subring cannot contain divisors of zero, p is a prime ideal. Each residue class of 0 / q naturally determines a residue class of 0 , / q P . On the other hand, let a E D,. As in the proof of (h), find an a E o not divisible by p but with a a E 0 . For a b E o with ab = 1 mod p one has a' = aab E 0, and a' = a mod q. Thus each residue class of D,/q, is really determined by one of D/q. It is obvious that the correspondence D/q 0 , / q P behaves properly under addition and multiplication. 7 *-)

70

11.

IDEALS AND DIVISORS

$3. Ideals in Different Fields; the Norm 1. EXTENSION OF AN IDEAL

Let o be a principal ideal domain with quotient field k and K be a finite extension of k. In case the extension is inseparable, we again make the limiting assumption of §1,2 regarding 0 . Thus, the theory developed in $1 holds for ideals in K. In addition to K we consider an intermediate field L, denoting the principal orders by O K ,DL, respectively. The theorems of $1 hold, of course, for the ideals of 0,, too. The subject matter of §3 will be two homomorphic mappings of the ideal groups of D, and O K ,each onto a subgroup of the other. The first of these mappings is an extension of an 0,-ideal a, to an DK-ideala, = O,a,. For another ideal, 6, ,

(ab), = DKaLb, = a,bK

(1)

clearly holds. Thus, we have a homomorphism. But, the mapping is even an isomorphism. In fact, if a, = D, then all elements of a, depend integrally on D,, especially those of a,, and so a, E DL.Now we compute D&'aK in two ways, using (1) and the hypothesis:

DKai 'aK = T ) K a i 'DKa, = 0,= DKai '.OK = DKaL

9

'

By the previous argument aL E D, . Therefore a, = 0,. Whenever only multiplicative relations between ideals are dealt with, it is permissible to identify the ideal a, of L with its extension aK of K . Then, as a rule, their distinction by subscripts becomes superfluous. In particular, the unit ideal (1) is designated with the same symbol in K and L. Furthermore, an ideal a of K is said to already belong to L if it can be generated by extension of an ideal of L. This convention is based on the conception that the notion ideal is a concrete realization of an already existent idea, guaranteeing the possibility of unique prime decomposition.

2. THE NORM Our starting point is the component decomposition

D~=

n nLP, P

where p runs through all prime 0,-ideals. The DLpare principal ideal domains. D, is the principal order of K with respect to D,. Let DKPdenote the principal orders of K with respect to the DLp. By §2,2 these are also principal ideal domains, along with the 0,.

(13.

IDEALS IN DIFFERENT FIELDS; THE NORM

71

An D,-ideal a can be represented, according to §2,4,(f), as the intersection of its p-components ap, p running through the places of L. As the DKpare principal ideal domains there exist ap E K such that ap

= OKp'p

(2)

and the ap are almost always units of DKpby §2,4,(c). The &-ideal a is now assigned the DL-ideal nK/L(a)= ~Lpn,/L(ap) (3)

nP

as its norm with respect to L. The norms of units of DKpwith respect to L are units of DLp. Thus, the n K / L ( M p ) are units almost everywhere, so that (3) defines an ideal according to §2,4,(g). By §2,4,(e), the components ap are multiplicative functions of a. Thus, along with the multiplicativity of the element norm, we see that the ideal norm is a multiplicative function, i.e.,

n,/dab) = nK/L(a)nx/L(b). For a # 0 we have, in particular,

(4)

n , / d D , ~ )= DLnK/L(a). (5) I f O , is a principal ideal domain, then 0,and a each have buses with respect to DLconsisting of [K:L] elements mi and a i , respectively. Let

When the bases mi and ai are changed, the determinant laij[ is multiplied by a unit and the ideal on the right in (6) remains invariant. This remark will be used in the proof, which follows. By §2,4,

ap = DLpa. Hence, the ai are also a basis of ap with respect to DLP.Moreover, the mi form a basis of DKpwith respect to O L DUsing . the elements up defined by (2), then, the a p o i form another basis of ap with respect to DLP. Set apmj= 1 oia;i,

so that the determinants laijl and laijl differ but by a unit factor in DLp. But the latter determinant is exactly nKIL(ap). This shows that laijl is divisible by the exact same power of each prime ideal p of L as is nKIL(ap), which by definition (3) above is exactly as often as is nKIL(a).T An ideal a of L satisfies = a[K:Ll. (7) nKIL(a)

11.

72

IDEALS AND DIVISORS

For if in L, ap = DLptlp holds with certain ap EL, then in K correspondingly ap = D K p a pand , nKIL(ap) = a;KL1 yields (7).

3. THEPRIMEIDEALS Assume the prime ideal p of L decomposes in K to (8) DKp= pi1 ... p: with prime ideals pp of K. The exponents e p = eK,L(pJare called the ramification indices of the pp with respect to L. Another natural number, fp =fK/L(pp), called the residue class degree or simply the degree of pp with respect to L is defined by the following theorem: The residue class field O K / p pis an extension of finite degree ~K/L(PJ =

C - ~ K / V , : DLhl

(9)

over the residue classfield DJp. According to $2,4,(i) the residue class fields in question are isomorphic to the local residue class fields DKpp/ppp and -0,,/p,, respectively. It is therefore also possible to define the residue class degree locally as (10) f K , L ( p p ) = C D K p p / p p p :o L p / ~ p l Proof. The residue classes of the elements of D, mod ppand mod p coincide. Thus the former field of the theorem is an extension of the latter. And, there can be at most as many linearly independent residue classes in DK/ppwith respect to DL/pas there are linearly independent elements in DKwith respect to DL,i.e., [K:L]. 7 Given a prime ideal pK of K, there exists exactly one prime ideal pL of L divisible by pK, and then ~ K / L ( P K )= p L J ,

f =~ K / L ( P K ) .

(1 1)

Proof. Start with the residue class field DKmod pK. The residue classes representable by elements of DL form a subring without divisors of zero, isomorphic to DJpL, where pL = D, n pK. Hence pL is a prime ideal of L divisible by pK Moreover, for a prime ideal q, # pL we have (pL, 9,) = -0, and thus (qL, p ~= ) O K ,i.e., pKqL= DKqL.This implies that nKIL(pKqL) = E l L q L , so that (1 1) holds with some power f. It remains to show that f coincides with the residue class degree. For this aim we observe that the pL-components satisfy the corresponding equation nK/L(pKpL) = pipL with the same exponent. Therefore (I 1) need only be proved for the pL components which is done with the use of Eq. (10) for the residue class degree. Apply

.

$4,

THE COMPLEMENT, DIFFERENT, AND DISCRIMINANT

73

the elementary divisor theorem (1,51,2) to the O,,,-modules DKpL and pKpL, arriving at a basis wi of the former and elements t i E O L P Lsuch that ciwi form a basis of the latter. As DKpLpL is contained in p K P L , some of the ri are units in DLpL while the rest are divisible exactly once by pLpL.There are over DLpL/pLpL. Thus exactly as many of the latter as the rank of DKpL/pKPL the norm of p K P Lwith respect to L is, by (6), equal to pipL,where f is the number in (1 I), which is verified. 7 The definitions of the residue class degree and the ramification index lead immediately to the bansitivity formulas f K / k ( P K ) =fK/L(PKlfL/k(PL),

eK/k(PK)= eK,'(PKkL,k(PLh

(12)

where pK is a prime ideal in K and p L denotes the prime ideal in L divisible by p K . Furthermore, arbitrary ideals a satisfy the transitivity formula for the norm nK/k(a) = nL/k(nK,L(a)). (13) It suffices to prove this for prime ideals, and then apply (4). For them, (13) follows from a double application of (1 I ) taking into account the first of Eqs. (12). A most important formula is

where pq runs through all prime ideals appearing in (8).

Proot Take the norm on both sides of (8). Because of (7) the left side Q of (4) and (1 1). 7 gives p[K:L1,while the right side gives ~ ' Q J because

n

Exercise. Build up the theory of the norm of ideals developed in $3 on the basis of Kronecker divisors, in particular proving (14). To do so prove and use ',/PI = [Ddz)/DK(2)pQ .oL(z)/DL(z)Pl' ['K/pQ

'

This method is even a short cut. Our detour serves, though, to familiarize one with the arguments of local considerations. 94. The Complement, Different, and Discriminant

1. THECOMPLEMENT Retain the assumptions and notation of $91 and 3. The elements of K can be taken as vectors in a [K:L]-dimensional vector space over L. If o is an arbitrary pseudotrace of K / L (cf. 1,§4,4), then a/? = o(ap) defines a scalar

74

11. IDEALS AND

DIVISORS

product of these vectors with values in L. In particular, it has the properties of 1,§1,3, if K is simultaneously identified with its dual vector space. Now let a be an 0,-module in K and a&, be the complementary DLmodule in the sense of 1,§1,3. Then is called the pseudocomplement of a with respect to the pseudotrace used. For separable K/L the trace may be chosen for Q. One then speaks of the complement of a. Here, as for all other derived concepts, we use or omit the prefix "pseudo" according to whether a pseudotrace or the trace is being used. The concept of the complement is due to R. Dedekind. As was determined in 1,§1,3, if a and b are modules,

a G b implies a&, 2 b&.

(1)

Further, the following equation is evident: (aa)& = a&Lu-l.

(2)

The pseudocomplement of an ideal is an ideal. Proof. If a has a basis with respect to D, consisting of [ K : L ]elements ai, the complementary basis ai* can be calculated by the method of 1,§1,3. It is a basis of a:/, which is then, along with a, a finite D,-module, and even a finite o-module because b, is finite. But, in general, a has no such basis. Yet it has the rank [K:L] with respect to D,, for if al, ..., acKILl are elements of K linearly independent with respect to L, then there is an a E DL so that aal E a. Without loss of generality, then, let all the a1 E a. They span a submodule b of a, but then the ai* span the module b:/, containing a S L . The former is finite so that, by the finiteness criterion of I,§l,l, we have the finiteness of at/, . Arbitrary elements a, a*, w of a, a:/, DK,respectively, satisfy

a(a*a*w)= a(ao.a*) E DL, 7 Hence, as maintained, we have DKa:,, = a:/,. As a traverses an DK-ideal a, u(a) traoerses an D,-ideul in L. It can be denoted a(a), the pseudotrace of a. With this notation Q

= DL

(3)

holds. For certainly a(aatlL)= b is an integral O,-ideal. If we had b c D,, then a:/,b-' I> a:/,.. But, also

n(aa:/,b-')

= a(aa&,)b-'

= D,,

and thus a a;/,b-' 3 1 Let pseudotraces be given satisfying the transitivity formulas I, §4,4,( 10). Then the pseudocomplements with respect to the extensions K/k,K/L, and

$4. THE COMPLEMENT, DIFFERENT, AND DISCRIMINANT

75

L / K satisfy the transitivity formula: D&k = D;/LD:/k

(4)

*

Proof. Apply ( 3 ) to k instead of L,so that = oK/k(D&k) = 0.

oL/k(°K/L(D&k))

Thus o K / L ( D & k ) is an DL-ideal of pseudotrace oK/L(Di/k)

0,

i.e.,

O:/k *

This implies that and thus, on the one hand, D&kD:jkI s Di/L or

Dz/k c D):/LD:/k * By (3), on the other, oK/L(DE/LD?/k)

= oK/L(D&L)D:/k

= D:/k

so that, by taking o L / k on both sides, oK/k(D&LD&k)

= b L / k ( D t / k ) = O*

This gives

Di/L.D:/k

Di/k *

This, along with (5), demonstrates the assertion (4). 7 We finally mention the two formulas a&L = D&,a 0 it contains a lattice point in its interior aside from the origin. This lattice point cannot change as t + 0, and therefore

If we assign cvosuch irrational values that the corresponding equations cannot hold for rational xi, we have again the inequality sign and so (6). But E 3,so that n(t) is an integer, and by our inequality it would have to be 0. On the other hand, our construction shows that t is not 0, so that n(5) = 0 is impossible. This proves (3). 7

$1. THE FINITENESS THEOREMS

101

The approximation (3) can easily be refined t o t

10(1~)1 > (n/4>"(nn/n !)'. This is done by finding a convex symmetric domain as large as possible within the surface I n ( c xiii)l = 1 of affine space with coordinates x i and then, in connection with the general Minkowski lattice point theorem, finding a lattice point aside from the origin. The inequality (5) does, in fact, define such a convex region, but it leads to the approximation (3). There exist more favorable regions that, at the cost of their simplicity, permit the improved inequality.

3. THEDIRICHLET UNITTHEOREM As already noted, the concept of divisors and of ideals (3-ideals) is essentially identical here, so that the units of K in the sense of II,§5,3 coincide with the units of 3. Dirichlet could already analyze the structure of the group of units (5: Let there be rl real and 2r2 pairwise complex conjugate number fields algebraically conjugate to K . Then, there exist

r - 1 = rl

+ r2 - 1

(7)

units E~ in K , such that every unit of K can be written uniquely as a power product & = 58:' .. &,h,/ (8) up to a root of unity l. Such units are said to form a system of base units of K . Thus, drawing on the fact normally proved in algebra that all the roots of unity of a finite algebraic number field form a cyclic group, the Dirichlet unit theorem asserts that the unit group (5 has a basis of r elements. The proof must be preceded by several auxiliary considerations. In them a surface a,defined by

.

and to be considered as in the n-dimensional affine space R" with real coordinates x i called the "norm one surface," plays an important part. The points are' simply denoted 5 = xiii . Those points having rational coordinates are simultaneously elements of K , and form a group with respect to multiplication. If a is an element of this sort, then

1

t

Cf., for example H. Hasse, Zahlentheorie, p. 454 Berlin, 1949; p. 590, 1963.

102

APPENDIX TO CHAPTER 11. ALGEBRAIC NUMBER FIELDS

defines an affine coordinate transformation, with determinant laij!= n(a) = 1. This gives a faithful representation of the multiplicative group of these a by affine transformations. As multiplication and inverse formation are continuous operations in K n % , the points of % taken naturally as limit points of elements with rational coordinates become elements of an abelian group, which is also faithfully represented by affine transformations.

Lemma 1. In the bounded subset $Icof %, defined by lxil c c, there exist at most afinite number of E E CE such that the corresponding afine transformations map a point of aCinto the same or another pont of aC. Proof. Let 5, q bea pair of such points, with q = E( or E = q t - ' . This product can be calculated in the representation group of affine transformations. It is a consequence of the continuity that the coefficients of the matrices of q, 5, t-', E are bounded, the bounds depending only upon c. Thus, E also lies in a bounded part of %, while on the other hand its coordinates are rational integers. This leaves only a finite number of values for E . 7

Lemma 2. There exists a constant c, such that to every point u E % there exists a unit E with &aE aC. Proof. Use the notation of (10). As lai,( = i - 1 , 1,§2,1 assures us that we can find rational integers x i , not all vanishing, such that

holds for all j . Thus, q E g2.Let y denote the maximum of all In(q)l with rlE32.

1

We now have 5 = xiii E 3: and In(5)l = In(q)l 5 y. Thus 35 is an integral ideal, with norm in a finite set. Using the argument of the last section, 35 also belongs to a finite set of principal ideals 38,.Then 5 = &E with a unit E and /Iv belongs to a finite set. Using (lo), finally, gives &a= p;'q. As q belongs to and 8, to a finite set, there clearly exists a constant c, dependent only upon 3,with &aE aC.7 Based on the fact that the determinant Izivl Z.0(cf. §1,2), we have the (otherwise obvious) following lemma:

Lemma 3. Using the algebraic conjugate bases ziv ,form n

numbered so that ~ r , t v = ~ r , t , . 2 + v ( v= 1, associate the vector I({) with components

logltvl,

..., r2). Then, with every vector 5

v = 1, ..., r - 1.

$1.

THE FINITENESS THEOREMS

103

This maps euery bounded subset of % onto a bounded subset of the (r - 1)dimensional space R r - ' . As

the points of 9l mapped onto a bounded subset of Rr-' belong, conversely, to a bounded set. Proof of the Dirichlet unit theorem. Multiplication of the t becomes addition of the I(() E R r - ' . Lemma 3 carries Lemmas 1 and 2 over to Rr-', in the manner to be expected. By Lemma 1 the I(&) of units E have no limit point at the origin, thus they have no limit point at all. Construct the base units as follows: First choose an giving the vector I(&,) minimal length. If E , , ..., E , - ~ have already been chosen ( e < r - 1) find an E , such that I(&,), ..., I(&,) span a parallelepiped of minimal positive volume. If such an E, exists, then there can exist no E E (5 for which

I(&) = S l l ( E 1 )

+ + *-*

SPI(EP)

(11)

holds with nonintegral si . For otherwise, such an E would exist with 0 S si c 1. Say s, # 0, s , + ~= ... = 0. Then .. , I ( E ~ - ~I(&) ) , would span a parallelepiped of volume smaller than I(&,), ..., I(&,), contrary to assumption. The nonexistence of such an E, could have two possibilities as its cause. First, there could exist an infinite sequence of such E , ~ ,for which the volume decreases monotonely. To see that this is impossible, decompose Rr-' into the subspace R , spanned by I(&,), ..., I(&,-,) and its perpendicular space R , . In the first space, the I(&,) (v = 1, .. ., e - I ) also span a point lattice, and every vector of R , can be brought into the base parallelepiped defined by the same vectors by translation with a vector of the form (1 1) with integral si and e - 1 in place of e. This is done, in particular, for the components in R , of the vectors I(cUj),and these vectors are then replaced by -1 S J ( E ~ ) . This does not change the volume of I(&,), ..., Thus there is no loss of generality in assuming the components of to be bounded in R , . The volume sought is equal to the volume of I(&,), ..., multiplied by the length in R , , which must also be bounded. Thus, of the component of must itself belong to a bounded subset of R r - ' , so that by Lemma 1 there can only be finitely many such E , ~ . The second possible reason for the nonexistence of such an E, is that the volume of I(&,), ..., I(&,) is always zero. From what has already been shown, this would mean that I(&,), ..., I(&,-,) already generate all the translations associated with units. But these translations leave the vectors of R , invariant. Thus there would exist vectors that could not be translated into a bounded set in Rr-' by translations I(&). This would, however, contradict our lemma 2.

104


Thus it has been shown that there exist r - 1 units E ~ such , that for every unit E Eq. (11) holds with rational integral si and e = r - 1. Then, &

= [&?

... &T--i

is satisfied with a unit [ which, along with all its conjugates [, , has absolute value 1. This property is also held by all powers of c. By Lemma 3 they all belong to a bounded subset of a,and then by Lemma 1 there can only be a finite number of them. Thus [ is a root of unity, and the proof is completed.

4. THEREGULATOR Choosing a system of base units determinant lOglE1,lI R = 29 *

1

IoRIE,-I,II

we define the absolute value of the * * a

...

lOgl&1,r-11 (12)

lOgIEr-~,r-11

where for r2 = 0 and eij denotes thejth conjugate of E i , numbered by the convention of lemma 3 above, as the regulator of K. The fact that it is independent of the particular choice of base units or the numbering of the conjugates is almost self-evident. $2. Quadratic Number Fields and Cyclotomic Fields

1. QUADRATIC NUMBER FIELDS Quadratic number fields are discussed in detail in numerous elementary and algebraic number theory texts, so that we may limit ourselves to only the simplest facts, considering them primarily as exercises to the theory developed thus far. A quadratic number field arises when the square root of a nonsquare u E Q is adjoined to Q. There is no loss of generality in assuming u to be integral and free of square divisors. A basis of’ 3 is given by { 1, b( 1 + if u = 1 mod 4 and { 1, &} otherwise.

fi)}

+

Proof. Let a = a b Ju E 3, with a, b E Q . By lI,$l,l we know that a E 3 if and only if the trace s(a) = 2a and the norm n(a) = a2 - ub2 are contained in i = 2. It follows that either a and b are contained in 2, or that a = b = 4 mod 1 and ( 2 ~-) (2b)2u ~ E 0 mod 4; this latter case is thus only possible when u = 1 mod 4. The totality of the a is represented by the bases given. 7

105

$2. QUADRATIC NUMBER FIELDS AND CYCLOTOMIC FIELDS

The discriminant of 11,$4,(12) is BK,Q = iD, where

D = D(I,) =

v

4v

for v = 1 mod 4, for o f 1 mod4.

Decomposition of the rational prime ideals (prime numbers) 3 p : (a) If ( D / p ) = - 1 (Legendre symbol), then 3p remains a prime ideal. &)}, (b) If(D/p) = 1,thengp = p I p 2 ,wherep,,p,havethebases(p,h(x { p , g(x - &)Irespectively, or {p, x {p, x respectively, depending upon whether v = 1 mod 4 or v f 1 mod 4. Here x is a solution of x2 = v mod p if p # 2 and xz = v mod 8 if p = 2 . (c) If (D/p) = 0, then 3p = pz, where p has the basis {p, t ( p &)} if v = 1 mod 4, {p, &} if f p = v f 1 mod 4, and (2, 1 &} in the final case that v z 3 mod 4 and p = 2.

+

Jv>,

+ &I,

+

+

Proof. Let the decomposition be 11,§3,(8),withe, fl of 11,§3,(14).Three cases remain:

+ ... + erf, = 2 because

(a)r=l,e=l,f=2;

(c) r = l , e = 2 , f = l .

(b) r = 2 , e .1= f . 1= 1 ;

In case (a), 3 p is a prime ideal, as n(3p) = p2. In case (c), 3p = p2, and the Dedekind discriminant theorem gives (D/p) = 0. Conversely, (D/p) = 0 implies the case (c). For the case (b), certainly 3p = p 1 p 2 . Now, if a = $(a + bJu> (with a, b E 2 ) is not divisible by p but exactly once by p,, then its norm n(a) = *(a2 - b2v) must be divisible by p exactly once. Thus a and b are relatively prime to p if p is odd, and thus v , and with it D, are squares modp; for v f 1 mod 4, replace $0, $b in this argument by a, b. For u = 1 mod 4 and p = 2, v and thus D are even squares mod 8. Thus then, (D/p) = 1. Conversely, the existence of an a = $(x or = x E 3, with norm divisible by p, follows from (D/p) = 1. Then c1 is divisible by a proper divisor of 3p, and case (a) cannot apply, nor can (c), for then we would have (D/p) = 0. All has been proven. 7

+ 6) + 6

Exercise. Confirm the equations 3 p = p lp z and 3p = p2 by carrying out the product on the right. The units: (a) v < 0. The number r from the Dirichlet unit theorem is 1 . Thus all units are roots of unity. Save f 1, the only such roots which satisfy a quadratic equation are those of order 3 and 4 in the fields with discriminants -3 and -4. (b) v > 0. The number r is 2. No roots of unity can occur, aside from f 1. By the Dirichlet unit theorem, all units are powers of a single base unit E . It is even uniquely determined if one requires E > 1 . For small

106


values of u the most rapid way of finding it is by trial and error, seeking the smallest unit E > 1. 2. SPECIAL CYCLOTOMIC FIELDS We proceed to the consideration of the pth cyclotomic field K = Q(c), where [ is a primitive pth root of unity, and p an odd prime. It is well known that [K: Q ] = p - 1. K is a Galois extension of Q, the group being the multiplicative group of residue classes modp prime to p . It is cyclic. A basis of 3 is 1, 5, ..., The discriminant of 11,§4,(12) is IDK/Q = iD, where m rn/. vn-21 r\+n(n-lh n-2 , , (2)

cp-2.

I

Y

\ , - ,

I

/-\

\

-

Proof. The traces can be found directly from the defining equations:

-1

-1

-1 -1 -1

-1

-1

p-1

p-1 -1 =

-1 -1

...

... ... ...

...

-1 -1 p-1

p -1 =

-1

0

0

0 0 -1 - 1 0 0

a * *

... **-

. ...

0

p

* * a

0 -1 P , 0

['

The are contained in 3, so that a,,, is a divisor of D(1,c, ...). The element 1 = [ - 1 satisfies the equation

f (4=

(x

+ 1)P - 1 = xp-1 + px + + p = 0. X

Hence, n(1) = p, and 31 is a prime ideal. Furthermore, the quotients ("

- l)/([

- 1) = i l - 1

+ +1 * a *

lie in 3, so that all the prime ideals 3(c' - 1) conjugate to 3 A are identical with 31, and n(31)= (31)"- = 3 p . Thus, p = 31 ramifies with index e = p - 1 . The Dedekind discriminant theorem now states that is divisible by p exactly p - 2 times. This proves (2). 7

§2. QUADRATIC NUMBER FIELDS AND CYCLOTOMIC FIELDS

107

Decomposition of the rational prime ideals (prime numbers) 3 q : The decomposition of q = p has been given. Let q # p . Using the Dedekind discriminant theorem, no ramification can occur, so that

39 = q,

***

qr

with r different prime ideals q i of degreef, satisfying rf=p-1.

We maintain that f is the smallest natural number for which 4’-

1= O modp

(3)

holds. Proof. Let q be one of the q i . The residue class field 3/q is an extension of i/q of degree.f, and thus has q f elements. Its multiplicative group has qf - 1 elements, so that every i E 3 prime to q satisfies i 4 ’ - l = 1 mod q. In particular,

Cq’-’

E

1 mod q.

But then, were q f - 1 = 0 mod p not satisfied, all roots of unity would be congruent 1 mod q. Then q would divide ;1= 5 - 1, which is impossible as n(A) = p . Conversely, let q f - 1 = 0 mod p with a minimal f ’ > 0. Let (qf - 1) - (q‘f’ - 1) = q f f ’ ( q f - f J ‘ 1)

qf-tf‘- 1

0 mod p .

From this it is not difficult to see that the minimalf’ is a divisor off. It remains to be shown that, if qf’ - 1 = 0 mod p with minimal f’,then f’=f’. Now, letf’ be a proper divisor off. It is well known from the theory of finite fields that the residue class field 3/q is generated by adjunction of a primitive ( q f - 1)th root of unity to the prime field i/q. But, by our assumption, ‘4 is a primitive (ql’ - 1)th root of unity, and thus not a primitive ( q f - 1)th root. This contradicts the fact that 3/q is generated by adjunction of 5 to i/q. 7 Comparison of the decomposition laws of this and the last sections yields a proof of the quadratic reciprocity law. It is well known that (cf. 42,l of the Appendix to Chapter I)

Now, K , = Q(,/G) is a quadratic subfield of K. If (&p/q)= 1, then q decomposes to two prime ideals in K , . These are, in turn, decomposed into prime

108


ideals of some degree f, which divides [ K : K , ]= + ( p - 1). Thus by (3) we have q*(p-')

= 1 mod p .

The consequence is that the residue class of q mod p is a square in the multiplicative residue class group. This means that (q/p) = 1. The reciprocity law can be completely derived from this with the use of the first complementary theorem.

C H A P T E R I11

Algebraic Functions and Differentials The theory of algebraic functions can be approached in three ways. The first, which we followed in Chapter 11, erects the foundation of the theory of algebraic numbers simultaneously. The second proceeds along the framework of classical function theory, and can therefore only deal with fields of characteristic 0. We will develop this method in the next two paragraphs. This is done for various reasons. First, it is natural to connect our theory with the framework of classical function theory for the general educational value of doing so. Second, the simplest fields of characteristic p > 0 are generated from the rational integers by residue class formation with a prime p . It is possible and often advantageous to generate more general fields of characteristic p , and in particular algebraic function fields by similar residue formation. This process, which shall be carried out in $6,will be explicitly based on this second approach. Third, this approach is characterized by an intuitive clarity which serves to illustrate the abstract concepts. The third approach comes forth from the second; in principle it consists of the same formal arguments, but abstracted from the special assumptions of the classical case. It is the quickest, beyond doubt. We intentionally postpone it until the end of $2, because we believe the student can only appreciate the novelty fully when he has the means to associate concrete formations with the abstract concepts. A study of this book can be started with Chapter 111, as was already mentioned in the preface. At the beginning of $2 Sections I,$l,l and 3 as well as II,$l,I and 2 must be inserted, though. Further, 1,§3 is needed for the proof of the Riemann-Roch theorem in $3. The central problem of the theory is developed and solved in $3. It is the question as to the existence and multiplicity of functions with given zeros and poles. What remains then is the development of a differential and integral calculus. Even if only the classical case is to be considered, it is profitable here to develop first a formal calculus. In other words, differential and integral calculus can be pursued without the continuity concept. Not until Chapter IV will that concept occur. 109

111. ALGEBRAIC FUNCTIONS AND DIFFERENTIALS

110

$1. Power Series Expansions of Algebraic Functions 1. THEFIELDOF POWER SERIES First let ko be an arbitrary field; later we shall have to assume it to be of characteristic 0. By apower series with coefficients a,, E k, a formal expression m

f(x) = 0

or

f(x) =

1 a$',

a , # 0,

(1)

p=V

is meant. One way of defining calculation with power series is to state the rules for determining the coefficients of the sum and product of two power series, as is done for polynomials. Alternatively, the power series can be taken as limits of finite sums in the sense of the valuation of the rational function field k,(x) by which a polynomial f ( x ) # 0 of (1) has value If(x)l = 0'' with some constant t s between 0 and 1, fixed once and for all. f(x) is said to have a zero of order v at x = 0 if v > 0; if v < 0 then f ( x ) has a pole of order Ivl at x = 0. In that case -1

9 a,,xM

p=V

is called the principal part of the pole. If k, is contained in the field C of complex numbers the radius of convergence is defined for the power series. A concept formally related to this is the following: Let the ideal theory of II,§l hold in ko . The largest ideal a such that a'a,, is almost always an integral ideal is called the arithmetic convergence radius. For example, the series xN, l/p!xr have the arithmetic convergence radii (1) and (0). In the following we shall often make statements of the sort: A power series has a (respective) convergence radius fO. This means that if ko E C then the convergence radius is positive, and if the ideal theory holds for ko then the arithmetic convergence radius is not the zero ideal. AN power series with coeflcients in k , form a j e l d . The same is true of those power series with nonzero convergence radius. The simple steps necessary for the proof need not be carried out here. We will only show that the reciprocal power series g(x) of a power series f(x) with radius of convergence # 0 also has radius of convergence # O . Evidently it suffices to do this forf(x) = 1 cIx c2x2 ... ,and verification consists of the application with g(x) = 1 y and cx + cp(x,y) = (1 -f(x))(l + y ) of the lemma which follows. This lemma, due to Cauchy, not only will serve us on several later occasions, but its underlying principle has even proved useful in other branches of mathematics.

1

+ + +

Lemma. Let

+

91.

POWER SERIES EXPANSIONS OF ALGEBRAIC FUNCTIONS

111

be a power series, all of whose terms have degree 2 2. The equation Y

= cx

+dx, Y)

(3)

then has, as its solution, a power series m

y =

1 c,xp

with c1 =c.

(4)

,=1

If the

ideal theory holds in k, and there exists a nonzero a E k, such that c i j are integral, then (4) has an arithmetic radius of eonvergence # O . I f q ( x , y ) has a positive radius of convergence, so does (4).

ai+j-l

Proof. Equation (3) is solved by iteration. Set

y , = cx, *.., Y,+l = cx + 44x9 Y,). Here y, and Y , + ~do not differ in their first p terms. Indeed, by hypothesis, this is so for 11 = 1, and assume it to be so for p - 1 . The inductive hypothesis thus states that y , = Y , , - ~ xez, with a power series z, without negative exponents. Iteration yields

+

Y p +1

= cx

+ d x , Y,- + XPZ,)

= cx

+ q(x, y,-1) + xc+lz,+l

1

= y,

+

XP+l

Z,+l,

where the hypothesis concerning q ( x , y) assures that z a f l also has no negative exponents. Hence the formal limit y = limy, exists; it solves (3) and has the property (4). To demonstrate a nonzero arithmetic radius of convergence under the given assumptions, replace x and y by ax and ay. Then (3) becomes the corresponding equation with a i f j - l c i jin place of c i j . Thus there is no loss of generality in assuming the c i j to be integral. I n the process of the iteration it turns out that the arithmetic radii of convergence of all the series y,, and therefore also that of y , are divisors of the denominator of c. Only the question of the positive radius of convergence remains. Let q ( x , y ) converge absolutely for 1x1, lyl < y. Then, with some further constant y o , we have . . I C I < y o y - l = c, lCijl < )Joy-'-' = cij. (5) From their calculation above the c,, of (4) turn out as polynomials P,(c, c i j ) with positive integral coefficients. Replacing c and the c i j by their majorants ( 5 ) yields the majorants C,, = P,(C, Cij) of the c, . The C,, are coefficients of the power series m

z =

1 c,x,, ,= 1

112

111. ALGEBRAIC FUNCTIONS AND

which solves the equation z = cx

+ 1CijXiZ'. Li

Using ( 5 ) this equation becomes

-1

YZ = YO((y

DIFFERENTIALS

- x)(y - z )

--). Z

Y

This is a quadratic equation with solution

It can be expanded in the power series (6) and it is from this nature of (6) that we know it has a positive radius of convergence. The power series (4) then has a radius of convergence at least as large. 7 Exercise. Show similarly that if f ( x ) and q(x) have radii of convergence #O and q(0) = 0, then f(q(x)) also has a nonzero radius of convergence.

2. DIVISIBILITY, REARRANGING OF POWER SERIES Power series (1) with first exponent v 2 0 are called integral or holomorphic. They form an integral domain 3. The power series with first exponents v 2 vo form an %deal. The ideal for vo = 1 is a prime ideal La, and the elements of ko are a complete system of representatives of the residue classes of 3 mod La. Thus the residue class ring 3/Qis isomorphic to k , ,the proofs being obvious. A power series q(x) = q with first exponent v = 1 is called a prime element. For simplicity sake we assume that the coefficient of x in q(x) is 1. Letting f ( x ) be as in (l), we have fl(x) = f ( x ) - a&)" and further f2(4

= a,,,+ lx"+l

+

- a2,v+2xv+2 + =fdx) - al*,+lq(x) V + l

*.*.

This process finally leads to the identity

f(x)

= avq(x)Y

+ al,v+lq(x)y+l+ Q2,v+2q(X)Y+2 +

(7) which is still to be considered purely formal, that is, in the sense that the coefficients coincide. In the next section we will show that in the special case f ( x ) = x this series has a radius of convergence #O if this is so for&). Then, by theexercise at the end of the last section, this holds for the series (7)in general. We will illuminate the theory of power series briefly in the light of the results of Chapter 1I.t Let 3 be an integral domain with quotient field k. Let

t

The reader starting with Chapter 111 may skip these remarks.

***,

01.

113


there exist a single prime 3-ideal Q, which is a principal ideal, I, = 3q. Further, let the residue classes of 3 mod Q all be represented by elements of a field k, contained in 3. Any f~ 3 can then be expanded in a power series (7) with coefficients in k,: let a, be that uniquely determined element of k, which satisfies the congruencefr a, mod Q. Thenf, = (f- a,)q-' E 3 ; and sayf, = a, mod Q. Again let f 2 = (fl - a,)q-' = a , mod Q , etc. Call a, alq ... the power series expansion off. Iffis not contained in 3 , then with some exponent - v certainly f q - " E 0,and this element can be treated as described. Then f is associated with the power series beginning with avq". It is now easy to show that the correspondence between the elementsfc k and their power series is an isomorphism. The situation assumed always occurs if k is an algebraic function field, the constant field k, being assumed algebraically closed. For 3 choose the integral domain of functions of k whose denominator divisor is not divisible by some prime divisor q. (Compare this with the example a t the end of 11,92,1.) The prime ideal Q is the totality of those functions whose .numerator divisor is divisible by q. Ask, is algebraically closed all prime divisors, and in particular q, have degree 1, so that all residue classes of 3 mod Q are represented by elements of k,, . Therefore every algebraic function can be expanded with respect to a prime element q for an arbitrary place q. From this point of view divisibility comes first, and power series are a consequence. The foundation of the theory which we are now developing, though, starts with power series and bases divisibility upon them.

+

+

3. INVERSIONOF A POWERSERIES We start with the following lemma:

Lemma. Let f(x)=1

+ a,x + a2xz +

=1

+q(x)

(8)

and e > 1 be a natural number not divisible by the characteristic of k, then exists exactly one power series

. There (9)

g(x)=l +c,x+c2x2+-*

satisfying the equation g(x)' = f ( x ) . I f f ( x ) has a conoergence radius # 0, then so does g(x). Proof. In (8) let a, our equation

=

= ar-l = 0, a,

g(x)' - f ( x ) = exr-'(y

# 0, and put g(x) = 1

- (a,/e)x -

..a)

+ x'-'y.

Then

=O

is a special case of (3), and our lemma is implied in that of III,$l,l. 7

114


DIFFERENTIALS

With this preparation we can show: Let e be a natural number not divisible by the field characteristic and y = f ( x ) = a,xe

+ a,+ l x e + l +

..a,

,

a, # 0.

(10)

(ga-1,

(11)

There then exists a power series x = q(q) = c,q

+ c,q2 +

c1 =

. * a ,

with coeficients in the field k&&) which, substituted for x in (lo), yields the identity y = qe. After determination of an eth root of a, all the coeficients of (1 1) are uniquely determined. If (10) has a nonzero radius of convergence, then so does (1 1). For the case e = 1 where y is a prime element, (1 1) is the expansion of x with respect to it, which was shown to exist in the last section. For this case the coefficients ci of (1 1) can even be explicitly stated.t They are the coefficients of xi in the power series expansions of

Proof. Choose the value given in (1 1) for c1 and set indeterminate coefficients in a power series for q(q). Substitution of (1 1) into (10) yields a power series in q for y , beginning with qe and with ea,c;-'c,

+P,(c~,

cp-1;

a,,

..., a,+,-,)

as the coefficient of qe+lr-', thep, being polynomials in the variables indicated. Set these coefficients from p = 2 onwards to zero; this gives recursion formulas for the c,. They are solvable because e is not divisible by the characteristic, and the solution is unique. No generality is lost in assuming a, = c1 = 1 for the convergence proof. Using the above lemma the right side of (10) may be written y = g ( x ) , = (x

+ a2'x2+

.a*)',

where g(x) has a radius of convergence # 0. The series (1 I) must now be found so as to fulfill the identity q = g(x). This means that the convergence proof is only necessary for the case e = 1. Now write (10) as Y =x

+ 444

which is a special case of (3), and which was shown in III,§l,l to be solvable with a power series of radius of convergence # 0. 1

t M. DEURING, Eine Bemerkung iiber die Biirmann-Lagrangesche Reihe, Nachr. Akad. Wiss. Gottingen Math.-Phys. KI., 33-35 (1946).

$1.


115

4. ALGEBRAIC FUNCTIONS; REGULAR PLACES

Let K be a finite separable extension of the rational function field k = k,(x). (For ko of characteristic 0 the assumption of separability is trivial.) The theorem of the primitive element states that K = k ( y ) with some y € K satisfying the irreducible equation f(y, x) = y"

+ c,(x)y"-' + ..* + c,(x)

=0

(12)

with coefficients ci(x) E k. Its degree is n = [ K : k ] . If h(x) is the common denominator of the rational functions ci(x), then y , = h(x)y satisfies the equation y,"

+ h(x)c,(x)y;-' + h(x)Zc2(x)y;-2 + .*. + h(x)"c,(x) = 0,

with coefficients that are polynomials in x . Replace y by y , so that now y is a primitive element of K satisfying Eq. (12) with polynomials ci(x). We want to expand all the functions of K in power series in x - t or x - ' , where l is an arbitrary element of k,. We state beforehand that this is not always possible, but it is in a great majority of the cases. Clearly it suffices to so expand the single function y . For x trivially has such an expansion, so that then $1 ,I assures the same for all rational functions in x and y , that is for all functions of K. In a field extension Kl of k the polynomial (1 2) decomposes to factors of the first degree n

f(y, X> =

JJ (Y - wi)

i= 1

whose zeros, by our assumption, do not coincide. The discriminant icj

can, as is familiar, be calculated as a polynomial in the ci(x), so that it is itself a polynomial in x not identically 0. This polynomial has, at most, a finite number of zeros x = t e k , , called the critical places? of Eq. (12), other E k, being called regular places. At a regular place the equation f ( y , 5 ) = 0 has exactly n distinct zeros y = q i in an extension of k , , and we maintain : For a regular place 5 there exists, to euery zero y = q i o f f ( y , c), exactly one power series

0 which solvesf ’(y’, x‘) = 0. To simplify the notation we omit the primes and consider Eq. (12) under the assumption en-

,(O) # 0,

C,(O)

= 0.

It can now be written as

which is of the form (3). Thus the solvability now follows from the lemma in IlI,$l,l, as does the existence of a nonzero radius of convergence, provided k, satisfies the necessary hypothesis. Remember that we started with the substitution (14), we see that the coefficients of the equation solved belong to the field ko(qi). Following the solution of the equation in the proof of the lemma it is seen that the coefficients of the solution (13) also belong to that field. 7

5. CONTINUATION; CRITICAL PLACES We must now assume that the field of constants has characteristic 0. Let be a critical place. Again substitute as in (14), so that c,(O) = 0. In general, though, we now no longer have c,-,(O) = 0, but can only prove the following: There exists a natural number e and a power series with Coefficients in a finite field extension of ko ,

which solves (12). It has a convergence radius # 0 Proof. Replace (12) by the more general equation H

$1.

117


which can be decomposed into its homogeneous parts fr(y, x)

=

2criyixr-i i=O

and about which we assume nothing further than that it contain y and have no multiple solutions .The sum (16) is even permitted to be infinite. Now let h be the smallest degree for which some f r ( y ,x) is nonzero. This h is called the subdegree of f ( y , x), and the existence of the solution (15) is proved by induction on it. One first remark will be useful. Assume a power series x

+

= .,(.;/* a,+,(qy)e+l

+

a, # 0,

. a * ,

satisfying (16) has been found. The theorem of §1,3 then gives the inverted series

vi

+ + y = c,,(g/x)e’+ cep+l(fi)e’+l+

and

= bid;

62(fi)2

* * a ,

..*

is a solution of (16). With these manipulations a respective radius of convergence persists, so that we may interchange x and y in the course of the proof. This immediately settles the beginning of induction. By the last section h = 1 implies that a power series expansion exists either for y in x or for x in y , with first exponent v > 0. We may therefore assume the contention to hold for all smaller subdegrees than h, and let h > 1. By interchanging x and y if necessary, we may assume that #

fh(y,

In (16) set y

= xy,’

aXh*

and divide by x”, arriving at

fl’(Yl’7 x) = X-hS(xYl’, x) =fh’(yl’)

+ Xfh:l(yl’) + + X”-Yn‘(yl’)+ ”‘

***

= 0,

with fr’(y1’) = x-‘Yr(xY1’, x),

polynomials in y,’ alone. Our hypothesis assures thatf,’(yl’) is not a constant, so that this polynomial must have a zero Cl in some extension of k o . Set Yl’

= Yl

+ Cl

so that in the resulting equation, fl’(Y1

-k

Cl,

x) =fl(yl, x) =fh(y1)

+ xfh+l(yl) + “’ + x”-hf.(yl) +

*’*

= 0,

111.

118

ALGEBRAIC FUNCTIONS AND DIFFERENTIALS

fh(0)= 0 holds. Now, the degree offh(y,) is at most h, so that the subdegree hl of f,(y,, x ) must lie between 1 and h. If, indeed, we have attained h, c h, then we know by inductive hypothesis that y is a power series in (x)'le with some natural number e, starting with Clx. This would complete the proof. If, on the other hand, we still have h, = h, we repeat the argument. It must be shown that the subdegree h, if 2 2 , cannot remain unchanged infinitely often. If h, = h, thenfh(yl) has a zero of multiplicity h at the place y, = 0. This implies (writing partial derivation with respect to the first variable as an overbar) f,(O, x) = 0 mod x. fl(O, x ) = 0 mod x , Because f l ( Y 1 , X ) = x-hf(x(y1

+ el), X I ,

f l ( Y 1 , X ) = X'-hf(x(Yt

+ Cl),

XI,

(17)

it follows that

f(Clx, x ) = 0 mod xh+',

f ( C , x , x ) = 0 mod xh .

As we have assumed that the subdegree does not decrease at the next step either, etc., these same congruences also hold forf,(y,, x) and the polynomials obtained from them in like manner. We maintain that the congruences

+ + Cqx9,x) = 0 mod x q h + l , j(Clx + + Cqxq,x ) = 0 mod xqh, f(Clx

(18)

hold for q = 1, 2, ... and also for the polynomials f l ( y l ,x), etc., formed as described above. We prove this by induction on q, the first step being already taken. Our inductive assumption, which we formulate for f,in place off, is fi(c2x + ... + CqXQ-l, x ) 3 0 mod x(Q-')~+' ,jl(c2x CqXQ-', x ) 3 0 mod ~ ( 4 - l ) ~Using . (1 7) this gives (18). But, the congruences (18) would show that the infinite power series Clx c2x2 is a double root of (16), which would contradict the hypothesis that (16) is irreducible. 7

+ +

+

+

6. PUISEUX'STHEOREM Let K be an algebraic extension of the rational function field k = k,(x) of degree n = [K : k ] , where ko is the exact constantfield of K and of characteristic 0. Then, with every element of k, there are associated r = r( g ( W )

(11)

(Riemann’s part of the Riemann-Roch theorem). If the constant field k , E C, then we shall see in IV,§l that g is even a topological invariant of a certain two-dimensional manifold associated with the field K independently of the model used. This would be another proof of the invariance of the genus. As for the canonical class, we finally have: W is the only class with the properties (6) and (7). For, if W‘ is some class, satisfying these equations, then (2) reads dim( W /W ) = 1. But as g( W / W )= 0 the Riemann-Roch theorem shows that W ‘ /W = (1). 4. EXTENSION OF THE CONSTANT FIELD

Let k,’ be an extension of k , obtained by the successive adjunction of finitely many elements. We want to adjoin them as well to K and thus obtain a function field K‘ over k,‘, and to study the behavior of the divisors under the extension. But this meets with difficulties if the characteristic is a prime. We shall avoid these difficulties by imposing the following restriction on the field K which does not exclude the most important applications of the theory.? K has the exact constant jield k , and is separably generated over k , . This means, there exists at least one “ separating” element x such that K is a finite separable extension of k = k,(x).

t For a more thorough treatment see the book by C. Chevalley cited on page 142 and J. Tate, Genus change in inseparable extensions of function fields, Proc. Amer. Math. SOC.3, 400-406 (1952).

136

111.


Given a finitely generated extension k,’/k, there exists exactly one minimal field

K’= Kk,’ over ko containing both K and k,’ as subfields with K n kot = k , . K is a finite algebraic function field with the exact constant field k,‘. If k,’/k, is finite then

[K’:K] = [ko’: ko].

(12)

Proof. The adjunction of a single element to k , and K restores the situation and it suffices to treat such an extension. The contention is obvious for the adjunction of a transcendental element 9. Now let 9 satisfy the irreducible equation f(9) = 0 in k, . If it remains irreducible in K we can adjoin 9 to k, and K without ambiguity, and (12) holds. Iff (9) =fl(9)fi(9)were a proper decomposition in K , then K would contain the coefficients of fl($). These are algebraic over k , and thus, as k, is the exact constant field, are contained in k , . So, contrary to the hypothesis, the decomposition would already hold in ko In order to show that the exact constant field of K’ is k,’ we assume at first that k,’ is separable over k , . Let

.

a = Po

+ P19 + + Pm-19m-1 . - a

with

Pi€ K

(13)

be a constant in K($), i.e., algebraic over k,’ and therefore over k , , where m is the degree of f(9). Replacing here 9 by its conjugates 9, = 9, ..., 9, with respect to k , , we get m constants ai = Po + P,S1 + .-I. Using the Vandermonde determinant we can compute the Pi as

Pi = c i p j

with cij E k , .

i

Thus the P r are algebraic over k , and therefore lie in k , , it being the exact constant field of K . Thus, also a E k,(9) = k,‘, which is what was to be proved. Finally, let 9 be an arbitrary algebraic element over ko and a a constant in K(9) satisfying an algebraic equation g(a) = 0 in k,’. Furthermore let k,’ be the largest subfield of k,‘ which is separable over k , . With some power pe of the characteristic the peth powers of the coefficients of g(o1) lie in k,’. Now ape satisfies the equation gp’(aP’) =(g(a))Pe= 0 in k,’. Using what already has been proved ap must lie in k,’ and a fortiori in k,’. So a lies on one hand in k,‘ or a purely inseparable extension of k,’ and so in a purely inseparable extension of k,’(x), also. On the other hand, a lies in ko’(x,y ) , which is, by assumption, a separable extension of k,’(x). So a lies in k,’(x) and, being a constant, in ko’. This completes the proof. 7

$3.

137

THE RIEMANN-ROCH THEOREM

The divisors of K occur among those of K in a natural manner. We now study the behavior of the degrees and dimensions of the divisors of K under such an extension. Under the assumptions made above the degree of a divisor of K does not change with the extension of the constant field, and the dimension does at least not decrease. If k,' is separable over ko , the dimension is an invariant. Proof. Let x be 51 transcendental element of K . Then, by 11,§5,(16) or III,92,(20), the degrees of a divisor a in K' and K are those of both sides of nK'/ko'(x)(a)

= nK/ko(x)(a)?

whence the invariance of the degree. Next we study the dimension of a under a finite extension of k , . Let a, be multiples of a - ' in K , linearly independent, with respect to k , . If they are also linearly independent with respect to k,', the dimension of a in K' is at least as large as that dimension on K . This is indeed the case, for a linear relation in k,' would mean c a p , = 0 with a, = wiai,, tli, E ko and wi being a basis of k,'/k, . This would lead to wi ~ a i , a ,= 0. But because of (12) the wi also form a basis of K'/K. So we find aiva, = 0 and, because the a, are linearly independent with respect to k,, aiv= 0 or tl, = 0, as contended. If k,'/k, is even separable, all multiples of a- in K'are linear combinations of those multiples in K. In fact, let k,' = k,(9) and

c c

a = b,

+ b19 + + b,-,9"-'

3

bjEK,

be a multiple of a - ' in K'. The divisor aa remains integral if 9 is replaced by its conjugates S i with respect to k,; in this way we get m integral divisors aia in some further extension field with ai = bo + b,Si , Using the Vandernomde determinant we see that then the bia are integral, which is what we proposed to show. Lastly we adjoin an indeterminate 9 to k, and K. The substitutions 9 + 9 + y with constants y E ko are automorphisms of K' which leave K fixed and take integral divisors into integral divisors. Thus, if a(9) is a multiple of a - ' in K', so is a(9 7). a(9) is a quotient of polynomials in [ K 9 ] . Now, if k , is an infinite field, and a(9) were not the product of an element of K (which could be taken into the numerator) and a polynomial in k,[9] we should get infinitely many multiples a(9 y) of a - 1 linearly independent with respect to k,(9), which contradicts the Riemann-Roch theorem. If ko is a finite field we only need extend it finitely which leaves the dimension of a unchanged, in order to arrive at the same contradiction. So a(9) has as its denominator a polynomial in k 0 [ 9 ] . Multiplying a(9) by it we may assume a(9) to be in K [ 9 ] . Because a(9 y ) is always a multiple of a-I,

+

+

+

+

138


all the coefficients of a(9) must be multiples of a-'. With this the proof is complete. 7 Under the assumptions made above the genus of K' is at most that of K. If k,'lk, is separable, K and K' have the same genus.

Proof. Take a divisor a of sufficiently large degree such that (1 1) holds both in K and K'. Because g(a) is invariant and dim(a) does not decrease, the genus g does not increase. It even remains invariant if the dimension of divisors does also, and this is the case with a separable extension of constants. 7 If the above assumptions hold, and if K and K' have the same genus, then the dimension of a divisor of K is invariant under the extension. Proof. Let a be some divisor and b an integral divisor of so large a degree that, for ab, (11) holds both in K and K'. Since the degree is an invariant and the genus g is supposed the same, the dimension of ab does not change with the extension. The multiples of a - ' are among those of (ab)-'. Now, if the extension would lead to an increase of the number of multiples of a-' it would do the same for those of (ab)-'. This is not the case. 7 The genus can, indeed, decrease under extension of constants as is shown by the following example. Let k, be a nonperfect field of characteristic p > 2 and a an element of k , not a pth power. Let K = k,(x, ,)-, a separable extension of k = k,(x) as p # 2. The numerator divisor of x p - a and the denominator divisor of x are squares of divisors in K . By the Dedekind discriminant theorem the discriminant 3K,k is then divisible by the product of these divisors (and is, in fact, exactly the product). By ( 5 ) then, g 2 (p - 1)/2. If thepth root b of c1 is adjoined tok, then K' = ko'(x, ,/x - b) = k , ' ( G b ) is the rational function field in the variable Jq. Its genus is g' = 0. We conclude this section with an example where K' has an exact constant field larger than k,'. Let k,, be a field of characteristic p > 0 and A,, A, algebraically independent indeterminates over koo. Set k, = koo(A,,A,) and K = k,(x, Adjoin 9 = The element

q m ) .

fl.

m x-

9X

=

6

of K' is a constant, but not contained in k,(9). An alternative treatment of the questions of this section would use the following assumptions: K = k,(x; y,, ..., y,) where y i is defined by an irreducible equation f,(yi ;x, y,, ..., y i - ,) = 0 with coefficients in k, which remains irreducible under any algebraic extension of k , . Such equations are called absolutely irreducible. But it can be proved that this assumption amounts to the same as made above.

63.


139

5. THEFIELDS OF GENUS of If and only if K is of genus g K = k,(x) with a suitable x E K.

=0

and has a divisor class of degree 1 is

Proof. The fact that k,(x) really has genus g = 0 follows from (8) with K = k. Both denominator and numerator of x are divisors of degree 1. Conversely, let g = 0 and A have degree 1. As dim(A) = 2 by (2), there must exist two linearly independent integral divisors 3 and n in A. This independence assures that the principal divisor (x) = 3/11 is not the unit divisor, and then x $ ko . Thus the field K is a finite extension of k = k,(x) of degree, say, n. By II,§5,4 or 111,§2,3 the numerator and denominator divisors of ( x ) in K have degree n. But these divisors 3 and n have degree 1, so that n = 1 and K = k,(x), as was to be shown. Either a function jield K of genus g = 0 with the exact constant j e l d ko contains a class of degree 1 and is then K = k,(x) with suitable x, or only divisors of even degree occur. In the latter case K = k,(x, y ) where an equation with coeflcients in k, of the sort

+

+

+

a l l x 2 a 1 2 x y azzy2 a o l x

+ ao2y + a,, = o

(14)

holds between x and y , it having no solutions in ko . Proof. Now assume there to be no class of degree 1 in K. Clearly all degrees are multiples of some smallest degree g o . But g( W) = 29 - 2 = - 2 assures go = 2. By (2) a divisor class A of degree 2 has dimension 3; let x, q, it be three linearly independent divisors of A, and (x) = xn-’, ( y ) = qn-’. We maintain that, for any two elements a, /?E ko except tl = fl = 0, K is a second degree extension of k,(ax by). Indeed, the denominator of the principal divisor (ax by) is ti, and is of degree 2. By the same argument as above, then, [K: ko(ax b y ) ] = 2. Thus, every linear combination a’x p‘y # 0 satisfies an equation of degree 1 or 2 over k,(ax by),

+

+

+

+

(a’x

+

+ B’y)’ + (a’x + p ’ y ) f i ( a x + By) +f2(ax + py) = 0.

All of these equations must originate in a single equation, which is only possible when it is of the form (14). In particular, a z z # 0. For u z z = 0 would imply that either K = k,(x) or x and y were independent. The latter is impossible, and the former contradicts the hypothesis that K has no divisor of the first degree. We must finally show that (14) has no solutions in k , . If t, q were a

t To gain practice in the basic concepts it is advisable to read IV,52,1 after the present section. There fields of genus 1 are similarly considered.

140

111.


solution and p a divisor by which the numerator of x p would also divide the numerator of

- II were divisible, then

a Z 2 y + ( a d + ao2)y+ a l l t 2 + a o l t + aoo = 0. But by assumption this polynomial in y decomposes into two factors of the first degree aZ2Cy- q)(y - q’), with aZ2# 0. The numerator of 0,- q ) or (y - q’) would be divisible by p, say (y - q). Now, the numerators of ( x - t) and (y - q) are both of degree 2 and, as there are no divisors of degree 1, the divisors of both these numerators would be p and g(p) = 2. On the other hand, ( x - t) and 0,- q ) have denominator n, which would imply ( x - 5 ) = 0,- q), contradicting the linear independence of x, I), n. Conversely, let (14) be unsolvable in k , . Then either a22 # 0 or aZ2= a12= aO2= 0. The latter would make y independent of x . Now form the field K = k,(x, y) with this equation. Let us first determine the genus of K. The principal order 3 of K with respect to 2

i=

n i,=

k0[x]

P+,

has the basis 1, y, as aZ2# 0. The same argument, after (14) has been divided by x 2 , shows that 1, y / x is a basis of 3, ,the principal order of K with respect to i, . Define a pseudotrace o(a1

+ a2.Y) = a2

(a19 a2 E

k,(x)).

The pseudodiscriminant of 3 is then

9,= i A(1, y ) = i and the pseudodiscriminant of 3, is a divisor of

i, A(1, y / x ) = i,n2, where n is the denominator divisor of ( x ) in k = k,(x). Then gkPK,k)

5 2.

Equation ( 5 ) for the genus then assures g = 0, for g < 0 is impossible. There can be no divisor class in K of degree 1, for then we would have K = k&) with some z. This would make x =f ( z ) and y = g(z) with rational functions over k, . Setting this into (14) with an arbitrary z in k, would lead to a solution of (14) in k , , contrary to hypothesis. The assertion is thus proved in full. 7 6. LUROTH’STHEOREM Let K have the exact constant field k , and genus 0, and L be an intermediate jield not equal to k , Then L is a function field with exact constant field k , of genus 0. If K = k,(z) then L = k,(t) with a suitable t E L.

.

83.


141

Proof. Let x be some element of L not in k , , and k = k,(x). L and K are then finite extensions of k . Applying the Hurwitz genus formulas (9) and (10) with g = g ( K / k ) = 0 gives [ K : L ] g ( L / k )= [ K : L ] - $gL(BK/L) - 1 < [ K : L ] .

The genus g(L/k) of L must therefore be smaller than 1, i.e., 0. If K = k,(z), and were L not generated in this manner, then L = k,(x, y ) would hold with x, y satisfying Eq. (14). Now, x and y are rational functions of z with coefficients in k , . Setting these into (14) with an arbitrary z in k , leads to a solution of (14) in k , , contrary to hypothesis. 7 The extensions k,’/k, in which (14) becomes solvable coincide with the splitting fields of a certain division algebra of degree 2 with center k, . There is a similar connection between division algebras of degree m > 2 and center ko and certain function fields K/ko in m - 1 variables. See P. ROQUETTE, On the Galois cohomology of the projective linear group and its application to the construction of generic splitting fields of algebras, Math. Ann. 150 41 1 4 3 9 (1963). PROOFS AND GENERALIZATIONS OF THE RIEMANN-ROCH THEOREM 7. FURTHER It is impossible to give a complete survey of all proofs and generalizations here. Of course, we may omit all extensions to functions of several variables, especially as the notion of a generalization of the Riemann-Roch theorem is interpreted with some generosity there. We also omit methods leading from the theory of algebraic curves (found, for example, in [I I]?). Basically, three proofs then remain. The very first proof dates back to Riemann. It stays in the framework of classical function theory, and is based on the so-called Dirichlet principle, that is, on existence theorems for solutions of the partial differential equation A U = 0. Modern treatments are found in [l] and [12]. From the point of view of the algebraist today, this access is rather a field of application and exercise for certain methods of analysis than a constructive proof, as is wanted for any development of the theory. These intentions also appear to motivate Teichmueller’s proof [2], based on the uniformization of algebraic functions and the theory of Poincare series. Otherwise one would have to call this foundation of the theory the longest detour h u s far to the Riemann-Roch theorem. Two substantial extensions of the Riemann-Roch theorem, based on generalizations of the concept of divisor, fall into connection here. They are related to each other, but each surpasses the other in one direction. One of them is due to Weil [lo] and the other to Peterson; the latter will come up in IV,$4,9. ~~~

~~

t The numbers in brackets refer to the bibliography at the end of this section.

142

111. ALGEBRAIC FUNCTIONS

AND DIFFERENTIALS

While the proof of Dedekind and Weber consists of substituting the function-theoretical concepts into an already standing general statement, Weil's proof [9]starts with a new idea (cf. also [3], [4], [7], [8]). The residue theorem resp(al du) = 0 (cf. $4) gives, for every differential du, a linear relation between the local power series expansions of any function ct E K. Inverting this fact, define a system of such relations (for all a) to be a differential. It is easy to associate a divisor with a differential in this sense; the divisors of all differentials lie in the same class, the canonical class. The RiemannRoch theorem is a statement concerning the number of solutions of two associated problems of linear algebra. These problems are, despite formally different presentation, the same here and in the proof of Dedekind and Weber. Thus, seen from a higher vantage point, the proofs do not differ essentially. It should still be mentioned that the Dedekind-Weber proof is to be found in Schmidt [6] in somewhat modernized form. The Riemann-Roch theorem has been generalized in three senses. Rosenlicht [5] based his divisor concept on ideals of more general rings of K than our 3, (cf. also [4]). Witt [13] considered the theory for hypercomplex systems in place of fields. The third generalization has already been mentioned. All these generalizations can easily be derived from the theorem in I,§3,3. REFERENCES [l] M. SCHIFFER and D. C. SPENCER, Functionals on finite Riemann surfaces. Princeton, 1954. Theorie der analytischen Funktionen, p. 530. Berlin[2] H. BEHNKEand F. SOMMER, Gottingen-Heidelberg, 1955. [3] C. CHEVALLEY, Introduction to the theory of algebraic functions of one variable. New York, 1951. [4] P. ROQUEITE,ober den Riemann-Rochschen Satz in Funktionenkorpern vom Transzendenzgrad 1, Math. Nachr. 19, 375-404 (1958). [5] M . ROSENLICHT, Equivalence relations on algebraic curves, Ann. of Maths. 56, 169-191 (1952). [6] F. K. SCHMIDT,Zur arithmetischen Theorie der algebraischen Funktionen I , Math. Z. 41,415-438 (1936). On the theorem of Riemann-Roch, J. Fac. Sci. Univ. Tokyo Sect. I, [7] T. TAMAGAWA, 6, 133-144 (1951). [8] B. L. VAN DER WAERDEN, Algebra 11,4th ed. Berlin-Gottingen-Heidelberg,1959. (91 A. WEIL,Zur algebraischen Theorie der algebraischen Funktionen, J. Reine Angew. Math. 179, 129-133 (1938). [lo] A. WEIL, Gdndralisarion des fonctions Abdliennes, J. Math. Pures Appl. [IX], 17, 47-81 (1938). [ l l ] A. WEL, Sur les courbes algdbriques et les varidtds qui s'en ddduisent, ActualitCs Sci. Ind. 1041 (1948). [12] H. WEYL,Die Idee der Riemannschen Fluche, 3rd ed. Berlin, 1955. [I 31 E. Wrrr, Riemann-Rochscher Satz und Zetafunktion im Hyperkomplexen, Math. Ann. 110, 12-28 (1934).

w. DIFFERENTIALS

143

$4. Diffeientials

I. DIFFERENTIAL QUOTIENTS From now until the end of Chapter IV we will always assume K to be separably generated. If a field has the characteristic 0 then of course all its elements are separating. But, even in the case of prime characteristic, we have:

If the field of constants is perfect, then K is separably generated. Proof. Let the characteristic be p > 0, and let K be generated by adjoining the finite number of elements x , y , , .. ., y , to k , , among which the equations f ; ( y i ,x ) hold. First, let h = 1. Iff ( y , x ) is not a polynomial in y p and x p , then f(y, x) is separable either with respect to x or with respect to y. Thus either K/k,(y) or K/k,(x) is separable. If, however, f ( y , x ) = F(yp,xp), let Fp-' be that polynomial whose coefficients are the pth roots of those of F. By hypothesis these coefficients lie in k , . Then f(y, x ) = (FP-'(y, x))' = 0, so that FP-'(y,x ) = 0. If this polynomial again contains only powers of x and y divisible by p , the same argument can be repeated. After a finite number of steps an equation,f'(y, x) results, separable in y or in x . Thus the equation of minimal degree between y and x is separable. Now, say we have already proved that K , = k,(x, yl, ..., yh-,) is separably generated, and x, is a separating variable. Another irreducible equation f ( y h , x , ) = 0 holds, and is separable in either yh or x 1 by the above argument. Depending upon which of these two is the case, ko(xI,y,,)/k,(x,)or k,(x,, Yh)/kO(Yh)is separable. But, as K , / k , ( x , ) is separable, K = K,(yh)is separable over either k o ( x l )or ko(yh). This completes the proof. Using the theorem of the primitive element, a separable extension K/k,(x) can be generated as k,(x, y ) . We will always assume this to be the case, with the irreducible equation f(Y, x) = 0

relating x and y . Let x = ( # 03 be a regular place of k, in the sense of $1,4. Then y and with it every function u E K can be expanded in a power series m

u

=

C a,,zcw

(zc = x

- 5)

p=V

at a place 5, over (. This expansion is, by $1, an isomorphic mapping of K onto a subfield of the field of power series in zc . A differential quotient is defined by

144

111.


The familiar sum, product, and quotient rules hold. In particular,

-af- +dY - = f yaf- + f x =dY0 . ay d x

dx

ax

Because x is a separating element andf is irreducible, we havef, # 0, and thus dyldx =

-fxlfy.

(1)

This equation shows that the differential quotient again belongs to the field K, and that it can always be determined by (1). Formula (1) is correct, by its derivation, for all y E K. The detour to power series was necessary only to demonstrate that the formal rules of differentiation hold. Also from (1) it is seen that the differential quotient does not depend upon the place t. We end this section by proving the chain rule d u-d-v = dudx

dv dx’

To do this we use the equations relating u, u, and x: Let u = u l , ... be the critical places of the first equation. Further, let x = t be a regular place of the second equation and f&, t )# 0, ... . Solve fz(u, 0. It is familiar that raising to the power p is an isomorphic mapping onto a subfield usually denoted by KP.We introduce the field

.

K O= KPko For a separating element x of K

(2)

K = K,(x)

(3)

holds. Proof. As K/ko(x)is separable’there exists a primitive element y , such that K = ko(x,y ) and y satisfies an irreducible equation f ( y , x ) = 0 of degree

145

#4. DIFFERENTIALS

n = [ K :ko(x)] over ko(x). Then, over ko(xp),y p satisfies the equation f ( y , x)’ = F(yp,xp) = 0. It is irreducible in k0(xP). For otherwise f ( y , x ) would become reducible, also, afterpth roots of certain elements were adjoined to k, , which would contradict the separability of K/k,(x). Now, FQP, xp) has the same degree n in y p , so that [KO: ko(xP)]= [ko(yP,x’): k0(xP)]= [ K : k o ( ~ )= ] n.

(4)

As x satisfies an irreducible equation of degree p over ko(xp),we have Cko(x) : kO(X)P1= P.

Equation (4) leads to

[ K : ko(xP)] = [ K :K o ] [ K o : ko(xP)]= [ K : K o ] [ K :ko(x)] and ( 5 ) leads to [ K : k o ( ~ ’ )= ] [K:ko(~)][ko(~):k= o (~~[~K):]k o ( x ) ] .

Comparison of the two finally gives [ K : K O ]= p .

(6)

Now, x also satisfies an irreducible equation of degree p over K O ,so that CKo(x):KO1 = P.

But Ko(x)E K. Comparison of the last equation with (6) then yields the assertion (3). 7 Because of (3) we can write every element u E K in the form u =UO

+ ulx + + u p - IxP-l, *..

U( E

KO,

(7)

uniquely. The concept of differential quotient can now be introduced in yet another way, by setting du/dx = U I

+2

+ +

~ ~ 2 ~ ( p - 1 ) ~ IxP-’.

(8)

The differential quotient thus defined also satisfies the familiar sum, product, and quotient rules. The chain rule, du do - do __ -dodx dx’

follows immediately, even if o is not in K O .The verification actually remains in the realm of polynomials (7), the key to the argument being that dxp/dx = 0 not only from the definition (8), but also as a result of the product rule and the fact that the characteristic is p . The derivatives of elements a E KO are always 0, and these are the only elements with vanishing derivative. We therefore call them the p-constants,

146


while we call a u 4 KOa p-variable. For the case of characteristicp , therefore, the field KO defined by (2) plays the part played by the exact constant field when the characteristic is 0. For convenience we also speak of p-constants in that case, setting KO = k, . All p-variable elements are separating elements. I f x and y are p-variable then the first partial derivatives f, and f , of the irreducible equation f(y, x) between them do not vanish, and dy/dx can be computed by (1). This also shows that the differential quotient defined by (8) corresponds to that of the last section. Proof. Let x be a separating variable and u ap-variable of the form (7). Then K 2 K,(u) 3 KO.By (6) we then have

K = K0(u). Now, let K/ko(u)be inseparable, contrary to our assertion, and K, the largest subfield of K separable with respect to ko(u). Then K , # K, and K is obtained from K , by the adjunction of some p'th roots. Therefore even K,Ko # K. But K,Ko z KO@)= K. This is a contradiction. Let x and y be p-variables and f(y, x) the irreducible equation between them. As K/ko(x)is separable so is ko(y, x)/k,(x), so thatf, # 0. By symmetry f, # 0, so that (1) must hold. I Later we shall need the following lemma: I f x and y are p-variables, and the expansion

holds, then go=

xdy

--

(y dx)

xPdyP =-y p dxP'

where the differential quotient dyp/dxpis to be taken in K p as the derivative of y p with respect to xp. Proof. With p-constant coefficients write y=aox"+...+a,,,

.

ao#O,

ncp.

In some separable (because n c p ) extension KO'of KOthis polynomial decomposes into the product n

We also define a derivation of the elements of Ko'(x) with respect to x by treating the elements of KO'as constants. As x is not in KO'(it is inseparable

#4.

147

DIFFERENTIALS

over KO while Ko’/Ko is separable) this is possible. A simple calculation then yields x dx

X

=i v= 1

(1-

1 (1 1 - (X/BJP

+ x/pv + ... +

C x w p - ~ ) ) .

The asserted equation

is contained therein. 7 Exercise. Prove that if K is not separably generated there can exist no element x for which (3) holds. There then always exist several differentiations for which the elements of KO have derivatives 0. 3. THECONCEPT OF THE DIFFERENTIAL

By a diferential dw=udo

we mean a pair of elements u, u E K with u $ KO,where two such differentials dw and dw’ = u’do’ are said to be equal whenever

u = U’ do’/dv. Obviously the differentials form a one-dimensional K-module. A divisor can be made to correspond with each differential, but the procedure depends upon the nature of the constant field ko . In this section we assume ko to be algebraically closed. Let p be a prime divisor, and q an element of K with the order v,(q) = I , i.e., a prime element for p. It cannot be a p-constant. This is self-evident for characteristic 0. For characteristic p > 0 and algebraically closed constant field, KOconsists of the pth powers of elements of K. A prime element cannot be a pth power.? Now, set v,(dw) = V,(U do/dq).

(13)

This definition is independent of the arbitrary choice of q. Proof. If ko is algebraically closed, then by 11,553 every prime divisor is of the first degree. By §1,2,then, every element of K can be expanded in a power

t This argument does not hold if ko is not closed. For example, q = x p - a with a E ko can be a prime element.

148


series in q with constant coefficients. In particular, any other prime element can be written as 4‘ = c1q

+ cZqz +

~1

-**,

# 0, c,, E k o .

Using the reasoning of #,l we have

dq’/dq = c1

+ 2c,q +

a * - ,

so that v,(dq’/dq) = 0 and

1

The orders of (13) are, save for afinite number, all 0. Thus (dw) = p v d d w )

n P

defines a divisor. Given two diflerentials, (dw,) -=-=

(u1 do,) (u2

dvz)

(::::j --

(15)

holds so that, in particular, the quotient of the divisors of the differentials for two p-variables is equal to the principal divisor of their differential quotient. Proof. The chain rule yields (15). To show that vP(udv) = 0 for almost all p, remember from the last section that K is a separable extension of ko(v). Let y be a primitive element of K with respect to that subfield, and f (y, u) = 0 the irreducible equation between y and u. By §1,4 almost all the places v = are regular, so that y, as well as any function of K, can be expanded in a power series in v - 5 at all places 5, over 5. Thus v - 5 is a prime element for almost all and hence for almost all p. Settingq = v - 5 in (13) gives du/dq = 1. This proves the assertion, as vP(u)= 0 almost always. 1 Prime divisors p with vP(dw)< 0, or the places corresponding thereto, are called the poles of dw. In the corresponding sense we may speak of the zeros of dw. For algebraically closed ko any p-variable x satisfies

= 0 for all p. Clearly this differential dwh is uniquely defined by its principal part system dhn, up to the addition of another differential dvh for which a- '(dvh) is an integral divisor, that is, up to the addition of a differential of the first kind with respect to a-'. For instance, if h = 0 and a = (l), dwo = w is determined up to a constant. For, if simultaneously vP(wP- w ) 2 0 and vp(wP- u) 2 0 everywhere, then v,(w - u) 2 0 everywhere, so that w - u E k, .

-

3. THESCALAR PRODUCT

Let dw" be a differential of the hth degree and first kind with respect to a, and d l - b a principal part system of degree 1 - h for the divisor a. Then (dl-"m, dwh) =

1res,(dw:-" dw")

(1)

P

defines a scalar product. On the right we have differentials do, =dwi-h dwh of degree 1, of which at most a finite number have a pole at p. If d l - h w = (dw:-h) = (du:-h), our definition of principal part systems in the last section yields V,,((~W:-~

-

d d ) = v,(a-'(dw:-" = vP(a-'(dw:-h

- d ~ : - ~ )dw") a - dui-"))

+ v,(a d d )

with both summands on the right 20.Thus (1) really depends only upon the principal part system d1-'tD, and not on the representatives dw:-h used. Clearly the scalar product is a bilinear function, that is,

+ a2 d1-hn,2,dw") = al(d'-hml, dwh)+ a2(d1-hn,2,dw"), ( d l - " ~a1 , dwlh + a2 dw,L) = a1(d1-"n,,dwIh)+ a2(d'-hn,, dw?)

(a1d'-hn,l

hold for a l , a2 E ko . Using §4,6 we see that the scalar product is always an element of the field of constants ko . Let a principal part system dl-hn, for the divisor a be given. If the scalar products with all diflerentials dwh of the first kind with respect to a vanish, and only then, is dl-hn, exact.

$5. DIFFERENTIALS AND PRINCIPAL PART SYSTEMS

161

Some preparations are needed for the proof. As differentials and their divisors, principal part systems and their classes, and finally the scalar product do not change by an extension of the field of constants, we may assume that field to be large enough to assure that all poles p occurring are of degree 1, and that there exists at least one further divisor of the first degree q by which a is not divisible. Let such a divisor q be chosen and held fixed in what follows, and let q be a prime element for q. A special basis of the differentials dw" of the first kind must now be produced. By subjecting the power series expansions d W h = (cO

+

+ .-.)dqh,

~ 1 4

ci

E ko,

to a suitable linear transformation, a basis dWih

= (q"'

+

'+

~ i , ~ IqWi+ ( +

a * * )

dqh,

Cij E

k,

9

(2)

such that

0 5 p1 < p 2 < '.' < p G , = gh,a, (3) is found, and clearly uniquely determines the p i . The number g h a computed in §5,1 is here abbreviated G. The divisors a(dw,h) lie in the class a Wh.By the existence of the basis (2) we can deduce dim(aWh) = dim(q-'aWh) = = dim(q-%Wh) = G, dim(q-"-'aWh)

dim(q-"---'aWh) dim(q-"-'aWh)

=

... = dim(q-"aWh)

= G - 1,

... = ... = dim(q-"GaWh) = 1, =

(4)

... = 0.

Proof. The divisors of all dw" of the first kind with respect to a can be written as (dw") = a- 'g with integral g which is even divisible by qpl because of (2). This divisor g is in the class aWh, in which there are G linearly independent divisors. The divisors gq-', where v = 1, ..., pl, lie in the classes q-'a W h . As they are all integral, the number linearly independent among them is dim(q-'a Wh).But there must be just as many linearly independent divisors as there are g. This verifies the first equation (4). Omitting dwlh and thus reducing the dimension by 1 so that the remaining dw" are of the form a-'g with integral g divisible by qp2 gives similar arguments for the other equations. Two lemmas now follow.

+

Lemma 1. If for some i ( i = 1, ..., C ) p i 2 5 p i + , , then to every v in the sequence p i + 2, . .., p i + there exists a diyerential dv'-h of degree 1 - h such that the divisor q'a- '(dvl-') = b is integral and prime to q.

111.

162


Proof. Such a divisor b belongs to the class q'a-' W l W hIf. b = qb', then b' lies in the class qV-'a-l W'-h. The existence of a differential du'-h as described is dependent upon the truth of dim(qv-la-lW1-h ) < dim(q'a-'W'-h). By the Riemann-Roch theorem this inequality is equivalent to

v - 1 - g(a) + (1 - 2h)(g - 1) + dim(q'-'aWh)

< v - g(a) + (1 - 2h)(g - 1)

+ dim(q-'aWh).

But this inequality holds whenever dim(q'-'aWh) = dim(q-'aWh), and this is so, under our hypothesis, according to (4).

7

Lemma 2. A principal part system class of degree 1 - h always contains a principal part system diwhn,= with orders vp(a-'(dui-h)) 5 0, except possibly at p = q, where then v,(a-'(d~:-~)) 2 -pG - 1. Proof. Let ( d ~ : - ~be) a principal part system of a given class and vp(a-'(dwi-h)) = -1, I >= 1, for somep # q. We seek a differential dw'-'with vp(a-'(dw'-h)) = - I ,

v,,(a-'(dw'-h))

2 -pG - 1,

vp,(a-'(dw'-h))

2O

for all p' # p, q. If this d ~ ' -is ~multiplied by a suitable constant, then ) 1). The principal part system ( d ~ i - ~ ) vp(a-'(dwi-h - d ~ ' - ~5) -(I- ( d ~ ' - is ~ )equivalent to ( d ~ k - ~ and ) , we operate similarly on it, etc. A finite number of steps leads to an equivalent principal part system d' -htu of the sort required. The differential d ~ ' sought - ~ above must have as its divisor (dwl - h ) = p - l q - ~ ~ - l a c (5) with some integral divisor c prime to p . It belongs to the class W ' - h , so that the class of c is p'qPGo+'a-'W'-h. The Riemann-Roch theorem gives the dimension of this class:

6l = I

+ pG + 1 - g(a) + (1 - 2h)(g - 1) + dim(p-'q-PG-'aWh).

The divisor of a differential ( 5 ) for which c is divisible by p can be written p'-'q-pG-'ac'withintegral c', this c'belongingto the classp'-'qPGG'a-l W'-h of dimension 61- 1 = 1 PG - g(a) (1 - 2h)(g - 1) + dim(p'-'q-'C-'aWh).

+

+

Thus the existence of a differential ( 5 ) with integral c not divisible by p is assured whenever 6, > dI-', and this is the case if dim(p-'q-'G-'aWh)

= dim(p'-'q-J"c-'aWh)

= 0,

85.

DIFFERENTIALS AND PRINCIPAL PART SYSTEMS

163

As I? 1 this is an immediate consequence of (4), where we saw that dim(q-'G-'aW") = 0. This proves our lemma. 7 Little remains to prove the theorem. If d'-"n, is exact, i.e.,if d'-"w = (du'-") then for any differential dw" we have (d'-"w, dw") equal to the sum of residues of the differential dv'-"dw", a differential of the first degree, which sum must be 0 according to the residue theorem of §4,7. Conversely, let ( d ' - " ~ dWh) , =0 (6)

for all dwh with integral a(dw"). We have already seen that (6) also holds for equivalent principal part systems, so that we may choose for d'-*tu the system of Lemma 2. For the q-component we have the power series expansion dvt-" = LPGq-PG dq,

+

+

- * a )

while for p # q we have vp(a-'(du:-")) 2 0. Now using Lemma 1, we can even improve our principal part system with these properties within the class so that at most the exponents -pG - 1, -pC-' - 1, ..., -pl - 1 occur with nonvanishing coefficients. Then, by (2) we have

which must vanish by our assumption. Hence, the dot-" are integralat q and, as q does not divide a, u,,(a-'(du:-")) 2 0. We have thus verified that (dvk-") = (0), proving the theorem. 7 4.

THERELATIONSHIP TO INTEGRALCALCULUS

For this section we assume the constant field to be algebraically closed, so that all divisors are of the first degree. Further, let Q = (1). To any principal part system dow = (w,) there exists in K a diferential du of the second kind such that vp(dv- dwp)2 0 for all p. This dv is even uniquely determined up to the addition of a diferential of the first kind. The mapping of don, onto the residue class of dv modulo the diferentials of thejirst kind is a linear function. Conversely, there exists such a dow for every du of the second kind. For K of characteristic 0 there exists only one such dOw. Proof. Say some p has the prime element q and wp = c - , q -

+ + c-,q-' + .**

(7)

-**.

We first seek a differential of the second kind, dv = du(p), possessing the expansion dv = (-mc-,,q-"'-l

- ... - c-lq-'

+ +

- 9 . )

dq

(8)

164

111.


at po and no other poles, the pc-,,q-#-' for which the characteristic divides p dropping out. Now, a differential du, with a pole of order p 2 2 at p and no others has a divisor p-'b, with integral b not divisible by p and in the class p" W. As in the last section, use dim(p@W)= p + g

- 1 > dim(p"--'W) = p + g - 2

to show that such a differential exists. Then it is not difficult to construct a differential (8) with the dull. In turn, addition of these du(p) for the finite number of places at which w p is singular leads to a differential du satisfying the theorem. Clearly a differential of the first kind may be added to du, and the difference between two such do is of the first kind. Furthermore, the construction clearly shows the linearity of the mapping. Conversely, let du be a differential of the second kind and let (8) represent the principal parts of the finitely many poles. If p is divisible by the characteristic, no terms 4-I-l dq occur. Thus, as far as negative exponents are concerned, (8) represents the differentials of the elements (7). This shows that the w p are components of a principal part system don,that maps onto du. If the characteristic is 0 the principal part of (7) is uniquely determined by that of (8). In cases with prime characteristic, though, terms c-,,q-P c - z p q - z p ... with arbitrary coefficients may be added. 7 The relationship of the theorem may be represented symbolically by the indefinite integral

+

+

don, = j d u .

(9)

Moreover, this permits us to define a scalar product (du, do) =

resp(jdu.do) P

between any two differentials of the second kind. The power series expansions

at a place p (in which c, = d,, = 0 for p divisible by the characteristic because du and du are of the second kind) show that Cm d-m c d- + ... + res,(jdu-du) = - c - m dm - ... - c- d1 + a 1 1 m is skew symmetric in du and du, so that


165

This scalar product is clearly 0 if both du and du are of the first kind. It also vanishes if either du or du is exact, that is, if du or du is a differential of a function in K. Let du, be the basis (2) of first kind differentials with h = 1, and use its expansions at a place q to define principal part systems dooi by stipulating Dip =

for p = q for p # q

q-pl-l

(0

so that (dooi,duj) = 0 or 1, depending upon whether i # j or i = j . The corresponding differentials of the second kind dui are determined only up to differentials dw, = cij duj. Remembering (11) we can adjust the c i j and thus the dw, in a manner that the expression

1

(dv, + d w i , doj + dwj) = (do,, duj)

+1

((hi,

duJcj1

1

+

always vanishes. Again writing dv, for du,

(dui,duj) = 0,

du1)cii)= (do,, duj)

( d ~ j ,

+ cji - cij

+ dw, we now have

( d ~ iduj) , = 0,

The theorem of the last section together with the correspondence between principal part systems and differentials found here finally give the result: Every diferential of the second kind is a sum of a differential of a function of K and a linear combination of dui and dv, . 5*. THEDIAGONAL

The Green function of classical analysis depends upon two variables and has a singularity at a variable place. The rest of $5 will be devoted to finding an algebraic analog to the Green function. To this end, consider two isomorphic function fields K and K in one variable over the exact field of constants ko . Say K = ko(x,y) with f (x, y) = 0, where f ( x , y ) is separable in y ; take another variable x’, algebraically independent of k, and let y’ be a solution off (x‘, y’) = 0. Then K‘ = ko(x’,y’) is such a field. KK‘ = k,(x, y ; x’, y’) is a field of functions of two variables over k , , which can be considered a field of functions of one variable over K and also over K’ as constant field. In the first case it arises from K‘ by extension of the field of constants k, to K, in the second from K by extension of constants to K . KK‘ has an involutive automorphism, given by interchanging x, y with x’, y’. In the following it will always be denoted by priming.


166

Lemma. If x1 and x2 are any two separating elements of K then there exist two polynomials hl(xl, x,; xl‘, x,’) and h,(xl, x,; xl’, x,‘), with hl(xl, x,; xl, x,)h,(x,, x, ;xl, x,) # 0 and such that

Proof. A separable equation f(xl, x,) = 0, irreducible in both variables, holds. Hence, the equationf(xi, x,‘) = 0 has x2’ = x2 as its only solution in K, and thus f(x1, xz‘) = (x2 - xZ’)gz(xlY x,; %’I, where g , is a polynomial in x,’ with functions rational in xl, x, as coefficients, and for which g2(xl,x,; x2) = -af(xl, x,)/ax, # 0. The same argument shows that S(x1, xz‘) = (XI - xl’)gl(xl; Xl‘, X Z I ) , with a polynomial g 1 in x1 whose coefficients lie in K’ and such that gl(xl ; xl, x,) # 0. Division of these equations yields -= - x,’

gl(x1; XI‘, xz‘) gz(x1, xz; xz’)

x2

x1

- XI’

which goes into the representation desired when the fraction is extended by the denominators ml(xl’, x,’) and mz(xl,x,) of g1 and 9,. 7 Corollary. If thefields K and K’ are identiJed in the sense of the isomorphism given above, then x

- X I

( s ) K * = K

=-*

dx2 dx1

Proof. The assertion follows immediately from Taylor’s theorem for polynomials, which states that gz(Xi,

XZ; XZ)

= -fJXi,

xz) and 9i(xi; X i , xz) =f.,(xi, xz).

T

This taken care of, consider the integral domain K x K‘ of all finite sums xlyl’ + x2yz’ + with arbitrary x i E K and yt’ E K . The expressions (x - x’)Y with arbitrary x E K and Y EK x K‘ form an ideal DK K , , which is even a prime ideal as the residue class ring is isomorphic to the field K . KK‘ is the quotient field of K x K‘, so that in view of the lemma every A E KK’ can be represented as A = (X - x‘)’(X/Y)

with X and Y in K x K’ but not divisible by DK K,. The A with I > 0 form an which is even a discrete local ring in the sense of 11,52. integral domain 3=,


167

The prime divisor ID of K corresponding to the ideal I D K x K , is called the diagonal divisor, or simply the diagonal of K K . If KK‘ is taken as a field of functions of one variable over the constant field K’, then 9 becomes a prime divisor of the first degree, for the residue class field of Z$, mod 9 is the constant field K’. Interchanging K and K’ leads to corresponding statements. By the lemma x - x’ is a prime element for 9,where x is any separating element of K. 6*. THEANALOG OF THE GREEN FUNCTION

Let dw,h be the basis (2) of degree h differentials of the first kind with respect to a. Then

defines gh,aprincipal part systems d ’ - h ~for i a, with (d’-hq,

dw;)

=

for for

1

0

i=j i#j

(13)

according to (2). In view of @,3 these form a basis of all the principal part system classes, the complementary basis to dw,h. Contragredient linear transformations of both bases leave the relation (13) intact ;the bases remain complementary. Such transformations also leave the bilinear form dd” -h&joh = d” - h q ’ dW,h (14)

c i

invariant. Here we denoted by d‘l-”q’ the principal part systems carried from K to K‘ by isomorphism; priming the d will turn out to be practical. Equation (14) is to be read as between principal part systems of degree 1 - h in KK’ taken as a function field over K , multiplied by a differential of degree h of K. According to the second lemma of 45,3 there now exists an exact principal part system d‘’-h d&jIhof degree 1 - h in KK‘IK, multiplied by a differential of degree h in K, such that the equivalent principal part system d’1-h d&jh = d‘1-h d&joh

- d‘1-h

d(fjlh = (d’G:;h dxh)

(15)

(with some separating x of K ) has the property vp’(a’-’ d’Gi-”) 2 0 at all places p‘ of KK’IK except for p’ = ID, where that order is 2 - p c - 1. The notation pG is that of &3 with q = 9. At the place p’ = 9,

d‘Gk-h dxh = ( c - ~ ~ - ~ (-xx‘ ) - ” ~ - ’ + with coefficients c-

1,

+ c - ~ ( x ’- X ) - ’ ) ~ ’ X ’ ’ - ~ ~ X ~(16)

.. . E K. As in §5,3, the first lemma even permits a choice

168


DIFFERENTIALS

with coefficients c j = 0 with the possible exception of the c - , , , - ~ . Moreover, we now assert that all c, = 0 with the exception c- = 1. To prove this we note that d'1-hd(61his exact, so it does not contribute to the scalar product; by (15), (14), and (13), then (d"-h d B h , d'Wih) = (d"-'

dB,h, d'Wih) = dWih.

(17)

Clearly (17) holds for all differentials of the first kind, for the dwh form a basis of these. In order to evaluate (17) we make use of a third field K" isomorphic to K and algebraically independent of K and K'. For the dwh we choose the basis (2) of differentials of the first kind of KK"/K" using the diagonal of KK" as prime divisor q. This means that dW,h = ((x

- X")"

+ Y~,,,,+,(x- x")""' +

* a * )

dXh,

yij

E K".

(18)

The exponents pi are those of (16). An easy computation gives res,,=,

(XI (XI

- xn)v d'x' = (L)(x - X)P+l

- x")"-".

Now let i be the largest index such that the coefficientc-,,Eqs. (16), (1 8), and (19) yield ( d ' l - h d(fih,d'Wih) = ( ~ - , , , - l

# Oin(16).Then

+ * * * ) dXh,

the dots indicating a function which vanishes on the diagonal of KK".But, by (17), this is possible only if p i = 0, that is, if i = 1 and c - ~= 1. This proves our assertion. Remembering our notion of equality of principal part systems we have l d ' x ' l - h dxh for p ' =

with any separating variable x. (20) may be considered to be an analog of Green's function. The following theorem will show that the principal part system (20) is, up to minor alterations, uniquely determined. We state this fact in a different way, however, more adapted to an application in V,#2, distinguishing the two alternative assumptions on a made in &1. The exactprincipalpart system drl-h d B l h= G d r ~ " -dxh ~ has thefollowing properties as a divisor of K and K'. For every divisor p' of K K I K other than the poles of dl-hw,' and the diagonal, v,.(a'-'(G

d'x'l-h)) 2 0

(21)

g5.

DIFFERENTIALS AND PRINCIPAL PART SYSTEMS

169

holds. At the diagonal,

the dots indicating a function holomorphic here. Furthermore, under assumption (1) v,(a(G dxh)) 2 0 (23) holdsfor all prime divisors p of KK'IK' other than the diagonal. Under assumption (2) there exists (up to constant multiples) exactly one differential du'-h with du" with a suitable differdivisor (dv'-h) = a in K. Now the addition of ential duhof K brings G d'x"-h dxh into a normalization for which (23)also holds save for one arbitrarily chosen place po of K at which there is a pole of principal part shown by

po denoting a prime element at po . Proof. Equations (21) and (22) hold by our construction. The field KK'IK' has two sorts of prime divisors. The first are already prime divisors of K / k , , while the others are generated in the algebraic extension KK'/k,(x, x') by prime polynomials p(x, XI) dependent upon x as well as G dxh can have no pole of the latter sort aside from the diagonal, for it would occur in the denominator of G. The corresponding prime divisor of K K / K would divide the denominator, contradicting (21). To show that in general aG dxh can have no poles of the first sort either, we note that if such poles existed they would remain if the field of constants k, were extended. Thus, ko can be assumed algebraically closed. Let p be a prime divisor of the denominator of aG dxh. Choose some differential dw" of K whose divisor has numerator and denominator both prime to p. Also, let x be some element integral at the place p. We now show that G dxhldw" is, in general, integral at p. Assume the contrary, and let q be a prime element for p and 1 > 0 the least exponent such that q'G dxhldw" is integral at p. We defined G to be such that d(iihare identical (in the sense of &2) G d"-hdxh and C d " - h w i dw? as principal part systems of K K / K . Then the difference of XI.

multiplied by a'- is everywhereholomorphic in KK'IK. Residue formation for p-integral elements of KK' modulo p is a homomorphic map of the integral domain of those elements onto K . The latter principal part system being

170


considered is clearly mapped to zero. Thus the residue class of the former, namely, q'G d"-hdxh/dw", with respect to the module p is a differential of degree 1 - h in KK'IK, whose divisor is divisible by a'. This divisor then is b'a' with some integral divisor b', and it lies in the (1 - h)th power of the canonical class W'. So b' E a'- ' W" - h . Under the first assumption on a the class a'-' W''-hhas dimension 0 according to the Riemann-Roch theorem. So in this case we have a contradiction, and (23) is proved. The second assumption on a implies that b' lies in the principal class and, being integral, b' = (1). Now furthermore there exists (up to a constant multiple) exactly one differential du'-h whose divisor is a. With the corresponding differential d ' ~ ' ' -in~ K' our result that 6' = (1) means that, at p, G d'X'1-h dXh = d'u''-''(qq-' + ...) dqh

with c l ~ k OAfter . subtraction of d ' ~ ' ' -c,q-' ~ dqhthe same reasoning can be repeated. Thus G d ' x ' l - h dx" may possess poles at such places, but their principal parts must be of the form d ' ~ ' ' -(c,q-' ~ + + c l q - ' ) dqh. These poles can yet be compensated by subtracting from G d'x'' - h dx" differentials d'u'l-h dub, where duhhas one pole of exactly order 1 at p, and so forth. A differential duh of this property exists if and only if dim(p'aWh) = g(plaWh) + 1 - g

> dim(p'-'aWh)

+ dim(p-'a-'W'-h) + g(p'aW) + 2 - g + dim(p'-'a-lW'-h),

and this is the case if and only if dim(p-la-

1 ~1

-h) =

-1

a - 1 ~1

-h

).

For 1 > 1 this is indeed so because of our assumption on a. For 1 = 1 the same reasoning gives a differential duh with poles of first orders exactly at p and an arbitrary further place po, and we subtract it from G d ' ~ ' ' - ~ d x "So . we eventually arrive at a form of G d'x'' - h dx" where this differential has two poles at 9 and po . According to (22)

has the residue 1 in 9.By virtue of the residue theorem, this differential then has the residue - 1 in the pole po . Thus, in po ,

and the proof is complete. 7

$6.

171

REDUCTION OF A FUNCTION FIELD

NOTES The principal theorem of §5,3 is sometimes called the " inhomogeneous Riemann-Roch theorem." A generalization is given in the paper [lo] of $3,7, but only for the case of zero characteristic. The theorem was first stated in our form by Teichmullert and proved in a different, somewhat quicker manner. The proof given here is by Kappus.J Another proof is due to Peterson, 7 the theorem there being formulated within the framework of the theory of automorphic forms. §6*. Reduction of a Function Field with Respect to a Prime Ideal of the Constant Field

I . THEIRREDUCIBILITY THEOREM

Let the function field K be separably generated over its exact constant field k , , that is, let K = k , ( x , y ) with y satisfying the irreducible separable equation f ( y , x) = a,(x)y"

+ a , ( x ) y " - + + a , ( x ) = 0, *.*

a,(x) # 0 ,

(1)

over k,(x). The exactness of the constant field assures the continued irreducibility of (1) in k , ( x ) for every extension k , / k , , and conversely. Of course, this need be confirmed only for finite extensions, and then [ K k , : k,(x)]

=

[ K k , : k , ( x ) ] [ k , ( x ) :k,(x)]

= [ K k , :K I C K : k , ( x ) ]

By $3, (12), namely

[ k , : k,] = [ k , ( x ) : k , ( x ) ] = [ K k , :K ] , this leads to [Kki : k i ( x ) l = CK: k o ( x ) l ,

expressing the irreducibility of (1) in k , ( x ) . Let the constant field k , contain an integral domain 0, with quotient field k , and satisfying the ideal theory of Chapter 11. It is then no loss of generality to assume the ai(x) of (1) to be polynomials in oo[x]. We now want to investigate the congruence f ( y , x) = 0 modulo prime 0,-ideals po under these

t Drei

Verrnutungen iiber algebraische Funktionenkorper, J. Reine Angew. Math. 185,

1-11 (1943). $ Darstellungen von Korrespondenzen algebraischer Funktionenkorper und ihre Spuren, J. Reine Angew. Math. 210, 123-140 (1962).

7 Konstruktion von Moditlformen etc., S.-B.Heidelberger Akad. Wiss. Math. Nat. KI., 417-494 (1950).

172

111.


assumptions. Residue class formation mod po will be indicated throughout by overlining. Thus, in the following we are to consider the equation f(y, x ) =f(jj, Z) = 0

o,/p, = li.

in

As a first example, let k , = Q, 0, = Z. We then have an equation in the finite field Z / p which, if it is irreducible, defines a function field over that finite field. Second, let k , = koo(z),where z is some variable (independent of x ) and oo = k,,[z]. Taking residues modulo z - with E k,, becomes the familiar operation of substituting z = into polynomials. The theorems to be developed thus include the definition of such substitution in algebraic functions.

Irreducibility Theorem. For only a finite number of po in ko d o e s f ( y , x ) become reducible or even inseparable in the residue class field k , modulo po or in an extension of that field 17,’. Proof (Deuringt). The finiteness of the number of inseparable cases is obvious. If, after a finite extension k,’/ko of the constant field, the polynomial is irreducible modulo a prime divisor p,’ of po in k,’, then it is also irreducible modulo p,. This permits finite extensions k,’/k, in the course of the proof. As in §l,4, replacing y by a,(x)y leads to an equation (1) in which a,(x) = 1 and the other ai(x)remain polynomials of o,[x]. Except for the finitely many po which divide all the coefficients of the original a,(x), irreducibility of the new and old polynomials mod po coincide. If k, contains only a finite number of elements the theorem has no substance, and we eliminate that case. Thus k, and therefore oo contain infinitely many elements. In §1,4 we saw that all y algebraic over k , for whichf(y, y) = 0 has multiple zeros satisfy an equation D(x) = 0 where D(x) E oo[x].Now take some y E oo with D(y) # 0. As we could substitute x + x - y we can immediately assume y = 0. Further, let qio ( i = 1, ..., n) be the solutions off (y, 0) = 0. Finite algebraic extension of k , is permissible, so that we may assume the qio to lie in 0 , . Then, by §1,4 there exist n power series y, = q i o

+ qi1x +

* * *

and an N E oo such that qi,,NpE oo , satisfying

t Redukrion algebraischer Funktionenkorper nach Primdivisoren des Konstantenkorpers, Math. Z. 47, 643-654 (1941).

$6.


173

After the finitely many prime divisors of N are eliminated, residue class formation modulo the remaining po yields power series

yi = ijl0 + i j i l X

+

* . a

with coefficients in I;, which then satisfy

Now, if (3) were reducible then one of its partial products would have to be a polynomial in j , X, whose degree in X is at most the degree n, of f b , x ) in x. The corresponding partial product of (2) would be a polynomial in y with power series in x as coefficients which are infinite because, otherwise, f ( y , x) would be reducible in k,(x). So these series would have coefficients which are divisible by po with but finitely many exceptions. Therefore form all the possible proper partial products of (2) and in each take the first nonvanishing coefficient of a power of x which is > n,, and multiply all these coefficients. The product is divisible by p, while by hypothesis on oo only a finite number of po can go into such a product. Hence f ( y , x ) is irreducible mod po save for finitely many exceptional po Any finite extension ko’/ko arises from a finite extension k,’/ko by taking residues of the principal order 0,’ of k,’ with respect to 0, modulo a prime divisor p,’ of p, . Replace k , in the above argument by all possible k,’; this gives the last part of the theorem. The set of p, to be eliminated is finite, for these are always divisors of a fixed element. T One consequence of the irreducibility of f ( y , x) mod po is the following.

.

Theorem of Inertia. Let the polynomial f ( y , x ) be irreducible and separable mod p, , oyodenote the integral domain of those a, E k , with denominators not divisible by p, , and 3,,be the principal order of K with respect to opo[x].Then 3,,p, is a prime ideal and the quotient field R of the residue class ring 3,,/3,,p0 can be.formed. This means that every K E K can be written as K

= Poh(dB>

(4)

with ci, fl e3,, and p;lci, p i ’ p E 3,,, where po is some prime element for po in k , . If h = 0 the element K is called p,-normed. Thus any principal divisor ( K ) can be represented by a p,-normed element.

Prosf. It suffices to show that if ci E 3,,has a norm nK,ko(x)(a) divisible by po , then p i lci E 3,,. We use the generating equation (1) and, as before, may assume that ao(x) = 1 and that the other ai(x)E o,[x]. Then y E 3,,. As we

174

111.


assume f ( j , X) to be separable, the discriminant D = D(1, y, f 0 mod p o , so that a can be represented as (cf. 11,§1,2)

..., y"-')

n- 1

a = D-

' 1 biyi, i=O

bi = SK/ko(x)(ayi)

E

opo[~l.

As D f 0 mod p o we may replace a by uD in our verification. So now let n ( c biyi)= 0 mod p o As the order of norm and residue class formation may bJ') = 0 . Further, the order of residue class be interchanged, we have and discriminant formation is interchangeable, so that D(i, j j l , ..., j j " - ' ) # 0. The vanishing of the norm then implies all 6, = 0. Thus all the 6 , are divisible by po , completing the proof. 'I

.

n(c

2. REGULAR PRIMEIDEALS

From now on we always omit the prime ideals p o excluded in the proof of the irreducibility theorem of the last section from our considerations. Let R be the field found in the inertia theorem. It is defined over I;, by the equation (X, j ) = 0, and is a separable extension of degree n over ko(F), the exact field of constants being I;, . Further limitations are desirable. Consider the following example. Let k,, be an algebraically closed field of prime characteristic p > 2 and set k, = k,,(ii) with an indeterminate u. Finally, let K = k,(x, JG). The genus of K is g = ( p - 1)/2. Modulo any prime ideal p o of oo = k,,[u], the residue class U is a pth power in k, = koo. Thus K = k,,(JP - $) is a field of genus = 0. We should like to exclude exceptional cases of such a nature. A field K has been called conservative if its genus never changes under extension of constants. The K of this example is certainly not conservative, as adjunction of $u to k, leads to a field with g = 0. A prime ideal po of oo is called regular if R is a separable function field of degree [ R :kO(X)] = [K:k,(x)] = n over ko(Z) with exact constant field I;, and with the same genus 3 = g as K . Let K be separable over k,(x) and conservative. Then, except for a jinite number of exceptions, all prime o,-ideals of k , are regular. For the proof? we again form the integral domain opo of all a E k o with be the principal orders of K denominators not divisible by p o . Let 3,,, with respect to opo[x],i = k,[x], and of K with respect to i = I;,[X]. Clearly

3,s

-

3 =, ~ p o l ~ , o P* o

(5)

Let wi be a basis of 3 with respect to i. In II,fj4,2 we saw that the discriminant

t Another proof is found in M . Deuring, Die Zetafunktion einer algebraischen Kurve oom Geschlecht 1, Nachr. Akad. Wiss. Gottingen Math.-Phys.-Chern.Kl.. 13-42 (1955).

$6.


175

D(wi) is the norm of the different ideal of 3. We now exclude the finite number of po for which m iE 3,, is not true, and then the discriminant of the residue classes of the wi mod 3popo,namely D(Wi), is a polynomial in x having the same degree as D(oi). Using ( 5 ) then, the degree in X of the discriminant of 3 is not greater than the degree in x of the discriminant of 3. The same argument is then repeated with x - l = x’ in place of x, but only insofar as to see what power of x’ divides the discriminant of 3’. Together with the degree in x of D(wi) this gives the degree of the discriminant divisor: almost always daK/fo(a)>

g(aK/k,(,)).

(6)

Thus, by §3,(8) the genera of K and K satisfy S g. To prove the equality for almost all po we must show that equality actually holds almost always in (9,and that the same is true when x is replaced by x’. Then equality holds for (6) as well, and the assertion = g is verified. Due to the symmetry of x and x - l we must actually only do one. Let the polynomials q j = 9 j ( x ) ( j = I , ..., h) be the prime divisors of D(wi). Immediately exclude all po by which the denominators or highest coefficients of the q j are divisible. Now choose some finite algebraic extension k,’/k, so that the q j decompose into linear factors (which we again call 9 j ) , and so that the prime divisors of the q j in K‘ = Kk,’ are of the first degree. The lemma in $4,4 makes this possible. Simultaneously, replace po by one of its prime divisors po‘ in k,’. K was assumed conservative, so the genus g remains unchanged by this extension. By $3,4the genus of K is at most reduced. If we can show that S = g after the extension, this was also so before. For simplicity we omit the primes, thus again writing k , , p,,, etc., for ko’, po’, etc. We also now exclude those po which lead to the same residue class for different q j . Now let q be one of the 9 j , and the numerator divisor q of q decomposes in the manner q = q“1

... e.

(7)

By assumption the qQ are prime divisors of the first degree and 11,§3,(14) then shows the ramification indices have the sum

We now show that, in general, the numerator divisor of ij decomposes in the same manner. To this end we apply II,$2,2. The prime divisors q i are represented by prime 3,-ideals where 3, denotes as usual the principal order of K with respect to i, . No confusion is to be feared if we denote these prime ideals by the same symbols q i . As 3, is a principal ideal domain, q i = 3 , ~ ~ . The elements K~ may be assumed to be p,-normed. We now form the domains 3,,,, consisting of all x E 3, of the form (4) with h 2 0 and furthermore the

176


domains of their residues mod po . Then the i j i =3,Ki are integral ?&-ideals satisfying 3qq- --el ....-9: (9) q1 We maintain that the ij, are prime ideals. For, if

were their decomposition and& the degrees of the pPv with respect to Lo@), then 11,§3,(14) would give

But this equation is only compatible with (8) if

c

fQVEQV

=

for every e, that is, if the ij, were prime ideals. If K has characteristic 0 then the discriminant D ( 3 ) = D(wi) is divisible by q to the power (e, - l), by §2,(36). But the Dedekind discriminant theorem assures us that, assuming (9), q divides the discriminant D(3) to at least that degree. If several of the ij, were equal, an even higher power of i j would divide D(3). But as the ij = i j i were assumed t o be different and because of (6), it is impossible for D(3) to be divisible by a single i j i to a higher power than D(3) by the corresponding q i . Thus the discriminants D(3) and D ( 3 ) are of the same degree, so that equality holds in (5). This argument fails, though, for K of prime characteristic, for then it is possible for D ( 3 ) to be divisible by a higher power of i j . But now, construct the K @ in order, making them pairwise prime. Then for any two of them there exist elements a,,, , &,, E 3,,such that

c

+ BQ~K,,= 1,

apu~p

4Z

(10)

6.

Again, a,,, ,fl,, E 3q,p, except for finitely many po . Residues mod po can now be taken for (lo), showing the ij, to be differing prime ideals. Finally, for each 4 = i j i consider the element

1P I = K-el1 ... p r- iK-, e p

9

j = O , 1,

..., e,-

1.

(1 1)

They all lie in 3q,po/3q,popo = 5, and have the orders

The method of §2,4, using (12), can now verify that the XQj are a basis of -3,with respect to 1,. Therefore, if both of the fmodules of (5) are extended to i,-modules by permitting denominators prime to i j , they coincide. But this is $0 for all q that divide the residue class of D(wi) mod po , which proves (5).

$6. REDUCTION OF A

3. BEHAVIOR OF IDEALS

UNDER

FUNCTION FIELD

177

RESIDUEFORMATION

From now on let po always be a regular prime ideal. As above, let opobe the integral domain of a E k, with denominators indivisible by po , a principal ideal domain, by 11,§2,2. Finally, let R,, be the integral domain of elements (4) with h 2 0. Note that the definition of R,, depends upon the element x, so that it is sometimes practical to denote it by R,, = A,,,, . This section will be devoted to the study of %ideals and their residue classes mod po . The latter have yet to be defined; to do so consider, to any ideal a,

R,, ,

in particular 3,,= 3 n R,, , (13) These a,, are o,,[x]-modules of rank n, and they satisfy the conditions of the basis theorem of I,§1,5. In fact, let a, b E op,[x] be two mutually prime elements and aa, ab E a,,. Then a E a. As a and b are not both divisible by po we see that a E R,, and therefore also a E a,, . Hence the a,, have bases of n elements, mi, with respect to o,,[x], which are clearly also bases of a with respect to i. Taking residues mod po gives a,, = a v

-

a = apo/apopo, These 5 are %ideals

in particular

3 = 3 p o / ~pop.,

(14)

3 is the principal order of R with respect to i = EO[E]. The proof uses the inverse of an argument of the last section. Were the principal order 3' 15 (the notation is different here), then the discriminant of 3'with respect to i would be of lower degree in X than that of 3 = 3p,/3,0p~, while the degree of the latter is at most the same as the degree of the discriminant of 3 with respect to i. But then we should have jj < g by §6,2, contrary to hypothesis. 7 The residue classes of the bases ai of a,, are bases of the ii with respect to i. From the definition of the norm in II,§3,2 we immediately see n(ii) = n(a); (15) that is, norm and residue formation for ideals are commutable operations. Any two ideals satisfy ab = 6 6 . (16) Proof. Clearly (ab),, z aPobpoand therefore

By setting b This implies

= a-'

we have

ab 1 66. -

3 2 i a-'. 0-1

because the ideal theory of II§,l holds for the %ideals.

(17)

178

111.


As the residues of the bases of 3,,and (aare bases of 3 and a’, a linear substitution with determinant n(a-’) transforms a basis of 3 to one of a?. By (15) we have On the other hand, a linear substitution with determinant n(5-’) transforms a basis of 3 to a basis of ii- The equality of the norms, together with (18), yields

’.

-

a-1

= a-1

Multiplication of (17) with

or

- _

-

aa-’ =3.

(19)

7then yields ~-

a,

But (17) can also be applied in the form 6 = a-’ .ab 2 a-1 and then equality prevails in (20). But now multiplication by ii gives the asserted equation (16). 7

4. BEHAVIOR OF DIVISORS UNDER RESIDUEFORMATION We first review, in a slightly revised form, the introduction of the notion of divisor. The group of divisors of K, of 11,§5,3, can be described as follows. Separate the prime divisors p into two sets, F and I, the “finite” and “infinite” prime divisors, on the basis of whether p divides a prime divisor 3q(x, of the rational function field k = k,(x) which is the numerator of a prime polynomial q(x) E k,[x], or whether it divides the denominator divisor 3, of x . Correspondingly, split an arbitrary divisor up into a “finite” and an “ infinite ” part.

The group of “finite” divisors a = aF is isomorphic to the group of %ideals, where 3 is the principal order of K with respect to i = k,[x]. This isomorphism can be materialized as follows. For any divisor a, take the totality a3 of all a E K with v,(a) 2 v,(a) for all p E F. Certainly (1)3 = 3, and the a3 are %deals. We call them the 3-multiple ideals to the divisors. Our mapping a + a3 is a homomorphism of the divisor group onto the group of %deals which maps precisely the “infinite” divisors a, those for which a = as, onto the unit element. In the same vein the mapping of the divisors onto the 3,-ideals a, is a homomorphism which maps the “finite ’’ divisors a, those for which a = aF, onto the unit element. Thus the groups of finite components aF and infinite components a, of the divisors are isomorphic to the groups of %deals and 3,-ideals, and the group of all divisors is the direct product.

56. REDUCTION OF A

179

FUNCTION FIELD

All this can be used to define a homomorphic mapping of the group of divisors of K into that of R. First, (14) defines a homomorphic [because of (16)] mapping of the group of %ideals into that of %deals. T o distinguish ideals from divisors we now write

a,b,=Gb, though. The “ finite ” divisors iiF are associated, as described, with the 3-ideals so that a a3 +a F (23) produces a homomorphic mapping of the divisor group of K into the group of finite divisors of R. A second homomorphism a -,iiI must now be defined. Just as the principal order 3 of K with respect to i = ko[x]was formed, now form the principal order 3 ’ of K with respect to i‘ = ko[x’],x‘ = x - ’ , and, as before, 3’ as well as the sets F , I’. We have I c F’ and 1 c F’. The homomorphism a -,iiFt corresponds to (23). The divisors TiF, can be decomposed by (21) to aF, = ( ( 1 ~ s ) ~ ’ ( g p ) ~ SO , that

G+h p + (iip), = ii1 (24) is a homomorphism of the divisor group of K into the group of infinite divisors of R. a -+ a,,

“

”

Finally, by composing (23) and (24) we find the mapping a + iiFii1 = h

(25)

which is a homomorphism of the divisor group of K into that of K,that is, -ab = ii 6. (26 I We call ii the residue of a mod p o . This mapping has the following properties. Integral divisors a are mapped onto integral divisors ii so that iidivides 6 if a divides b. It leaves degrees invariant : g m = g(4. (27) Norm and residue formation commute: nK/k(a)= nK/,C(S)*

(28)

The principal divisor of an element u E K is mapped onto the principal divisor of the residue class of u mod po if only u is assumed to be po-normed: ( u ) = (a).

(29)

In view of (29) the mapping a + iiis also a mapping of the divisor class group of K into that of K.

180


DIFFERENTIALS

Proof. The assertions concerning integral divisors are immediate consequences of the corresponding and obvious statements for ideals. Equation (28) follows from the analogous equation (15) for 3-ideals and 3’-ideals, the latter computation using only the “ infinite ” components. Remembering (28) and 11,§5,(16), it now suffices to verify (27) for rational function fields. Thus, let K = ko(x).Further, it suffices to use only prime ideals. The denominator divisor 3m of x is mapped onto the denominator divisor 3, of X; here (27) is self-evident. Now let a = 34(x) be the numerator divisor of the prime polynomial q ( x ) = a0xm

+ + a,xrn--‘ + + a,,

a, E opo, and assume a, to be the first coefficient not divisible by po . Then, the “ finite ” part of i iis the numerator divisor of the polynomial ii,Xm-r + ... + a, and is of degree m - r. Further, *-*

+ ... + a$‘ + + a,,,xrm), a3, = 3’(ii,+ + ZmX’m-r)Xrr.

as, = Y ( a 0

- -

..a

Thus, the “ infinite ” part of TI is& = 3% and of degree r. We have shown that m is the degree of 6 as well as of a, proving (27). Equation (29) can be found directly from the equations

%=5a,

3‘a = y a

for principal ideals, by assuming a to be p,-normed (cf. 56,l). 7 The homomorphic mapping a + ?of i divisor groups described coincidesfor the two models of the field, K = ko(x, y ) with f (y, x) = 0 and K = ko(x’,y’) with f‘(y’, x‘) = 0 if the rings SZ,,,, and Rx+,poassociated with these models in §6,3 coincide.

Remark. The hypothesis of the theorem is not satisfied by the two models ko(x) and ko(x’) with x’ = p o x . For, the numerator divisor of pox + 1 is mapped onto the denominator of X by the former but not by the latter. Proof. A divisor a can be uniquely characterized by the totality of principal divisors (a) by which a is exactly divisible in the following sense: the product of the numerator and denominator of a is prime to a-’a. This characterization is an immediate consequence of the ideal-theoretical introduction of divisors in II,§S. Now, under the hypothesis, residue formation a -+ a leads to the same result in the two models. This is then also the case for principal ideals, by (29), and therefore even for arbitrary ideals. 7 It is worthwhile mentioning that it is not always known whether every divisor ii of R has an inverse image in K, as would be the case for residue formation in rings.

56.


181

5. CONTINUATION; BEHAVIOR OF DIFFERENTIALS UNDER RESIDUE FORMATION For the residue class 7i of a divisor a of K,

dim (5) 2 dim (a).

(30) Of course, the equality sign holds for all divisors whose degrees are g(ii) = g(a) 2 2g - 2, due to the theorem of Riemann-Roch. If (30) is an equality the multiples of 5 - l are residues of multiples of a-l mod po . Proof. The p,-normed multiples of a-1 generate a finite o,,-module. Let ai be a basis of it. The Ei are then multiples of 7i-I. They are linearly ciai = 0 mod po with independent, because a nontrivial linear relation ci E op0would contradict the assumption that the ai form a basis. 7 The commutability of trace and residue formation applied to the discussion of §6,3 implies that the different ideals satisfy

1

&-F - 3,

-

-

b3, = b3,,

the second equation implying that

Together they give the different divisor:

A consequence of (31) is: the residue of the canonical class W of K is the canonical class W of R. Let the residue class ii of the element u be a separating element of R. Then certainly u is a separating element of K . According to §4,2 there exists an irreducible equation in x and u, g(u, x) = 0, whose partial derivatives with respect to u and x do not vanish. Now, U generates a subfield of R over k,(X). Because [ R : kO(X)]= [K : k,(x)] we also have [k,(X, U) : k,(X)] = [k,(x, u ) : k,(x)]. This means that g(i,X) is also irreducible over k,, so that it is the minimal equation between U and X. We can now again cite §4,2 in that the partial derivatives gx and Q,-do not vanish. §4,(1) then states that

-

- -

duldx = -Sj=/g, = -g,/g,

= du/dx.

This equatidn also holds for the derivative of any other separating element B with respect to X, so that application of the chain rule in K and K finally gives residue formation and differentiation commute. Our treatment of differentials (of the first or arbitrary degree h) is based on


I82

(32). Associate with the differential u dvh, where u and v are p,-normed and 6 is a separating element, the differential

u dvh = ii dijh

(3 3)

as its residue mod p o , The independence of this definition of the particular choice of the variable v follows from (32) as in 94,3. If”the different divisors bKlkand g K , are ~ divisible by separable prime divisors only (this is particularly so if the constant fields ko and k, are perfect), or i f the fields K and R are conservative, then the divisors of differentials satisfv

-

(u dvh) = (ii doh).

(34)

Remark. The assumption is essential. As an exercise the reader can construct the example in which k, is of characteristic 0 while I?, is of characteristic p > 2, R being the field in the example of §3,4. Then, not even (dx) = (dX) holds. Proof. Using $4,8, the assumptions imply that

- (ax) = b~/k3;’ = ~ K R / L ~ , ’ = (dx). With this and (32) we can now compute

-

(U

(35)

-- -

--

doh) = ii(dv/dx)h(dxh)= U(dv/dx)h(dxh)= (U dijh),

which proves (34). 7

Let K and R be conservative (cf. §6,2). Let the residue class of a prime divisor p of K decompose in R into

-p =

$1

... P7,

where these prime divisors are separable. Then, if the residue of u dv at p lies in opo,we have r

res,(u dv) =

res& dij). i=1

Proof. If p is a first degree prime divisor then so is p, and (36) is trivial. In the next section we will show that extension of the constant field and residue class formation mod po are commutable operations. By choosing an extension in which p decomposes to first degree prime divisors we derive (36) from the sum formula §4,(31). 7 The notion of residue of a differential leads naturally to the residue of a principal part system. For these (36) shows that ( d ’ - h ~dWh) , = (d’-hiii, dW”).

(37)

56.


183

6 . BEHAVIOR OF THE FIELDUNDER RESIDUE FORMATION AND EXTENSION In this section we assume p, to be regular and both K and R to be conservative. Apply some extension of constants k,'/k, and let 0,' be the principal order of k,' and p,' be a prime ideal of k,' dividing po . The prime will be applied correspondingly elsewhere too, e.g., K = Kk,'. The following is obvious. If K and K are conservative fields then the regularity of po carries down t o all prime divisors p,' of po in k,'. The ideals in K and R occurring in $6,3 can be extended to ideals in K' and R', residues being taken mod pol after the extension. Extension and residue formation thus commute for ideals. The results of $4,4 carry this over to the divisors. Let K and R be conservativejields. The homomorphism a ii of the divisor group of K into that of K commutes with the embedding of these groups into those of' K' and R'. The same is also true for the groups of divisor classes. We finally take up the question whether all divisors of R are really residues of divisors in K. In general the answer is negative. For example, let ko = Q, 0 , = 2, po = 2 3, and K = k,(x, y ) with x2 + y 2 + 1 = 0. This equation has no solutions in k , , so that by $ 3 3 all divisors are of even degree. On the other hand, the equation is solvable in k , = 213,so that divisors exist in K with arbitrary degrees. Let K and R be conservative jields and ii be a divisor in R. Then for some finite extension of constants k,'/k, there exists a divisor a in K' = Kk,' which is carried into ii by residue formation modulo a prime divisor pol oj' p o t . ProoJ First choose some extension such that iidecomposes completely into prime divisors of the first degree. Then we need only prove the theorem for such a prime divisor. Choose an element ti E R with numerator divisible by ii. ti has an inverse image a E K. The numerator divisor b of a has an image 6 divisible by ii in R. Now extend ko so that b decomposes into prime divisors of the first degree; ii will keep this property. One of the prime divisors a of b must have the image ii. 7

NOTES Deuring uses a different definition of the residue class mapping a -+ ii for divisors in the paper cited on page 172, which is continued and supplemented by Lamprecht [4]. The problems we have touched have been extended in varying directions. If the residue class f(j,5) of the polynomial in (1) is ____

t Such an extension is not even necessary if ko is perfect with respect to a discrete valuation with the same prime ideal p (cf. Ref. [7]).

184


reducible, then po decomposes to po p;'py ... in K. The irreducible factors x(J, X) of f(J, X) defme fields of functions Ri/ko(X),the genera of which are =

ei(gi - 1) 5 g - 1. (For this, in particular, limited by the inequality cf. Lamprecht [5].) Lamprecht [6] considers to what extent the field R is dependent upon the choice of the variable x used in its definition. Roquette [9] investigates reduction modulo p o in function fields of arbitrarily many variables. The notion of the function field is subordinated, however, to that of the algebraic-geometric variety. The definition given there of the homomorphism a + ii does not require K to have the same genus as K. Numerous further references to the literature are given. In several cases it has been possible to gain insight into all regular prime ideals po of oo as well as to completely characterize the behavior of the irregular po in K (cf. Deuring [l], Hasse [2], and Igusa [3]). REFERENCES [l] M. DEURING, Die Zetafunktion einer algebraischen Kurve vom Geschlecht 1, Nachr. Akad. Wiss. Gottingen Math.-Phys.-Chem. Kl., Part I, 85-94 (1953). Part 11, 1 3 4 2 (1955), Part IIZ, 37-76 (1956), Part IV, 55-80 (1957). [2] H. HASSE,Zetafunktionen und L-Funktionen zu zeinem arithmetischen Funktionenkorper vom Fermatschen Typus, Abh. Deutsch. Akad. Wiss. Berlin, K1.Math.-Naturwiss.No. 4 (1954). [3] J.-I. IOWA,Kroneckerian model of fields of modular functions, Amer. J. Math. 81, 561-577 (1959). Restabbildung von Divisoren I, Arch. Math. 8, 255-264 (1957). Part I l , [4] E. LAMPRECHT, ibid. 10,428437 (1959). [5] E. LAMPRECHT, Bewertungssysteme und Zetafunktionen algebraischer Funktionenkorper I , Math. Ann. 131, 313-335 (1956). Part II, Arch. Math. 7,225-234 (1956). [6] E. LAMPRECHT, Zur Eindeutigkeit von Funktionalprimdivisoren, Arch. Math. 8, 30-38 (1957). [7] A. MATTUCK, Reduction modp of p-adic divisor classes, J. Reine Angew. Math. 200, 45-5 1 (1958). [8] E. D. NERING, Reduction of an algebraic function field modulo a prime in the constant field. Ann. of Math. (2) 67, 590-606 (1958). [9] P. R O Q U E Zur ~ , Theorie der Konstantenreduktion algebraischer Mannigfaltigkeiten, J. Reine Angew. Math. 200, 1 4 (1958).

CHAPTER I V

Algebraic Functions over the Complex Number Field Now that the methodical foundations of the theory have been developed in Chapter 111, their applications to specific cases and the problems involved therein take over the foreground. Here we encounter the oldest parts of the theory. The impulse for the creation of a new mathematical discipline is, in the course of history, always given by the desire to solve certain problems, a minimum of methodical apparatus being ready at hand. It is only at a later stage in the development that a broader conceptual foundation is laid. Chapter IV primarily introduces two special classes of functions, the elliptic functions ($2) and the modular functions (&I), to which the deepest applications of our theory are bound. This serves, on the one hand, to illustrate and establish firmly the general concepts, and on the other to lay the groundwork for the last chapter in which the theory is again developed further, but which also introduces applications to number-theoretical problems. $1 deals with the notion of Riemann surfaces, and is not more than a brief account. There is n o shortage of detailed expositions of this theory in the usual texts. The famous Abel theorem is dealt with in $3, but the modern theory of abelian varieties, to which it leads, can no longer be subject matter for this book. $1. Riemann Surfaces 1. THERIEMANN SURFACE OF AN ALGEBRAIC FUNCTION We assume throughout $1 that the constant field k, = C, the field of all complex numbers. Let an algebraic function field K = k,(x, y ) be given by Eq. 111,$1,(12), which is assumed to be irreducible. Let its degree in y be n. We will first consider y as a function of x . According to Puiseux’s theorem (111,$1,6) one can, for every place x = 5 (t E ko = C or 5 = a),find r = r(5) 5 n power series 111,$1,(21)in the variable zc = x -

5

or

zc = x - l ,

each satisfying Eq. 111,$1,(12). The “ramification indices” eQ= e,(t) are limited by Eq. 111,$1,(19), and all e,({) =1 with at most a finite number of 185

186

1V. ALBEBRAIC FUNCTIONS OVER THE

COMPLEX NUMBER FIELD

exceptions. These power series all have positive radii of convergence, the minimum of which we denote R(t). If n = 1, so that y is a rational function of x , the Riemann number sphere fi0 is associated with the function y as its Riemann surface. On it, the totality of Euclidean circles about the points with radii less than some now arbitrary limits R(5) represents a system of neighborhoods, giving it the structure of a Hausdorff space. (The reader should consider the full meaning of this statement, which includes the point 5 = 00 along with all others.) According to the covering theorem of Heine-Borel, a finite number of these circles already forms a covering of so. For n > t the Riemann surface % of y is defined as a covering space of ‘$ asl follows: o to every circular disk U(5)of the number sphere associate r = r(5) disks U(rl), ..., U(t,), some of them “wound ” or “ramified, ” with ramification indices el( 0, % can be cut open to a simply connected surface section by 2g closed incisions. The boundary then consists of 4g curves, which we denote a,, b,, a,', bl', a,, 6, ,a,', . .., b,, a#',b,' in the sense of positive traversal. To reconstruct % one must attach the a, with the a,' and the bi with the b,', always in the opposing sense. (All corners then meet at a single point.) These 29 incisions a,, b, which leave % simply connected are called a canonical system of incisions. The a i , b, form a homology basis. Any two canonical incision systems satisfy the homologies

where

is a symplectic modular matrix in the sense of §1,4 of the Appendix to Chapter I, that is, G E re. Conversely, a canonical system of incisions is associated with every such G (cf. 43,l). Any'closed path on % is homotopic to a combination of the paths a i , a,', b , , b,' (that is, it can be deformed to such a combination). The a, differ from the a,' (and the bi from the b,') only in their orientation. The combination a,b,a,'b,' a,b,a,'b,' is homotopic to 0. For g = 1 this last statement implies that ab and ba are homotopic. This means that any path is homotopic to p traversals of a and then v traversals of b, where p and v are rational integers which are taken negative for traversals in the opposite direction of a and 6 .

1%

Iv. ALGEBRAIC FUNCTIONS OVER THE COMPLEX NUMBER

FIELD

$2. Fields of Elliptic Functions 1. INTRODUCTION

By a field of elliptic functions K we mean a separably generated and conservative function field of genus g = 1. Throughout $2 we assume K to have at least one prime divisor o of degree 1, which would not necessarily be true.f For the start we make no assumptions concerning the base field k , . The canonical class W of an elliptic function field must always be the principal class, for being of dimension 1 and degree 0, it must contain an integral divisor of degree 0. The class of the divisor o2 is of dimension 2, according to the RiemannRoch theorem. Then let x be an integral divisor of this class linearly independent of 0 2 , and let x be an element such that (x) = x / o 2 .

(1)

The numerator divisor x of x is of degree 1 in k,(x) and of degree 2 in K. Using the concluding remark of 11, §5,4 we see that K is then an extension of k = ko(x) of the second degree; the same could be seen from Eq. III,$2,(20) by letting a there be the denominator divisor o2 of x. A11 multiples of o-’ are now linear combinations of 1 and x. Again using the Riemann-Roch theorem, the dimension of the class of o3 is 3. Thus there exists a y E K with irreducible divisor representation ( Y ) = 4b3.

(2)

1, x, y are now a basis of the multiples of o - ~ The . element y is not contained in ko(x) so that, as [ K : k,(x)] = 2, necessarily K = ko(x, y). An irreducible equation f ( x , y ) = 0 over k, must hold between x and y , naturally, but its exact form is of no interest now. In the case of an algebraically closed constant field k , of characteristic 0, all prime divisors p correspond to pairs of solutions (xl, yl) of the equation f ( x , y ) = 0 in k , , in the manner of 111, $2,2, with the one exception of the prime divisor o. We now want to investigate the general case of first degree prime divisors p # o where the constant field is arbitrary. By II,§5,5 such

t I. R. SAFAREVI~ [Exponents of elliptic curves, Dokl. Akad. Nauk SSSR 114, 714-716 (1957)] shows that there exists an elliptic function field over ko = Q with an arbitrary natural rn as the minimum degree of its prime divisors. Exercise. Show: (a) K = ko(x,yl/?) with Eq. 111,§3,(14), unsolvable in k o , holding between x and y, is an elliptic function field with 2 as least degree of a prime divisor; (b) K = Q(x, y ) where x3 + y3 1 = 0 is of genus g = I , with minimal divisor degree 3. Use the fact that the Fermat equation with exponent 3 has no integral rational solutions.

+

$2. FIELDS OF ELLIPTIC FUNCTIONS

191

a p produces a homomorphic mapping of every ring 3 of functions of K, whose denominators are not divisible by p, onto It,. If further 3 has K as its quotient field, this homomorphism can be extended to the ring 3, of all elements of K with denominators not divisible by p. One such ring 3 is that generated by x and y from above. The residues xl, y , in ko of x , y mod p are often denoted x'1

=xp,

y1 = y p

(3)

As f ( x , y ) p = Op = 0 they are a solution of the equation f ( x , y ) = 0 in k , . Conversely, any mapping of the pair x , y onto a solution x l , y l defines a homomorphism 3 -+ k , whose kernel is a prime %deal, which in turn defines a prime divisor p # 0. Hence, the prime divisors p # o of degree 1 are in one-to-one correspondence with the pairs of solutions (xI, yl) in ko of the equation f ( x , y ) = 0. Finding all the first degree prime divisors is a plausible task for the theory. The following theorem relates it with a deeper notion, which clearly goes to the core of the theory. Ecery dirisor class P of degree 0 contains exactly one divisor PO-', where p is a prime diiiror of degree 1. This correspondence P - p is a one-to-one mapping of' the dirisor classes of degree 0 onto the prime ditisors of degree 1.

Pro@ If p is such a divisor, then PO-' lies in such a class P. P is the unit class if and only if p = 0. For, there can exist no element z 4 ko with the divisor decomposition (2) = PO-'; the argument used right after Eq. (1) would then lead to K = k,(z). Conversely, let P be given. By the RiemannRoch theorem, the dimension of Po is 1, so that there is exactly one divisor p of degree 1 in the class Po, and, of course, it is a prime divisor. V

2. THEADDITION THEOREM Let two prime divisors of the first degree, p and q, be given; by the last theorem they define another such divisor r, with Pq

- - N -

r

O D

O

Writing this relation as p+q=r

(4)

we define an addition of prime divisors of degree I . Any such divisor could play the part of 0, but it must then be held fixed. The prime divisors of the first degree are thus made into an additive abelian group, isomorphic to the multiplicative group of divisor classes of degree 0. The neutral element (zero element) is 0 .

192

Iv. ALGEBRAIC FUNCTlONS

OVER THE COMPLEX NUMBER FIELD

Addition Theorem. Let pi, p 2 , p3 be three prime divisorsof the$rst degree, not coinciding with each other or with 0 , and satisfiing

1 1

XPl

YP1

XP2

YP2

=o.

(6)

1 X P 3 YP3 Ifsome of the pl coincide with each other or with supplements. When

0,

we have the following

When P1

=

-P2

- P2

= -2P2 #

then I1

XPl

YP1

1

XP2

YP2

dx dy -P2 dP2 dP2 with some prime element p 2 for p 2 . For 0

p

0,

Pz # 0 ,

(5'7

0,

(5"))

I = 0,

-P2

+ p + P = 3P =

0,

Pf

we have 1

0

XP

dx

-P dP

YP

dy

-P dP

=O

(6)

d2x d2y 0 Z P dp2P

dP

Finally, if (5y

then (6'"')

Conversely, Eqs. (6), etc., imply the truth of Eqs. (S), etc. Proof. In the general case, Eq. (5) states that P I P ~ P ~ O=- (z) ~ is a principal divisor. z is a multiple of o - ~ ,so that it is of the form z = a + fix + yy with 01,8,y E k,,. This means that we can now replace one column of (6) by zpi .

$2. FIELDS OF ELLIPTIC FUNCTIONS

193

But these are all 0-residues, which proves (6). An analogous argument leads from (5') to (6'). Given (5"), thereexistsaz = u px 3- yy with numerator divisible by p1p22. Then dzldp, is still divisible by p 2 , and as above we can replace a column by zpl, zp, ,dz/dp2 p 2 . The vanishing of these residues proves the assertion. A similar Y) argument holds in the cases (5"') and (5""). 7 The addition theorem can be visualized geometrically. We must assume k , to be algebraically closed and not to have characteristic 2 or 3. In $2,6 we shall show that x and y can be so chosen that (21) there is the defining equation; it represents FIG. 2 the curve of Fig. 2. Write xi, yi ( i = I , 2, 3) for short instead of xpi, y p i .Theseelement pairs lie on the curve. Equation (6) states that the three points (xi, yi) lie on a single line or, better, that (x3, y 3 ) is the third point of intersection of the curve with the line given by

+

/

Substitution into (21) of the value this gives for y yields

As we have two zeros, x

= x1 and x2, the

third is

It corresponds to

3. AUTOMORPHISMS

In this section we will discuss the automorphisms 0 of K which leave the elements of k, fixed. Application of such a 0 to the element z and divisor p yields the element z' and the divisor p'.

194

Iv. ALGEBRAIC FUNCTIONS OVER THE COMPLEX NUMBER FIELD

As preparation we need a lemma: An automorphism is uniquely character-

ized by its action on the prime divisors. In other words, we must prove that if p" = p for all prime divisors, then CT is the identity. The hypothesis implies (cf. 11,§5,3 or 111, $2,2) that c transforms all elements to multiples of themselves : z b = C,Z,

C,

E k, ,

If z l , z 2 are linearly independent over ko we then immediately see that cz,+z2= c,, = c Z 2 , and, since c1 = 1 must hold, c, = 1 for all z . Thus c is the identical automorphism. Let 0, 0' be two prime divisors, not necessarily differing, of the first degree, and let 3 be the integral divisor linearly independent of but equivalent to oo', which we know to exist by the Riemann-Roch theorem. Let 3(00')-' = ( z ) . The argument of 42,l then shows K to be an extension of the second degree over k,(z). This extension is always separable, even if the characteristic is 2. For, if this were not the case, the composition R = KI;,of K and the algebraic closure I;, of k, would be an elliptic function field on the one hand and an inseparable extension of I;,(z) of the second degree on the other. We should then have R = ko(z,&(T) with a polynomial a(z) = aOzn+ a,. Now, I;, is algebraically closed, so that = fio(,/Z)n + = b(Jz) is a polynomial in Ji over I;, , which would mean R E I;,(/.) Comparison of the degrees would lead to [I;,(&): k0(z)] = [ R : E,(z)], so that R = Lo(&). But this would give R the genus 0. As K/k,(z)is now a second degree separable extension, there exists an automorphism CT = CT,,,.not the identity, of order 2 of K , which leaves k,(z) and therefore ( z ) = 3(00')-' fixed. We call it the refection automorphism of o and 0' in K. It interchanges o and of if o # of was chosen, as is usually done in the decomposition of a prime divisor of k,(z) to two prime divisors in a quadratic extension K. Now let p be some divisor of degree 1. Then (p/o)(p/o)b is a divisor of degree 0 in the rational function field k,(z), and thus a principal divisor. In terms of the convention (4) this means

&

+ + ,/
1. Then, except for the cusps, z - to is a local uniformizing variable everywhere. Thus at most the prime divisors 5, of the cusps can occur in the denominator of the divisor of M ( r ) dth. By (13) and (16) we have, for the cusps, 53

M(r) drh = M ( r ) (dr/dq)hdqh = ( N / 2 ~ i ) ~cvqv-hdqh. v=o

This puts the divisor of M(r) d8’ in the form m(el , ... ,5,,)-’ with an integral divisor m. On the other hand, the divisor of M ( r ) dth must lie in the hth power of the canonical class W, which places m in the class Wh(el, ..., s,~)~.By the RiemannRoch theorem this class contains (2h - l)(g - 1) ha, linearly independent integral m.This then is exactly the dimension of the linear space of all integral modular forms M ( t ) of this sort. An explicit expression for this dimension is gotten from (5), (1 l), and (12). There exist

+

(2h - l ) N + 6 N ’ n ( 1 -q-’) for N > 2 and h + 1 for N = 2 24 q/N linearly independent integral modular forms of level N > 1 and of dimension -2h with the character 1. It is somewhat more complicated to determine this number for N = 1, as the local uniformization at the points and i deviate, as seen in §4,1. We have

c

-(t - 0’ dq, - 3 -- (22 + 1 ) dt

t(z

+ 1)

(7

z’(z

-

o3

+ 1)’ ’

3 = 2 - (z - - -* i ) dt z

(z - i)* 2’

Thus the divisor of a holomorphic modular form M(r) of this sort generally contains qc to a power 52h/3 and q, to a power 5 h/2 in its denominator, the. notation being obvious. The rest of the argument is as above, yielding the following. There exist [2h/3]

+ [h/2] - h + 1 = [h/6] = [h/6]

for h = 1 mod6

+1

for h f 1 mod6

linearly independent integral modular forms of level 1, dimension -2h, and character x = 1. We finally remark: for an odd dimension -k no integral modular forms of level 1 exist having characters with values & I exclusively. For if M(T) were

$4. MODULAR

225

FUNCTIONS

such, then M ( T ) , d t k would be a differential. Then M(T), (dr/dqi)kwould be a power series in qi at T = i. But then it can be read directly from yi and dq,/dr that M ( T ) cannot be expanded in a power series in T - i; therefore M ( T )cannot be holomorphic at T = i. 7. FOURIER EXPANSIONS OF EfSENSTEIN SERIES In this section we assume (u,, a , , N ) = 1. Starting point is the equation

which holds for k 2 2. On the left is a function periodic in z, whose Fourier coefficients are correctly given, as is easily verified. The sum (17), which we want to compute for k > 2, contains terms with m, = 0 if and only if a, = 0 mod N . By setting 6 ( a , / N )= 1 or 0, depending upon whether a , / N is integral or not, we have

+

'

ml E a l ( N ) 2' ml

#O

+m

1

,gm( m l t + rn, + mm,N)k'

where the sum C2 is taken over all m2 running through a residue system mod m,N with m, _= a, mod N . Now, the inner sum over m is ( m l N ) - k times the sum on the left of (22), with N - ' ( T m 2 / m l ) in place of T . By applying Eq. (22) we find that

+

The inner sum over m2 can be realized by setting m2 = a, + Nm,' and then letting m2' take on all values from 0 to rn, - I. It is lmll times nk-'rn;' exp[2ninN-'(~+ u2m; I ) ] or 0, depending upon whether n is divisible by m1 or not. Set n = tm,, so that, f o r k > 2,

226

Iv. ALGEBRAIC FUNCTIONS OVER THE COMPLEX NUMBER FIELD

In particular

Here c(s) denotes the Riemann zeta fur1ction.t The constant term of the Fourier series (23) is # 0 if and only if a, 3 0 mod N. Assume Gk(T;a,, a,, N) to be primitive and apply to it the substitution (18). Now let z approach ico, but by using the expansion (16) for the cusp r(N)a/c on the left and the Fourier series (23) on the right. This yields m

c,

exp[2niN-'~] = Gk(r; a , a

+ a2c, a , b + azd, N )

n=O

and the constant term of the expansion (16) of a primitive Gk(T; a,, a,, N) at the cusp r(N)a/cis co # 0 fi and only ifa,a a,c E 0 mod N. Now we can prove that all formally different Eisenstein series are linearly independent if k > 2. By the last statement it suffices to show that these series with a,a a,c E 0 mod N are linearly independent. Such Eisenstein series are uniquely determined by the residue of a,b + a,d = a,' mod N , and among them there are as many formally different as a,' takes values mod N with (a,', N)= 1 and 0 c a,' 5 N/2; the latter inequality takes account Of the fact that G k ( T ; a,, a,, N ) and G k ( T ; -a,, - a , , N ) do not differ formally (in the sense of §4,4). It amounts to the same to prove that all primitive Gk(T;0, a, N) with (a, N ) = 1 and 0 < a S N/2 are linearly independent. This we derive from their Fourier expansions (23) [where we neglect the common factor ( - 2 ~ i ) ~ / (k l)!Nk]. In the Fourier expansions we only consider the terms ck(mN; 0, a, N) exp[2nimz] with rn = 1 and further prime values of m by which N is not divisible. According to (23) the coefficients are

+

+

t The numbers 25(2/0 (2k)! ( 2 ~ ) -=~W~ are the so-called Bernoulli numbers. They are rational. In particular, 2[(4) = (2~)~/720,25(6)= (2+/42.720.

$4. MODULAR FUNCTIONS

227

the latter for a prime m; y(a) is an abbreviation. If our Fourier series were linearly dependent, so would the series

1y(ma) exp[2nimt]. At first let k be even. Then the y(a) form a basis with respect to Q of the largest real subfield of the Nth cyclotomic field Q N . To each residue m mod N prime to N there corresponds an automorphism u, of Q N taking y(a) into y(aYm= y(ma). Therefore the determinant Iy(ma)l with m and a running through a system as described above is # 0 which proves our contention. If k is odd the ?(a) form one-half of a basis of QN/Q, the other half given by exp[2niaN-'] exp[-2niaN-']. Now the same reasoning as before can be repeated. 7 Integral modular forms of dimension -2 can, by §4,6, exist only for N > 1. Then

Qr'

+

)

1 =

N-'p('

rN

C' rnr=ar(N)

1

((mlt + mJ2

-

1

((m,-

+ ( m z-

~2))'

1

(25) are such modular forms. These are the partial values of the Weierstrass p function of §2,(26), of periods t and 1. The case a, = a, = 0 mod N i s now excluded by assumption ; the prime over the summation indicates that whenever m, - a, = m, - a, = 0 that subtrahend is to be omitted. The easy verification of the functional equations (18) is left to the reader as an exercise. The Fourier expansion of (25) is found by summing first over m 2 and then over m,,preliminarily limiting m1 to a finite range:

W

+ n = l c,(n; a , , a , , N) exp[2ninN-'r],

(26)

the sums in the last equation having nt-' = m, or m, - a,, respectively, limited to that finite range. This range is then increased until all integers m, = al mod N are included. The result is a sequence of Fourier series, whose nth coefficients c, remain unchanged following some member depending upon n, and each of which converges absolutely and uniformly in Im(t) 2 E > 0. The same is true of their limit, which therefore represents the function (25).

t t takes on values of the positive and negative divisors of n.

228

IV. ALGEBRAIC

FUNCTIONS OVER THE COMPLEX NUMBER FIELD

The demonstration of the linear independence of the primitive G,(t; a,, a 2 , N) cannot be taken over to the partial p-values because of the extra terms 2N-’1(2) in their Fourier series (26). But the argument, applied to their differences, still shows that uN - 1 linearly independent modular forms (25) exist. Actually, the only linear relation between them is the vanishing of their sum. For this sum, being an integral modular form of level 1, must vanish according to the last section. To every integral modular form of dimension -k there exists a linear combination of primitive Eisenstein series ( k > 2) or of primitive partial p-values (k = 2) such that their difference vanishes in all the cusps. These differences are modular forms, called cusp forms. Proof. The assertion immediately follows the fact, for k > 2, that there are just as many linearly independent Eisenstein series as cusps. For k = 2 a linear combination of the partial p-values can still be found, such that the differencef ( r ) vanishes at all except at most one cusp. Now, by the last section,f(z) dz is a differential having at most a pole in that cusp, which at most is of the first order. By the residue theorem of 111, $4,7 the residue is then 0. Thus no pole occurs. 7 A cusp form f(z) of dimension -2 determines a differential f ( z ) dz of the first kind, and conversely, as follows immediately from the definition of the divisor off(z) dz of the last section. Another theorem of importance is found from the series (24): the discriminant A(z) = g2(z, 1)3 - 27g3(?, 1)’ can be expunded in the Fourier series OD

A(r) = ( 2 ~ ) ’ ~r(n)eZninr n= 1

with integral coefficients t(n) and with z(1) = 1. Furthermore,

with rational integers c, . Incidentally, the surprising formula m

can be proved, but it is difficult to interpret it in the framework of our considerations.t

t A simple proof of this is found in R. FUETER, Vorlesungen uber die singularen Moduln und die komplexe Multiplikation der elliptischen Funktionen, p.24. Leipzig-Berlin, 1924.

$4.MODULAR ProoJ

FUNCTIONS

229

By (19) and (24)we have

with rational integers yn , y n ’ , and further, 4,

A(T) = ( 2 n ) 1 2 1 n=l

15 + 712

pe2ninr +

t/n t>O

...

12

where the indicate a Fourier series with integral coefficients and starting with e4nir.But, by a theorem of elementary number theory, (5 + 7t2)t3= 0 mod 12 must always hold, so that the ~ ( n )are integers. As ~ ( 1 )= I , the c, are also integers. 7 1..

8. THETAFUNCTIONS Special modular forms of dimension - k and level N were already derived in the appendix to Chapter I, §2,3,as the theta functions

to definite quadratic forms F[m] in 2k variables and of level N . The c(n, F) were the numbers of ways the integer n could be represented by the quadratic form +F[m].It turns out that in certain cases these functions can be represented as linear combinations of Eisenstein series. The expansion coefficients of the latter were computed in the last section, and are functions of elementary number theory. In such cases, then, the c(n, F) can be expressed by such’functions. Otherwise one can at least find approximations of the c(n, F ) by elementary expressions. As an example we consider the form F[m]= 2(m12 + + m42) in detail. In this case it is best to verify directly from (27) and the appendix to Chapter I, $1 ,( I3), that $(7/2, F) is a modular form of dimension - 2 for the subgroup 0 of r generated by (A t ) and (-: A). §l,4of that appendix shows that 0 3 r(2). Thus, not only is $ ( T , F) a modular form of level 4 as seen in the appendix to Chapter I, §2,3,but 9(5/2,F ) is one of level 2. This simplifies our investigations.

230

Iv. ALGEBRAIC FUNCTIONS OVER THE COMPLEX NUMBER

FIELD

9(2/2, F) is a linear combination of the two linearly independent partial p-values

(the sums being taken only over the positive divisors t of n, here), and of a cusp form. Further, the cusp forms correspond to differentials of the first kind. Since, by (1 1) and (5), the genus of the field Kr(2 is ) zero, there can be no cusp forms. For the two coefficients c(0, F ) = 1 and c(1, F ) = 8 to agree, we must have

9(t/2,F ) = n-'(p(t/2

+ f-; t/2, 1) - @ ( ~ / 27,; 1)).

For the Fourier coefficients this implies

An argument from elementary number theory improves this: c(n, F) is 8 or 24 times the sum of the odd divisors of n, depending upon whether n is odd or even. This result was attained in essentially the same way by Jacobi. The quadratic forms with matrices

(; i 8 $ 2 1 1 1

(:; :p). (; : :) 2 1 0 0

2 1

0 0

0 0 1 2

0 1

1:5 4

have the determinants IF1 = 4,9,25 and levels N = 2, 3, 5. The corresponding 9(~F , ) are modular forms of that level and of dimension -2. By (1 1 ) and ( 5 ) the genera of these K r ( N )are still 0, so that no cusp forms occur as yet. Hence the functions 9(r, F ) can be linearly combined out of the partial p-values. An elementary computation shows that the number c(n, F) of representations of n by these forms F is equal to the sum of the divisors of n prime to 2, 3, 5, multiplied by 24, 12,6, respectively Quaternary forms F with larger determinants and forms in more variables lead to modular forms really including cusp forms. The functions 9(7,F ) are then sums of partial p-values or Eisenstein series and cusp forms. Denoting the Fourier coefficients of the latter by yk(n) we have, for the general case, c(n, F ) =

a(a1, a2)ck(n, a , , a2 9 N ) a1.a2

+ yk(n).

(28)

54. MODULAR FUNCTIONS

23 1

A simple approximation given by Hecket shows that the yk(n) are of the order (29) while the c,(n; a,, a 2 , N ) are, by (23) and (26), of the order O(nk-l+e)with arbitrarily small E . Thus the c,(n; a,, a2 , N ) prevail if k > 2. More advantageous approximations of yk(n) exist, but lie far deeper (see Chapter V). yk(n) = O(nk”),

NOTES The theory can be carried over to an essentially more general class of modular functions, which satisfy the functional equation M(-)(cr ar + b cr + d

+ d)-k =

f;)M(r)

instead of (14). An arbitrary complex number can even be taken for k. The no longer uniquely defined function (c7 + d), must be fixed if k is not an integer. The numbers u(: !) are called the multipliers of M(z). Divisors in a generalized sense can be associated with these modular forms, and a suitable generalization of the Riemann-Roch theorem permits the determination of the number of linearly independent modular forms satisfying certain integrity conditions, in many cases. The theory was built up by Petersson in a series of papers [4] which, incidentally, touch on the paper by Weil [lo] mentioned on page 142. In [8] Petersson discusses the possibility of a generalization beyond (30). Petersson [ 5 ] , using an idea which will occur in V,§3,4, proved that all modular forms of level 1 and of real dimension -k < - 2 can be represented by special PoincarC series G,(z, m) =

1(ct + d)-, exp[2nim(ar + d)(cr + d ) - ’ ] ,

(31)

where c, d run through a full system of mutually prime numbers and a, b are chosen correspondingly so that ad - bc = 1 . The corresponding generalization holds true for arbitrary subgroups of finite index in r, and even for the most general discontinuous groups of motions of the hyperbolic plane [6]. In another direction, Eichler [ l ] shows that the cusp forms of dimension - 2k can be represented by theta series $(r, F ) whenever N is a prime number and the functional equation (14) is assumed to hold with x(: :) = 1. A similar statement continues to hold for the generalized theta series (32) limited to quaternary forms [2]. The proof is of a number-theoretic nature, but enters upon algebraic properties of modular forms (see Chapter V). The method of the last section for constructing modular forms using quadratic forms has been generalized as follows. Let F = VVT with a real matrix V andf(x) be a homogeneous spherical function of degree 2r in 2k

t E. HECKE, Mathemarkhe

Werke, Paper 24, pp.461486. Gottingen, 1959.

232

IV. ALGEBRAIC FUNCTIONS OVER THE COMPLEX

NUMBERFIELD

variables. Then the function S(z; ~ , f=)C f(VT(m))enirF[ml m

satisfies the functional equation (14) if k + r replaces k. The proof, given by Schoeneberg [lo], uses a method essentially dating back to Hermite. It has finally become clear that the proof of Eq. §2,(15) of the appendix to Chapter I also holds for quadratic forms in odd numbers of variables. The determination of the sign, now less simple, is carried out by Pfetzer [3], immediately for the generalized series (32). Modular forms of positive dimensions and their Fourier series can also be applied in number theory [9]. Finally, we cite a paper by Peterson [7], which conceptually characterizes the Eisenstein series G k ( z ;a,, a,, N) as against other integral modular forms by one of its properties. Completely ignored in this survey of the literature are the various treatments of the question as to Weierstrass points on the modular surface !RtT"). Furthermore, all papers on the analytic number theory of quadratic forms were passed over, insofar as they offer no substantial contributions to ,the theory of modular forms as an essentially algebraic notion. No sharp boundary can be drawn, but were one to neglect any sort of division, the literature to be cited would swell substantially. The boundary between modular functions and automorphic functions is also vague, but literature on the latter has been suppressed as far as possible. Further literature on modular forms must still be discussed in ChapterV. REFERENCES [l] M. EICHLER, Ober die Darstellbarkeit der Modulformen durch Thetareihen, J. Reine Angew. Math. 195, 156-171 (1956). QuadratischeFormen und Modulfinktionen, Acta Arith. 4,217-239(1 958). [2] M. EICHLER, [3] W. PFETZER,Die Wirkung der Modulsubstitutionen auf Thetareihen zu quadratischen Formen ungerader Variablenzahl, Arch. Math. 4, 448-454 (1953). [4] H. PETERSSON, Zur mlytischen Theorie der Grenzkreisgruppen, Part I. Math. Ann. 115.23-67 (1938); Part 11, 175-204; Part 111,518-572;Part IV, 670-709. [S] H. PETERSSON, Ober die Metrisierung der ganzen Modulformen, Jahrb. Deutsch. Math. Verein. 49, 49-75 (1939). [6] H. PETERSSON, Ober eine Metrisierung der automorphen Formen und die Theorie der Poincardschen Reihen, Math. Ann. 117,453-537 (1939). [7] H. PETERSSON, Ober die systematische Bedeutung der Eisensteinschen Reihen, Abh. Math, Sem. Univ. Hamburg 16, 104126 (1945). [8] H. PETERSSON, Ober die Tramformatiomfdtoren der relativen Invarianten hearer Substitutionsgruppen. Monatsh. Math. Phys. 53, 17-41 (1949). [9] H.PETERSSON, Konstruktion der Modurformenund der zu gewissen Grenzkreisgruppen gehorigen automorphen Formen von positiver reeller Dimension und die vollstandige Bestimmung ihrer Fourier-Koefizienten,S.-B.HeidelbergerAkad. Wiss. pp. 3-80(1950). [ lo] B. SCHOENEBERG, Das Verhalten von mehrfachen Thetareihen bei Modulsubstitutionen Math. Ann. 116,511-523 (1939).

CHAPTER V

Correspondences between Fields of Algebraic Functions By a correspondence between two Riemann surfaces 'iRl and !R2 we mean a mapping of R1onto 'iR2 which is, along with its inverse, finitely multivalued and conformal, the latter condition being permitted to fail in finitely many exceptional points. $1 parallels this geometric notion with that of an algebraically defined correspondence, the Riemann surfaces being replaced by arbitrary algebraic function fields. The correspondences of an algebraic function field with itself form an associative ring. In $2 the representations of that ring by matrices are considered. A number of important applications of algebraic function theory to number theory are based on the notion of correspondences. $3 will deal with the correspondences of the field of modular functions with itself, in particular, the results including deep-lying facts concerning the number theory of quadratic forms. §4 returns to the main line of the general theory and includes the proof of an important principal theorem. Its meaning becomes particularly clear by its almost surprising implications in 55. A short survey of correspondences between elliptic function fields is finally given in $6. $1. The Correspondences

1. BASIC CONCEPTS Let K and K' be two fields of algebraic functions of one variable each, both having k , as exact constant field. K and K' are assumed to be separably generated, conservative, and algebraically independent of each other, the latter meaning that no two nonconstant elements x E K and y' E K' can satisfy an algebraic equation Ax, y') = 0 with coefficients in k , . It is obvious that the algebraic independence of K and K' is preserved under extension of the constant field k , . We agree to always distinguish elements of Kand K' by denoting the former without, the latter with primes. We follow the same convention for ideals and divisors. 233

234

v. CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS

If the constant field ko of K is extended to K', then the algebraic independence assures us that the function field KK' has the exact constant field K'. The field K'K can be formed analagously. Then KK' = K'K.

(1)

All this equation states is that the order of adjunction of the primed and unprimed quantities has no effect on the final field. This is also a consequence of the algebraic independence. The fields Klk, and KK'IK' have the same genera, as do also K'lk, and KK'IK, because K and K' were assumed conservative. According to 111, §3,4this implies that a dioisor ofKlk, keeps its same degree as a divisor of KK'IK', and then so does a divisor of' K'/ko extended to KK'IK. We shall denote elements of KK' by upper-case Greek letters, such as A, B, r. For two integral domains 3,3' in K, K', let the ring product

1

be the integral domain of all finite sums t i l l ' . In particular now, let 3 be the principal order of K with respect to k,[x]. Then 3 x K' is the principal order of K K with respect to K'[x]. For the proof denote this order by R ; clearly it contains 3 x K'. If R were larger than 3 x K', the determinant of a linear transformation (cij)transforming a basis Ki of R/K'[x] into a basis i i of 3 / k o [ x ]which is also a basis of 3 x K'/K'[x] would be a nonconstant polynomial in x. Thus the extension k , + K' of the constant field would decrease the degree of the discriminant divisor of K and consequently of the genus of K. This is a contradiction. We shall apply the ideal theory of 11, $1 to the principal order of KK' with respect to K'[x]. The 3 x K'-ideals will be denoted A S x K ,B, S x K ,etc. , By A ~ ~ K , + A K X= K KA3.K. ,

(3)

these ideals are mapped onto K x K'-ideals. The mapping (3) is a homomorphism. The kernel ofthis homomorphism is the group consisting of all ideals a x K' where a is any 3-ideal in K. An 3 x K'-ideal A, K' has a finite system of generators. But a system of generators of is also a system of generators of A K x K ebecause of (3). By calling a K x K'-idealfinite whenever it has a finite system of generators we may state: any finite K x K'-ideal is the image of an 3 x K'-ideal. The principal theorem of II,§l may then be carried over as follows. Thefinite K x K'-ideals form a group, each ojwhose elements can be uniquely decomposed to prime ideals. With the elements A K x K 'of this group we associate elements of an isomorphic abstract group, but now writing the group operation as addition

§ 1. THE CORRESPONDENCES

235

we call the elements correspondences between K and K’. The sum of two correspondences A and B is thus defined by stipulating ( A + B)KX,, =AKxK,BKxK,. (4) Correspondences belonging to prime ideals are called prime correspondences. There is an isomorphic image of the group of correspondences in the divisor group o j the field KK‘ oj’functions of one variable over the constant $eld K’. The same holds if K and K’ are interchanged. Indeed, we have already associated every correspondence A with an ideal A , , to the principal order 3 x K‘ of KK’ with respect to K ’ [ x ] ,and actually with the set of such ideals aA,.,,, where a is any 3-ideal. The 3 x K’-ideals factor uniquely to prime ideals. not containing any prime 3-ideal in Now, the set of 3; x K’-ideals their factorization form a group which is isomorphic to the group of correspondences, by definition. But this group is also isomorphic to a subgroup of the divisor group of KK’IK’ from the definition of the latter in 11, §5,3. The theorem just proved permits us to identify correspondences with divisors and to associate properties of those divisors with the correspondences. Thus we can first speak of equivalent correspondences A and B : A B if A K x K = , ABKxK, with an A E KK’ (moreover A = B if A E K or K’). The classes again form a group, and we can write A 0 for A K x K * = AK x K.‘? We can also associate two degrees, g(A) and g’(A), with each correspondence. We mean the degrees of the associated divisors in KK‘ with K’ and then with K as constant field. KK‘ is first considered an extension of constants of K and then of K‘. Addition of correspondences clearly leads to N

-

g’(A + B) = g ’ ( 4 + g’(B). g ( A + B) = g(A) + g(B), (5) One misunderstanding should be avoided. The degree of the principal divisor (A) of KK’IK‘ is g((A)) = 0. However, the degree of the correspondence represented by the ideal AK x K‘ = A, K‘ may differ from g((A)) = 0, as the ideal AZ x K’ = A l x K may , have prime factors of the sort a x K’. We frequently want to extend one of the fields K or K‘ algebraically to Kl or K 1 ’ .A K x K‘-ideal A K x Kthen 3 passes to

A:, X K’ = KIAgxK*, = Ki’AKxK,. (6) The associated correspondences are denoted A’ and A”. Equation (6) indicates isomorphic mappings A + A’ and A + A’’ of the groups of correspondences of K and K‘ into those of Kl and K’ and of K and Kl’. An

t We have seen that a divisor of KK‘ over the constant field K and a divisor of KK’ over K‘ is associated with each correspondence. Thus, the equivalence of two correspondences A, B really means: the divisor associated with A-B in KK‘IK’ is a principal divisor up to a factor of a divisor of Klko, and that associated with A-B in K K I K is a principal divisor up to a divisor of K’lko .

236

V.

CORRESPONDENCESBETWEEN FIELDS OF ALGEBRAIC FUNCTIONS

extension of K or of K' may imply an extension k l / k o of the constant field. In this case A' or A" clearly are correspondences between K, and K'k, or between Kk, and Kl'. Under an extension of the constant field the degrees of the correspondences remain unchanged, due to our assumptions. For extensions which leave the constant fields invariant we have g(A') = [ K , : K I g ( 4 , g'(A') = g'(A),

g(A") = g ( 4 ,

(7)

g'(A'') = [Kl':K']g'(A).

The first and last of these equations result from the last theorem of 11,§5,4. Taking into account that K' and K,' are the exact fields of constants of KK' and KK,', and that K has been assumed separably generated and conservative, we see by 111, §3,4 that g(A") = g(A). The other equation follows symmetrically. Examples of prime correspondences can be given. Let an isomorphism a + a' of K onto a subfield KO'of K' leave the field of constants, k , ,fixed. Then the differences a - a' for all a E K , together with their products with K x K', generate a K x K'-ideal PK K' which is prime and of degrees g ( P ) = 1, g'(P) = [K':K,']. Proof. Clearly P K x K is , an ideal. To see that it is finite remember that all we need are the differences a, - a,' with a system a, of generators of K / k , . For, if a =f(a,)g(a,)-' with polynomials f(a,), g(a,) in those generators, then clearlyfla,) g(a,)-' -f(a,') g(a,')-' lies in K x K'(a, - ",',az - az', ...). The residue classes of K x K' mod P , K , can be represented by elements of K'. The field of these residue classes is therefore equal to the constant field K' of KK'IK', so that g ( P ) = 1. But at the same time the residue classes of the subfield K x KO' mod P , x K t can be represented by elements of K, all residue classes of K x K' mod P , K , generating an extension Kl of K isomorphic to K'. This means that the latter residue field is an extension of degree g'(P) = [ K , :K ] = [K':K,'] of the constant field K of KK'IK. 7

2. MULTIPLICATION OF CORRESPONDENCES Let three algebraically independent fields K , K', K" of functions with common constant field k , be given. Let A and B be correspondences between K and K', and K' and K".The product AB = C will now be defined as a certain correspondence between K and K", this being done in several steps. The degrees of B will be denoted g'(B) and g'(B) in an obvious manner. (a) Let B be a prime correspondence of degree g'(B) = 1. By 11, §5,5 then, the residue class ring of K' x K" mod BK, is a field isomorphic to K". Moreover, every residue class has exactly one representative in K". By taking K,r

8I .

237

THE CORRESPONDENCES

the elements of K' in particular, we have an isomorphic mapping of K' onto a subfield of K". Thus, B has precisely the property of the example at the close of the last section. We also have an isomorphic map of K x K' onto a subring of K x K . For this case we denote the K x K"-ideal arising from A K x K , under the isomorphism by ( AB)K ,,,. Clearly

( A , + A z ) B = A , B + AzB. (8) (b) Let B be a purely inseparable prime correspondence with respect to K", that is, let the residue class field of K' x K" mod B,, ,, be a purely inseparable extension of K" of degree ph, where p is the characteristic. Then every a' E K' satisfies some congruence arph =a" mod B, , with a" E K". Thus there exists an isomorphic mapping a' --t a'"" + a" of K' onto a subfield of K", and B,..,.. is generated by the differences arph-a"as in the above example, again with only a finite system a,' of generators of K'beingnecessary. Extend K" to K," by adjoining all phth roots.? The extended ideal of B,, K,, then becomes B,,1" K P = ( G K *x K;""" and the corresponding divisors are B"' = phG, GK, ,;' denoting the ideal generated by the differences a' - $@. GK, maps K' xK; onto K ; and is therefore of degree 1. The residue classes of elements of A , x K , in KK; mod form a K x K;-ideal V K x K ; let ; r, G K S x K ;

be a system of generators. The rl" then generate a K x K"-ideal CKxK". The correspondence C between K and K" associated with C K x, is called the product AB. Again (8) holds. Later we will use a property of the product in this case. Clearly C , x K,, K; E (VK x K,)"". The opposite inclusion holds, too. To prove it, let 3, be a prime 3 x Krideal. The '$-components (in the sense of 11,§2) are principal ideals:

(vZfx K;)O = r,(3

x K;),,

rO= ~ A , r , , A, E (3 x

K;),.

But then, (VKxK;)< = Tg(K x K&,

T< = CA$T$

so that

((CK x K**)K;)q for all $3, of course. Thus we also have (VKx,;)ph c CKxKtvK; ( VK

x

t For this extension, as for similar cases which follow, the exact constant field is generally also extended. But, this in no way impairs the arguments which follow. When two, or more such extensions are carried out, however, the resulting constant fields should be checked to see if they are still identical.

238

V.

CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS

and therefore CK K"K; = (VK y K Y ) ~ ' . Remembering the definitions of C and V, we can write this equation as (AB)~''= P ~ A G . (c) Let B be a separable prime correspondence with respect to K".By the lemma of III,w,4 there exists a finite separable extension K;/K" such that B"' decomposes to prime correspondences all of degree 1. It is no loss of generality to assume K;'/K" to be Galois. Then, if B'" = C B,, we may use Step (a) and set AB'" = AB,. Under the automorphisms of K ; / K the ideals (AB,), K;' permute just as the (B,), K ; . This leaves their product invariant, which, we maintain, is generated by a K x K"-ideal. For, if that product is decomposed to its prime factors over KK;, the hypothesis assures that, along with any factor, its conjugates over KK" occur. Now, with a suitable basis the discriminant of K x K ; with respect to K x K is an element of K" and thus a unit of K x K". Hence, by the Dedekind discriminant theorem there can be no ramified prime K x K;ideals, so that the product of all the conjugate prime ideals of KK; is actually the corresponding ideal of KK". But this implies the assertion. Hence AB"' = C'",where C is a correspondence between K and K", which we define to be the product of A and B. This definition of the product is independent of the choice of K;. For, if K," is a larger Galois extension the B, can nevertheless be decomposed no further, as their degrees g'(B,) = 1. (d) Now let B be an arbitrary prime correspondence. Say K," is the largest subfield in the residue class field of K' x K"mod B K t x K ,separable , with respect to K". Let K ; be the Galois closure of K,". Then the ideal BAYxKT decomposes to a product of purely inseparable prime ideals. The definition of the product AB is now derived by combining Steps (b) and (c). (e) Finally let B be the sum of arbitrary prime correspondences B , . Use Steps (b) through (d) to define

c

A ( C m,B,) = * (9) The conventions of Steps (b) through (d) assure the invariance of the product AB under an extension of K" even if the B, decompose. Exercise. Let K =k,(x), K'=ko(x'), K" = ko(x"). Further, set A K x K = , A(x, x')KxK' and BKTxK,,=B(x', x")K'x K", where neither of the polynomials A(x, x'), B(x', x") are divisible by a polynomial in a single variable. Then, (AB),.,,,=r(x, x")K x K", where T(x, x") is the resultant of the polynomials A(x, x') and B(x', x") with respect to the variable x'.

$1.

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3. PROPERTIES OF THE PRODUCT As it is in effect derived from the first two steps above, the product always satisfies the distributive laws (8) and (9). Let k , / k , be an extension of the constant field and A’, B’ the extended correspondences between K k l , K’k,, and K ” k , . Then the definition of the product shows A’B’ = ( A B ) ’ ; in other words, the product is invariant under an extension of the constant field.

Lemma. Let K,’ be a finite extension of K‘ with the same exact constant fields and A“, B“ be the correspondences between K and K,’ and between K,’ and K defined by the ideals A , KPKl’ and K,’B,. K,,,respectively. Then A”l3’’ = [K,’: K’IAB. Proof. Indeed we may extend K” in such a way to K;‘ that B”’ decomposes to prime factors of degree g’ = 1. Assuming the lemma to be proved for such factors, Eq. (9) shows that ~ ” ~ 1 ‘ 1 ’=’

[K,’: K‘IAB“’.

The corresponding K x K;’-ideals are obtained from (A“B’’), ,, and ( ( A B ) , K , , ) [ K 1 ‘ : K ‘ ] by multiplication with K;’.Evidently this process cannot make different ideals equal. The proof is thus reduced to this case. Now let g’(B) = 1. By (7), then, g’(B’’)= [ K , ’ : K ’ ] .To carry out the construction of A“B” we must extend K” suitably to K ; so that B’””

=

1B,

with B, again of degree g’(B,) = I . Then we see by ( 5 ) that the number of B, occurring is exactly [ K l ’ : K ’ ] . Any system of generators of the ideal A K x K ,also generates the ideal Residues of these generators mod Kl’B,r.,,.K;’ = ( B 1 ’ l “ ) K I . x K i ’ in K x K ; are the same as their residues in K x K” mod B,, , . But residues of these generators modulo the (By)Klrx,y are also the same. The assertion now follows from the fact that exactly [ K , ’ :K ’ ] such Bp occur. 7 For the degrees we have the equations K 1 3 .

g(AB) = g(A)g’(B),

g”(AB) = g’(&’‘(B).

(10)

To prove the first of Eqs. (10) extend K ‘ until A decomposes to prime correspondences A , of degree g(A,) = 1. This extension of K may be effected in

two steps. In the first, only the constant field is extended, so that both sides of (10’) remain unchanged. Now we consider the second step which leaves

240


the exact constant field invariant. Here g(A) remains unchanged, while g'(B) takes on a factor [K,':K'], because of (7), and the lemma shows that the product AB behaves like the latter. Because of ( 5 ) we need only consider the A , , which amounts to assuming g(A) = 1. Similarly, a suitable extension of K" decomposes B to prime correspondences B, of degree g'(B,) = 1, so that we may assume g'(B) = 1. Now, A defines an isomorphic mapping of K onto a subfield of K',and B an isomorphic mapping of K' onto a subfield of K". From the example at the end of Ql,l we see that the composed mapping corresponds to the product AB, so that we have g(AB) = 1. The second of Eqs. (10) results from the same argument, again setting g(A) = g'(B) = 1. The isomorphisms A: a + a' and B: a' + a" of K onto a subfield KO' E K' and of K' onto a subfield K," E K", of indices [K': K,'] = g'(A) and [K":K,"]=g"(B),lead to a composed isomorphism A B : a + a" of K onto a subfield K,", of K," of index [K,":K,",]= [ K ' : K , ' ] . But then g"(AB) = [K": K i o ] = [K":K,"][K,": K,",] = g'(A) g"(B). We can now prove the associative law (AB)C = A(BC).

(11)

By the lemma both sides of (11) are multiplied by the same factor under finite extension of K' and K". Extension of K" is necessary in any case to form the product. Now extend K', K", and K" so that A , B, C all decompose to first degree prime correspondences with respect to their first fields. By distributivity we need only prove (1 1) for the resulting prime correspondences; i.e., we may assume g(A) = g'(B) = g"(C) = 1. As A , B, C induce isomorphisms of K , K', K" onto subfields of K , K", K , the composed mapping is associative, and both products are generated by the differences a - a"'.

Symmetry law for the product. Choose some extension K 1 of K such that the extended correspondence A' between Kl and K' decomposes to a sum A' = m,A, of prime correspondences A , of degree g'(A,) = 1. Now let the product A,B be that correspondence whose ideal is the residue class mod A , of the ideal of B in K l K". Further let A B be the correspondence whose extension is A' B = m,A,B. The product thus defined coincides with that of our definition above. The roles of A and B have been interchanged in this new product formation. The fact that this new product is at least a correspondence between K and K" is seen as above, in the original definition.

c

c

Proof: Extend K and K" so that A and B both decompose to prime correspondences of degrees g' = 1. We need only carry out our proof for these prime correspondences. Thus, let g'(A) = g'(B) = 1. This means that isomorphisms of K onto subfields of K and K" are associated with A and B, respectively. But this implies that to every M E K x K' x K" there exists exactly one NBE K x BKlxR"and one N, E AKxK'x K" such that M - NB

$1. THE CORRESPONDENCES

24 1

and M - NA lie in K x K".We denote them NB = q B ( M ) and N, = q A ( M ) ; these are linear functions. For any A E K x K x K" both A - qB(A) and qB(A)- q A ( q B ( A ) ) lie in K x K", and then so does A - q A ( q B ( A ) ) = A - qB(A) +&A) qA(qB(A)). Thus, the last difference is a constant in KK'K" taken as a field of functions over KK". But, choosing A E A K x K 'x K", this difference also lies in the ideal A K x, x K", which is now possibleonly if A - q,(qAA)) = 0. Interchanging A and B leads to a similar equation. Our results, stated together, are: q A ( M ) ~ & x r x K, (PB(M)EKxBK,XR", qA(qB(A)) = A

for A E A K x K , x K ,

rpdqA(B))= B

for B E K x B R , x K , r .

The product (AB), , is generated by all the differences A - q,(A), by definition, and the ideal formed by interchanging roles of A and B is generated by all the differences B - qA(B). But, among the former, the differences q,(B) - q,(q,(B)) = qA(B) - B occur, that is the latter. The converse is shown similarly. This completes the proof. 7 4 . CORREsPoNDENCES OF A

FIELDWITH ITSELF

A particularly important case is that in which K and K' are isomorphic extensions of the constant field. We then speak of correspondences of K with itself. Multiplication can be definedfor the correspondences of K with itserf so that these, using the addition already defined, form a ring with unity. This ring also has an involutive antiautomorphism, called the antiautomorphism of Rosati.

Proof. To define the product of two correspondences A and B between K and K consider also a third isomorphic but algebraically independent field K",and the ideals A K x K " and B R M x R isomorphic t to A K x Kand t B K x K * , along with the associated correspondences. These can now be multiplied as above. The associative and distributive laws are consequences of (8), (9), and (1 1). The unit correspondence D, taken as a divisor, is the diagonal divisor ID introduced in 111, $ 5 3 . By interchanging the fields K and K' in the ideals A K X , we are lead to an isomorphic involution of the additive group of correspondences onto itself. It is denoted A c-* A*, and satisfies (A@* = B*A*

because of the symmetry law of multiplication. 7

(12)

242

V.


We can also immediately see that g(A*) = g ' M ,

-

(13)

g'V*) =g ( 4 .

The correspondences A 0 form a two-sided ideal in the ring of correspondences, and the residue classes with respect to i t , that is, the correspondence classes, also form a ring with unity and with an involutive antiautomorphism.

-

Proof. If A - 0 , B - 0 then clearly also A - B - 0. If A 0, that is, AKxK' = A K x K', and B is arbitrary, our definition of the product in §1,2 assures A B 0. And, as A 0 implies A* 0, we have

-

-

-

BA = (A*B*)*

- o* -

0.

It is the classes of correspondences, above all, which have important applications. For characteristic 0 there are generally no classes other than the integral multiples of the class generated by the unit correspondence D.t Nontrivial correspondence classes occur in special fields only; some examples will be considered in 553 and 4.

5. EFFECT OF CORRESPONDENCES ON DIVISORS With a correspondence A between K and K' we now associate a mapping a + aA = a' of divisors a of K onto divisors a' of K'. It turns out practical to

write the divisor group additively. The steps of the definition are virtually a repetition of 51,2. (a) If A is of degree g(A) = 1, then aA is that divisor of K' arrived at from a by taking residue classes in K mod A , K t . Clearly (a

+ b)A = aA + bA,

(14)

so the mapping is a homomorphism. (b) For a prime correspondence A, purely inseparable with respect to K' of exponent p", let K,' be the field extended by adjoining all phth roots. Then = ( B K x K , f )with p h g(B") = 1, and BKxK,, furnishes a mapping of K onto a subfield of K,'. Denote the image of a under this map by a,' and set a' = aA = pha,'. This is a divisor of K'. (c) If A is a prime correspondence separable with respect to K' and A'' = A, in a suitable Galois extension K,' of K', set a' = aA = aA, . This is again a divisor of K . As in §1,2 the image a' is independent of the choice of the extension Kl'. (d) For an arbitrary prime correspondence A choose a suitable Galois extension such that A'' = A, with purely inseparable prime correspondences A , .

t A. HURWITZ, ober algebraische Korrespondenzen und das allgemeine Korrespondenzprinzip, Math. Werke, Vol. I pp. 163-188. Basel, 1932.

$ 1.

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243

2

Then define a A , as in Step (b) and let a' = aA = aA, , again a divisor in K'. (e) For a completely arbitrary correspondence A define a' = aA using Steps (b) through (d) in connection with a(2 m,A,) = m,aA,. As all other cases were referred to the first, the mapping a + a' = aA is always a homomorphism of the divisor group of K into that of K'. We also have a(A + B) = aA aB. (15)

+

Furthermore, with another correspondence B between K' and a third field K" we can show that (aA)B = a(AB). (16) Proof. As in §1,3 in the proof of (1 1) no generality is lost in assuming that g(A) = g'(B) = 1. But then we are dealing with an isomorphic embedding of Kin K' and K' in K", which also implies a homomorphic mapping of divisor groups. Equation (16) simply states that AB is the correspondence associated with the composed mapping. Similarly to $1.3, the operations of extension of the constant field and formation of the product aA are permutable. T The most important case is again that in which K and K are isomorphic extensions of k,, a fixed isomorphism K -+ K being given. The image aA = a' can then be carried buck to K in the sense of thisjixed isomorphism, leading to a ring of homomorphic mapping of'the divisor group of K into itself; this ring is homomorphic with the ring of correspondences between K and itself. In particular, the image a D under the unit correspondence is a itself. Actually, it can even be shown that the ring of these mappings is isomorphic to the ring of correspondences.? This is why these mappings are often taken as the correspondences themselves. Our procedure, however, leads to certain simplifications. The lemma of §1,3 can now be carried over. Let K,/K be a finite extension with the same exact constant field and a, the image of a under the natural embedding of the divisor group of K into that of K,. Then we have a,A' = [ K , : K ] a A .

(17)

Proof. Let K" be a third field isomorphic to K and D the unit correspondence between K and K". Then a D = a" is the image of the divisor a under the given isomorphism. Further, let K ; be the extension of K" corresponding to K , , and also rewrite A as a correspondence between K" and K'. The lemma of $1,3 then yields

[K,:K ] a A = [ K , : KlaDA

= a(D1"A1") = a,A',

which was to be shown. T

t M. DEURING, Arithrnetische Theorie der Korrespondenzen algebraischer Funktionenkorper I, J. Reine Angew. Math. 177, 161-191 (1937).

244


Now that we have (17) we can carry the proof of (10) over (considering aA as a divisor of K’), arriving at g’(a4 = g ( 4 g ’ ( 4

(18)

Correspondences map principal divisors onto principal divisors so that, with (14), they are mappings oj’divisor classes.

Dually, equivalent correspondences map a divisor onto equivalent divisors. This only holds, though, for correspondences A and B equivalent in a stronger sense than that of our definition, in that the r which generates (A - B ) K x K * must generate ideals in the rings 3 x K‘ and K x 3’ not divisible by any prime ideals p x K’ or K x p‘, respectively. (p and p’ are any prime ideals in 3 and 3‘).Using equivalence in this sense we have the theorem: If the constant field is algebruically closed, then equivalent correspondences map divisors onto equivalent divisors. Thus the mappings of divisor classes depend only upon the class of the correspondence. Proof. Because of (15) it suffices to prove the first assertion for prime correspondences A. To form (u)A we take an extension K,‘/K‘ in which A” decomposes to prime divisors A, of degrees g(A,) = 1. Then (a)A = C(a)A,. The (a)A, = (u,’) are principal divisors in K1’, and their sum is the principal divisor (n a,’). Discussing the cases of a purely inseparable, a separable, and a mixed extension K,’/K’ as above, we see that ll a,’ is an element in K‘, remembering that the a,’ are algebraically conjugate to each other since the A, are conjugate correspondences. Some preparation is needed for the proof of the second assertion. Let p be a prime divisor of K ; as we now assume k, to be algebraically closed its degree is 1. Let 3 be the principal order of K with respect to k , [ x ] , x being some element whose denominator is not divisible by p. An 3-ideal p, is thus associated with p. Also, let A be a prime correspondence of degree g(A) = 1 represented by the ideal A, K ’ . The divisor pA = p’ in K‘ is found by taking the residues of p, mod in K ; they form an 3’4deal p j . in a subfield KO’ of K’. Then p’ is the prime divisor in KO’associated with pj,, where 3’is the isomorphic image of 3 projected into KO’by A. In general p’ is not a prime divisor in K ’ . Our discussion of the law of symmetry in the multiplication of divisors can now be carried over. Form the 3 x 3‘4deal p, x 3‘.To every M E 3 x 3‘ there exists a cp,(M)~p, x 3‘ and a ~ ~ , ( M ) E A , n ~ ~3, x 3‘ such that M - cp,(M) E 3‘ and M - cp,(M) E 3’.As above we have cp,(cp,(A)) = A for any A E n 3 x 3’and cp,(cp,(n)) = n for any K E p , x 3’.The ideal pj, consists of all differences n - cp,(n), for these are the residue classes of the R in K . Also, as before, it is seen that this ideal is generated by the

$1. THE CORRESPONDENCES

245

differences A - cp,(A), and these are exactly the residue classes of the A E A 3 x , , n 3 x 3' with respect to the module p3 x 3'. This supplies us with another mode of calculation of p A . We have shown that for prime correspondences A , the divisor p A is given by the 3'-ideal consisting of all residues of A 3 x K 'n 3 x 3'modp3 x 3'. The modified equivalence concept assures us, that this representation holds for all correspondences with which we must deal. As in 111,§6,we here have a reduction of a field KK'/K modulo the prime divisors of the constant field K. We need not exclude any p. The divisors of KK'IK which are already divisors of K'lk, could cause trouble, though, as they are associated with the unit correspondence. Therefore our new mode of calculation is applicable if and only if A K x K ,n K x 3' contains no such factors in its prime decomposition. Little remains to prove the second assertion. It suffices to treat the case of a prime divisor a = p. Let A be the principal correspondence generated by the element A satisfying our conditions. The computation thus permitted shows p A to be the principal divisor (a') where a' is the residue class of A mod p in K ' . The assertion is proved. 7

6. PRIME CORRESPONDENCES In what follows the fields K and K' need not be isomorphic. Let A be a prime correspondence between them. The residue class field of K x K' mod A , , is an extension R of K and R' of K' of degrees

[ R :K] = g'(A),

[ R :K'] = g(A),

respectively. In the second of these equations we have abused the symbols and set R = R', which actually cannot be as K and K' are algebraically independent. But, there really is a field K" E l? isomorphic to K' and of index g ( A ) in R, and occasionally writing K' for K" can lead to no errors. In this sense a prime correspondence between two fields establishes a certain algebraic dependence between them, and it pays to investigate it from the point of view of field theory. In this section we assume the extension RIK' to be separable. According to the last section we need a Galois extension L'IK' to form an, the associated extensisn A'-' of A decomposing to prime correspondences of the first degree: AL' = A , , g(A,) = 1. (We had previously written K,' for L'; the symbol K,' must now be reserved for another field.) The residue classes of K x L' modulo all the ideals (A,)K ,*, can be represented by elements of L'. 'Thus exactly one x,' EL' is associated with any x E K , the mapping x + x,' being an isomorphism of K onto a subfield K,' of L'. All the x,' can never coincide for different e, as this would lead to equal A , , which is impossible

246

V.


due to the assumed separability of K / K ' and the Dedekind discriminant theorem. The Galois group of L'IK' permutes the A , . The totalities of the xp' are permuted in the same manner, and must therefore be algebraic conjugates with respect to K'. Thus, any function symmetric in them must lie in K'. Now, in the last section we had set a' = aA = aA,. From our present vantage point we have the following interpretation. First make a a divisor in the extensionfield R = K x K'/AK K , of K in the natural manner and denote it ii. Then a' is the sum of the divisors conjugate to ii with respect to K'. This is exactly the norm: a' = n K & i ) . In conclusion consider the classical case, k , = C. The prime divisors of K now correspond one to one with the points of the Riemann surface % of K . Let % and %' be the Riemann surfaces of K and K'. As a covering space of % and of %', the number of sheets of % is g'(A) and g(A). The image p' = pA of a point p of 8 is found by first finding all the points p i of % lying over p, counting them with correct multiplicity if p is a ramification point. The g'(A) points pi thus found for p satisfy the equation p = pi. Each of these pi is then mapped onto the pi' of %' over which it lies. The prime divisors thus denoted satisfy pi' = nK,,.(pi). The image p' = p A is now the group of points p i . The following is also clear: The Rosati4onjugate correspondence A* maps a point p' of %' onto the totality of points p of % whose images under A include p'.

7. INSEPARABLE EXTENSIONS Let the field of constants k, be an arbitrary field of prime characteristic p. Let A be a prime correspondence between K and K' and use the convention of the last section which makes the residue class field K = K x K'/AKXK*to a finite extension of both K and K'. R is also a minimal field in this respect. In fact, any subfield o f R containing K and K' would contain the residues mod A K x K tof all finite sums xiyi' with xi E K, yi' E K', but these form the field K by definition. If k , is perfect then R is separable over either K or K', or over both. Proof. Use the considerations of 111,§4,2. As ko is perfect, formula (3) there goes into K = KP(x)with any p-variable x in K . This means that there can only be a single inseparable extension of K of degree p. that being K($), and only a single subfield of index p over which K is inseparable, that being KP. The same is true of all finite extensions of K. But now, if contrary to the assertion R were inseparable over both K and K', then Rp would contain both K and K', a property shared by no field smaller than R as we have seen before. 7

51.

THE CORRESPONDENCES

241

Assume now that K‘ has the property that the field K Pof its pth powers is an extension of the constast field k, isomorphic to K . This means that if K is generated as k,(x, y ) with a polynomial equation f ( x , y ) = 0 over ko , then there exist two elements x’, y’ of K‘ such that KIP = ko(xIP,fP) with f(x’P, y’”) = 0. This assumption is equivalent to: K ’ = ko(x’,y’) with f p - l (x’, y’) = 0, where f is found by extracting pth roots of the coefficients off. Remember that taking pth powers is an isomorphism of K’ + KIP. Hence the assumption implies the isomorphism of K and K‘. In general, however, K and K‘ are not isomorphic extensions of k,, this being true only if (in a suitable model) the coefficients offp-’ andfcoincide, that is, if they lie in the prime field. Under our assumptions a prime correspondence P between K and K‘ is given by P,

K,

= K x K ’ ( x - x ‘ ~JJ, - y’”).

The residue class field K x K‘/P, K ‘ is the field K ‘ ; we haveg(P) = 1 and, with 111, $4,(6),g’(P) = p. It is easy to see that this correspondence does not depend upon the model of K used. Now choose another field K ” isomorphic to K ‘ , but independent of K and K ‘ . We form the product with the Rosati adjoint P i t p x= KK“ x K(x’”’ - X , y”” - y ) and find that (P*P),.,

K’

= K“ x K ‘ ( X ” P G (K” x

- x r p , y”P - y’P)

K’(x”- x’, y” - Y ’ ) ) ~= (pD)KtrxK,

holds. Because of (10) the degree of the left side is g(P*P) = p , and g(pD) = p as well. We therefore have equality throughout, so that P*P = p D .

(19)

We also prove the opposite equation PP* = p D noting, though, that where before D was the unit correspondence of K’ it is now that of K. The ideal associated with P* in (20) is generated by the elements x’” - x”, y’” - y“. Following the second step in §1,2 we extend K” by adjoining all pth roots, the resulting field K ; containing, in particular, x;= p px y, , = p yp * might appear arbitrary, inasmuch as it also arises Our notation for P,. from PIC , by the isomorphism K“ E K without the antiautomorphism. Our product establishes an order between the fields, though, and the notation I’

248


expresses the intermediate position of K. It even becomes imperative when K and K' are isomorphicextensionsof ko . We leavethe verification of the relation Pix

Kt

= K x K'(x

- x"', y - fP')

for the product P2 = PP which can then be formed to the reader as an exercise. I f , for a prime correspondence A , the field extension RIK defined in the last section is inseparable, then A = BP with the correspondence P just defined, and some other prime correspondence B. IfKIK' is inseparable, then A = P*B with a prime correspondence B. Conversely, products of this sort always have the given inseparable extensions. Proof. Let K = ko(x,y ) and K = ko(x', y') withflx, y) = 0 andf'(x', y') = 0. As elements of R, x' and y' satisfy certain equations F(x, y ; x') =0, G(x, y ; y') = 0, and the functions F and G generate a K x K'-ideal BKxK'. This ideal contains A K x K , since x' and y' satisfy the above equations; on the other hand, BKxK#is a prime ideal because its residue ring is a field. So BKxKt = A K x K , . Now we assumed KlK to be inseparable. Therefore F and G are inseparable with respect to x' and y', and the ideal A K x K p is generated by functions of the form F(x, y ; xf Pylp). , These generate a K x K'"-ideal whose extension is exactly A g x K , . Choose a third independent field K" which is an extension of ko isomorphic to KIP. The correspondence P is then defined between K" and K', and the F(x, y ; x", y") generate a K x K"-ideal BK K,,. It is a prime ideal, and associated with a prime correspondence B between K and K". It can now be seen that A = BP. If K/K' is inseparable the above result yields the second assertion by the use of the Rosati antiautomorphism. The converse, in the form asserted, is obvious. 7 Let Kand K' be isomorphic extensions of the constantfield. Prime correspondences A = BP then generate a two-sided ideal in the correspondence ring. The same is true of prime correspondences A = P*B. Proof. We must show that, for another prime correspondence C, the prcduct BPC is again in the ideal. This is obvious for C of the form C = EP. For C not of this form, though, the last theorem shows the residue class field K' x K"ICK. to be separable over K". Then C can be decomposed to a sum of prime correspondences C, of degree g'(C,) = 1 in a separable extension KfIK". If we can show that for each v there exists a prime correspondence E, such that BPC, = E,P, our assertion is proved. Thus we need only consider the case where g'(C) = 1. By hypothesis and the previous theorem certain functions F(x, y ; x l P ,y t P )generate BP. Say C is associated with the isomorphism a' + a" of K' onto a subfield of K"; then BPC is generated by the functions F(x, y ; x"P, y"P). A final application of the same theorem completes the proof. 7

$1.

249

THE CORRESPONDENCES

All assertions and proofs of this section retain their validity i f p is replaced by any power q = ph. 8. THEFROBENIUS AUTOMORPHISM In addition to the assumptions of the last section we now also let K and K' be generated as K = k,(x, y ) and K' = ko(x', y') with the same equation, f(x, y ) =f(x', y') = 0, the coefficients off being elements of a finite subfield of k , with q = Iph elements.Thus K and K' are isomorphic.Also, let k, be perfect. Exponentiation with q is an automorphism of k,; denote it by the symbol K , so that a" = a'. As ko is perfect the inverse automorphism ic-l exists. By setting x" = x'-' = x, y" = y"-l = y we can extend K and K - ~to automorphisms of K , and similarly for K', for our hypothesis assures the invariance of the-coefficientsof the defining equationJ'= 0 under K, K - ~ . The same argument shows that K'' is generated as Kt4 = kO(x"J, y'') with f(x'', y'') = 0. By the last section then, a correspondence F can be defined by the ideal F K ~ K= ,K x K'(x - x", y - y"). (21) It is called the Frobenius correspondence of K with itself, is of degrees

g ( F ) = 1,

g'(F) = q = Ph,

and satisfies the equations

FF* = F*F = qD. An element a E K, expressed as a rational function p(x, y ) , is mapped onto a' = ( ~ ( x 'y'') ~ , E K" by F. The principal divisor (a) is also mapped onto (a'). In the sense of the isomorphism x -XI, y c)y' between K and K',the image in K of a' is a, = d x ' , Y 9 = (pK-l(x, Y))'. This equation is valid for principal divisors, and then applies for all divisors. We thus have the following. Denote the image of aF in K' (cf. §1,5), carried back into K by the above isomorphism K z K', by aF as well. Then

(23) To be able to compute aF* we must extend K' to K,' by adjoining all qth roots, according to our rule. The extended ideal is then aF = (aK-I)'.

FK

Kl'

= (K x Kl'(x -

q?,y - VJY'))" =Gi

x

K1' = (qG),

fl)

x

K1

I

An element a = e(x. y ) E K is mapped onto a, = e ( d z by the correspondence G thus defined. Hence the divisor (a) is mapped onto (a,)' = (e"(x', y')) by F = qG. Carried back into K this gives the principal divisor

250


(a"). Of course this result is valid for arbitrary divisors:

(24)

aF* = a".

9. CORRESPONDENCES OF A FIELDOF AUTOMORPHIC FUNCTIONS WITH ITSELF

We finally consider certain examples of correspondences that will occur in 43. Let a group 8 of fractional linear substitutions t -+ (at

+ b ) / ( ~+t d ) = M(T) = t'

(25) of a complex variable T be given. Consider the field K of all meromorphic functions f ( t ) satisfying the functional equation f(z') =f(r) for all substitutions (25) of 8, and assume that 6 is the greatest group under which K is invariant. In brief, let K be the field of automorphic functions of the group 8. The field of constants is C.Assume further that K is generated by two such functions x(T), y ( ~ ) between , which an algebraic equation f(y(z), x ( t ) ) = 0 holds identically in T. We will not discuss the nature of the group and the range of the variable T leading to the realization of this hypothesis here. It suffices to say that the modular functions represent an example, choosing 8 = T ( N ) , the congruence subgroup of the modular group of some level N >= 1. Now, say R is another linear substitution not necessarily belonging to 8, but such that the group @ = 8 n R - ' Q R has finite indices g'(R) and g(R) in 0 and 0'= R-'QR. Let

v= 1

V=I

be decompositions into right cosets. The notation of g'(R) and g(R) should be noted, although it appears inconsistent. The functionsf(R(t)) form a field K' of automorphic functions with respect and K is clearly isomorphic to K. Because of the nature of to the group 8', the M , , the functions f(RM,(z)) (v = 1, ... ,g'(R)) suffer the same permutations as the cosets E M v under substitutions in 8 for 7. The symmetric ) functions, being invariant under 8, must lie in K . As the ~ ( R ( T ) occur among thef(RM,(z)), every function of K' satisfies an equation of degree g'(R) in K. Adjoining x(R(z)) and ~ ( R ( T )to ) K gives a field R containing K' and with [R:K ] 5 g'(R). The degree of this extension is really [R:K ] = g'(R). For were it smaller, a subset of the f(RM,(?)) would be permuted into itself under 8.But 8 permutes the cosets E M v transitively. This would lead to a G E 8 and a pair of indices v, p such that for all functions f ( ~ ) f,( R M , G ( t ) ) = f(RM,,(?)) would hold, while MvGand M,, would lie in different cosets. Thus all ~ T Twould ) be invariant under the element RM,GM;'R-', not in 6, contrary to hypothesis.

01.

THE CORRESPONDENCES

25 1

Hence the situation of §l,6 applies. It only remains to be shown that the field R can be defined by residue class formation in K x K’ modulo a finite ideal A K w K ,To . this end choose a duplicate z’ for the variable z for the field K‘ and subject it to the substitutions of 0’.Further, choose two indeterminates u and u.

n

e‘(W

(ux(R7‘)

+ vy(Rt‘) - ux(RM,7) - uy(RM,z)

v= 1

is a polynomial in u, u. The coefficients A i ( ~T’), are invariant under the groups 0 and Q’, and therefore belong to the composite field KK‘, and even to the ring product K x K‘. Set = ux(Rt’) + uy(R7‘), then (27) becomes a polynomial A( (16) R 3 ,

also holds with the exception of the poles of the d’l-hmi’and a single exceptional p = p o , the latter only under assumption (ii). Represent H in the form

H=

i

with H i E K, Hi’ E K ‘ .

HiH,‘

(17)

We maintain : there exists a representation whose summands satisfy the inequalities Vp,(Q’-lHi’dx’1-h ) > = 0, v,(aH, dxh) 2 0. (18) To prove this, expand Hi‘ = civx’v, civ E ko ,

c V

in power series in the prime element x‘. Were not all of the first inequalities (18) true, i.e., if smaller powers of x’ occurred in these series than (18) permits, these would have to cancel out in the sum (17) because of (16). This would imply that certain H i differ but by a factor in k,, , so that H could be represented as a’sum of the sort (17) with fewer terms. By assuming that (17) can no longer be simplified in this sense, we have proved the first inequalities of (18). The same argument proves the other inequalities of (18). The second inequalities of (18) state that aHi dx” satisfies the conditions for a differential of the first kind at the place p. Then, by the proof of the theorem in $2,1 we also have Vp’(a’sK/K’(Hi(dx/dX’)h) dX’h)2 0.

262

V.


By applying (17) and the first inequalities of (18) we see that

so that H contributes nothing to the trace. Finally we must discuss the exceptional place po under assumption (ii). Now G dx'l-h dx" has a further pole of order 1 at a place po which can be chosen arbitrarily. We choose po in such a way that its images under A fall neither on the singular places of the d"-htvi, on the fixed places, nor on the prime factors p,,' of a'. There is (up to a constant multip1e)exactly one differential dul-h with the divisor (du'-h) = a. According to 111,§5,6 the residue of G dx'l-h dx" at the pole po is made up of the contributions of the fixed points to the trace and the sum of the residues of the differential

at the images phi ( i = 1, ..., g'(A)) of po in K'. Because the phi differ from the p,,', each place phi is the image of exactly one place For of R lying over po . We can apply Eq. (31) in 111,$4 and compute the sum of residues in K. Now this sum is equal to dv'l-h dpoi Ci res,, dv'-h poi with a prime element poi at qOl.But, the poi are all the places lying over po, so that the values ofdu'/du at these places are all the conjugates of du'/dv with respect to K. Thus we get

where the subscript i numbers the different conjugates of du'l-h/dx'-h with respect to K. But in this form it is seen to be

E

R

The corollary in 52,l states that dv'lPhA* is of first kind with respect to a w l , so a-l(du"-h A*) is an integral divisor. But by assumption (ii) a is contained in the (1 - h)th power of the canonical class. Therefore a-l(dv'l-h A*) is the unit divisor. This, in turn, identifies the differential dv"-hA* up to a constant multiple as du' - h , so that dufl-h A* = Sl-h(A*) dvl-h with a constant sl-h(A*). Thus we have a first degree representation A --t s,-,(A) of the correspondences by the differentials of degree 1 - h and of first kind with respect to a-'. In the case h = 1 it is reduced to s,(A) = g ( A ) or so(A*)= g(A*) = g'(A).

52.

CORRESPONDENCES IN THE SPACE OF DIFFERENTIALS

263

We can now compile our results as follows

The summation is over all fixed places of A , which are thus incidentally shown to be of finite number. As we stipulated, x‘ is a prime element Tcr p’ and a‘ an element such that vp.(a’) = vp,(a’), while x and a are the isomorphically corresponding elements of K .

5. EVALUATION OF THE TRACE FORMULA It becomes useful to remember 111,§4,(31) and write (19) as a sum of residues in R : in case (i) s,,(A) = T res, x -x aa ’ ( dx’ dx)hdx’)+r s1 -h(A*) in case (ii) (20)

1

(

We now define fixed points of A in K as those places j j of K whose prime divisors simultaneously divide p and p‘, where p and p’ correspond by the isomorphism. This convention permits the decomposition of a fixed point in K or K‘ into several fixed points in R. Let X be a prime element for a p, and let

with cog, cb, # 0. Our m = mp is the smallest exponent such that cAP# cmp. We introduce the distortion ratio (22) of A at a fixed point p. Choose 1 or xh-l for a, depending upon whether p divides a or not. We must compute the residues of y-P = co-/cb, P

dX --

x dX x ‘ x - ~- 1 ’

1 dx dX x dX x ’ x - ’ - 1

at a place p. Two points must be noted. (a) For h # 1 the assumption of bounded ramification includes regular ramification in the sense of 11, §6,2. The ramification indices e, and e,’ cannot be divisible by the characteristic. Thus for h # 1 we have 1 dx -x dX = 4 -

+ ...) 1 dx’ e,‘ -dX the dots indicating a power series in 2, starting with the first power. For X I

264


h = 1, though, irregular ramification can occur. But in that case these quotients, as well as ( x x ' - ' ) ~ - ' , contribute nothing to the residue. (b) The same reasons show that h must equal 1 in the cases 1,2 and 1,3 below. For the assumption of bounded variation would assure e, = e,' = 1 for h # 1.

Table I. Sh(A)= R, + s, -h(A*) or 0for a separable prime correspondence A # D, taken over allfixed points @ of A in K. We exclude the cases ee = eel, ys = 1, while e, = e,' = 1 must hold whenever p$ a and h # 1. We see that

..

p I a: p I a:

(2,2) ee > e,' (2,3) e, < ep'

Rg=O Re = -(e-6/er')h-'e-6or -e, ifh = 1

Additional coefficients of the power series (21)go into the residues of the excluded cases ee = eb', ye = 1. But then sh(A + A*) can be found; for, the terms still needed are the residues, at a place p, of 1 a' l a -dXh dX'1-h - -- dx" dx'-' x'-xu x' - x a' =-

d(x' X I -

-x) a'( dx)h-' x

dx'

a dx'

. .

For e-P = eel, ye = 1 this is

-(2h - l)(me

+ eV)

or

-(2h - l)m, - e,,

depending on whether p $ a or p I a.

TableII. Sh(A + A * ) = C R , + g ( A ) + s , - , ( A + A*)orO foraseparableprime correspondence A # D, taken over thejixedpoints of A in R. Whenever p # a, h # 1, then necessarily eV = ey' = 1. We see that

(1,2)

(1,3) (2,l) (2,2) (2,3)a

eV = e,', y, = 1 eii # erl el =ee ye # 1 e, = eP' yI = 1 e, # e-6'

pta: p $ a: p I a: p I a: p I a:

+

RV = -(2h- l)(ms ee) Rp = - min(ee ,es') Re = -es Rs = -(2h - 1)mp - ep Re = - min((ee/eg')h-' ee , (ep'/e-6)hlev')

aIn this case (ee/e5')*-1 must be replaced by 1 if h = 1.

52.

CORRESPONDENCES IN THE SPACE OF DIFFERENTIALS

265

Consider the case h = 1 in more detail. Let us say that a fixed point has the order for yp # 1 or es f eS’ for y, = 1 and e, = e,’.

min(e,, e,’)

fg =

[rn, + e,

(23)

By the trace formula

+ A*) = -Cfp + g ( A ) + g’(A), for g ( A ) and g‘(A) interchange when A is replaced by A*. The quantities appearing on the right of (24) were defined as natural numbers. Because of the left side, though, they actually represent elements of the prime field of the characteristic in question. It seems plausible, nevertheless, to consider them to be natural numbers in any case. Then (24) associates a natural number with the correspondence A. In 9this possibility wilI have important consequences. Exercise. Let K be the field of elliptic functions over ko = C, generated by the Weierstrass p-function p(u) and @’(u), and A the correspondence given by the “natural multiplication” u + nu = v , where n is a natural integer. A and A* are unramified correspondences. Set the divisor a = (1). The duh(h = 0, 1, +2, ...) are all differentials of the first kind; they lie in the hth power of the canonical class which is the principal class. So we are always in case (ii). The number of fixed points is (n - 1)2 both for A and A*. Prove s,,(A) = n2-h,s,(A*) = nh by using the definition of du A and du A* and by applying the trace formula (Table 1).

NOTFS The calculation of the trace essentially reproduces Kappus’s paper cited in 111, @,7. The present author had previously? determined the trace, limiting himself to correspondences of the sort in §1,9. It was incorrectly asserted, though, that the traces .$,(A) and $,(A*) were always equal. Also, the contribution of a fixed point of the type pla, eS = e,‘, ya = 1 wasincorrectly computed. For the classical case ( k , = C) §2,3 gives us S,(A*) = s,(A).

(25)

The left side of (24) then becomes $,(A*) + s,(A*). By §2,2 this is the trace of the topologically defined representation H(A). Thus, its trace is s(H(4) =

t M. EJCHLER, Eine (1957).

-Cf* + g ( A ) + g ’ ( 4

(26)

Verallgemeinerung der Abelschen Integrale, Math. 2. 67, 267-289

266


This formula was first proven by Hurwitz.? It is actually a special case of a formula of far greater generality, valid for finitely multivalued mappings of arbitrary manifolds onto themselves. Any textbook of topology gives further information concerning this formula, the Lefschetz j x e d point theorem. Also noteworthy is a theorem of Deuring’s:$ for a correspondence A - 0 (that is, generated by a principal ideal) we have dw A = 0 for a l l j r s t kind dzferenriuls dw. As Deuring points out, this theorem has aclose relationship to the Abel theorem. It no longer holds true for h # 1. Thus, a completely different grouping of correspondences into classes is found by calling them equivalent if they have the same effect upon first kind differentials dw” with h # 1. Exercise. Prove Deuring’s theorem as follows. Use Green’s function d 6 = dd’O6 and show that (du’, d 6 ) = du

for all first kind differentials. Thecorrespondence A applied to both sides yields (du’, d 6 A) = du A . For A generated by a principal ideal (A), it must now be shown that d 6 A = A-’ dA

(K’being held constant in differentiation). This implies (du‘, A-’ dA) = 0.

§3. Modular Functions 1. THEMODULAR CORRESPONDENCES The notation of IV,W will be used. According to the lemma of IV,§4,3, every residue class M =(; t) of determinant I modulo N contains a matrix (; i) E r. It can be uniquely determined by M as the coset T(N)(; i). By

it defines a correspondence A, of &(N) with itself. Moreover, A, is an automorphism of Kr,*r,/Kr, so that both degrees g(A,,) = g‘(A,) = 1. This method gives us all the elements of the Galois group of Kr,N,/Kr, which is then isomorphic to the factor group YJl(N) = r/r(N). For each residue class r prime to N we need a matrix

which we choose arbitrarily in the coset T ( N ) U , .

t LOC.cit., p. 242, Eq. (29). M. DEURING, Arithrnetische Theorie der Korrespondenzen algebraischer Funktionenkdrper II, J. Reine Angew. Math. 183, 25-36 (1940).

$3.

MODULAR FUNCTIONS

267

Throughout $3, n will be a natural number prime to the level N and R,

=

(; n").

(3)

By §1,9 a prime correspondence T, = A,"

(4)

is associated with the substitution R = R,; it is called the modular correspondence for n. In a wider sense, the notion of modular correspondence is extended to include all correspondences of the field with itself composed of the prime correspondences of the type given in $1,9. These then also include the automorphisms A,,, given above. An outstanding role in analytic number theory is played by the T,; the presentation of the foundation of their theory which we will give now is due to Hecke. Calculations with the T, must first be simplified. Several arguments using elementary number theory are needed. Set 6 = T(N) into $1,9. To represent T, as §1,(28) [or to generate the associated ideal as §1,(27)], the M , of the decomposition $1,(26) need not be explicitly known, the cosets T(N)R,M, sufficing. We now maintain

with a , 6, d taking on all values satisfying ad = n , a > 0, b = 0 mod N b traversing a complete residue system mod d .

(6)

Thus the number of cosets in ( 5 ) is the sum of divisors

u

Proof. We have T(N)(t)T,= T(N)R,M,(z) from $1, (28), the M , coming from T ( N ) = ( T ( N ) n R;'T(N)R,)M,. The B, = R,M, thus all satisfy the congruences B = R, mod N and IBJ = n. Furthermore, no V E T ( N ) can exist for p # v with R,M,, = VR,M,. For such an equation would imply R,M,,M;'R,-' E T ( N ) , contradicting the nature of the M , . Now, for a B = Ua-'(; t) on the right in ( 5 ) we have B = R,, mod N and IBI = n. The quotient of any different two of these B cannot lie in T ( N ) . For, Ua-'(; :)($ ::)-' UatE T ( N ) would imply (g :)($ ::)-' E r and then ad = a ' d = n, a > 0, a' > 0 would lead to a = a', d = a. U,T(N)U,-' = T ( N ) along with the restriction of b to 0 5 b < d would also give b =b'. Thus, the right side of ( 5 ) represents at least a part of the right side of $1,(28). On the other hand, let B =(a, f ) be given with IBI = n and B = R, mod N . Then find an (;: $) E r such that

u

268


that is, such that the product has the lower left coefficient c = 0. Let the g.c.d. (a, y) = a > 0. Set 6' = aa-' and y' = -ya-', then determine a", B" such that a"8' - 1"y' = 1. Now setting a' = a" + ty' and B' = 8" + rd' gives a solution of the equation dependent upon the parameter 1. Our assumption that a 3 1 and y = 0 mod N assures that 6' -= a-' and y' = 0 mod N.Choose t such that fi' = 0 mod N. Further, since a'd' - B'y' = 1 we have a' = a mod N. Thus

This means that we can write (;: that

$1)

=

U,($ $1) with ($

i;:) E T ( N ) , and

By now choosing a suitable integer u and multiplying from the left by U; '(i ;")U,,necessarily contained in T ( N ) ,we can have 6 match any given residue class system modulo d. This shows that the sums in §1,(28) can be placed in the form of (5), so that ( 5 ) is proved. 7 Similar to ( 5 ) we also have

with a, b, d taking on the same values as in (6). Proof.

By §1,(29) we see that

T(N)(r)T: =

u T(N)M,'K;

'(z),

V

where the M,' are such that R ~ T(N)R, = U(r(iv) n R; lr(N)R,)M,'. V

This implies that

r(N) = U(r(N) n R,r(N)R; l ) ~ n ~ f lv. f ~ ; V

As the same substitutions of the z-plane are given by Rn-' and R, = nR;' we may write r(N)(z)Tn* = Ur(N)R,RnMv'R; '(TI. V

93.

MODULAR FUNCTIONS

269

The same argument as above now shows that the cosets on the right are represented by a system of integral matrices B = R, mod N and with IBI = n, which cannot be transformed into each other by left multiplication with an element of T ( N ) . Such a system can be put in the form B = U,& :) with a, b, d again as in (6). 7 In particular, (8) gives us g'(Tn*)= g(Tn) = g(Tn*) = g'(Tn)*

(9)

2. PRODUCTS OF MODULAR CORRESPONDENCES The rules Au,Au, = Au,,

9

A",Tm = TmAu,

(10)

hold, as well as

so that the T, and the AUngenerate a commutative ring. Proof The first of Eqs. (10) is obvious; for the rest we remember the convention at the end of §1,9. For example, the sum tT(N)(z)A,'T,,,,,-z which will occur on the right in (1 1) is taken as the union of the T(N)B(z)associated with the individual summands, these being given the proper multiplicity. Leaving out the variable z simplifies the notation. The right side of the other equation of (10) becomes a b ad = m, etc. TmAun= T(N)U; (o d ) U,,

u

The matrices B , = U;'U,'(, a db ) U ,

again traverse a full system with the properties: B = R, mod N , lBll = n. such that no two differ only by left multiplication by an element in T ( N ) . Thus

which is exactly the assertion of the second equation (10). The same argument places the left side of (1 1) in the form

the a, ,..., and a2 ,..., traversing solutions of (6) for nl and n 2 . Ce r(N)U,; u,; = r(N)U,;f,, SO that

270


If (nl, n2) = 1, then the product on the right, ( a 1 b1)(a2 ' 2 ) = ( a b ) 0 d, 0 d, 0 d '

traverses a system of solutions of (6) for n = n l n 2 . Thus (11) is proved for this case. We have reduced (1 1) to the special case of powers of primes, which remains to be proved: rnin(r,s)

TFTp,=

C

S=O

paA;jaTF+.-2m.

First show, more special yet, that TrTp = TF+l pAi,'TP,-l.

+

(14)

According to (12) the product on the left is found by multiplying the matrices (al

0

bl) =

dl

("0

'I)

p"

with u + v = r, 1

b2

where bl traverses a residue class system mod p", and b2 a residue class system mod p. Thus the matrices

(where u > 0 has been assumed in the latter) occur, each of them once. Those of the first two kinds yield precisely T P r +The ' . matrices of the third type may be divided by p , this factor being without meaning in their application as substitutions in the t-plane. Note that r(N)u,' = u;lr(~)u;5, showing the contribution of that type matrix to be

The b, + p U - l b 2 each traverse the residue classes mod p" exactly p times. Thus (14) holds. It is now seen that the Tp' (r = 1, 2, ...) can be written as polynomials in Tp, meaning that they commute.

$3.

27 1

MODULAR FUNCTIONS

Equation (13) is finally demonstrated by induction on the number s. No generality is lost in assuming s 5 r. The correctness of (13) for some s implies, by multiplication with Tp and application of (14), that = TF(Tps+l +p A ~ ~ T p s - ~ ) Tp,TpsTp

=

C ( P ~ A ; ; ~ T ~ ++. p+a ~+ 1- ~~; ~p a + l ~ r + . - l - 2 u )

a=O

and then an easy calculation gives

the last summand being omitted in both equations if s = r. This, however, is the desired equation (14) with s replaced by s + 1. 7 3. REPRESENTATIONS OF MODULAR CORRESPONDENCES BY DIFFERENTIALS The applications of modular correspondences in analytic number theory depend upon their representation §2,(12) by first kind differentials. For the case of modularfunctions of level N > 1 the AM and T, are correspondences of bounded ramiJicationin the sense of §2,1, the exceptional set permitted there being the prime divisors 5, associated with the cusps. Proof. This is clear for the A,, as they are automorphisms. The T,, , and as a matter of fact all correspondences of type §1,9, map the neighborhood of a point t = to onto neighborhoods of a finite number of points by

+ b,

+ + +

avrO b, avdv- bvcv (T - 5 0 ) *** C , T ~ d, (c,zo dJ2 according to §1,(28). Then we know from IV, §4,2 that, under our hypothesis, N > 1, t - to uniformizes locally whenever to lies in the interior of the upper half-plane. This means that the mappings are locally single valued everywhere, ramification points being possible only at the cusps. The AM and the T,, map each cusp into (generally several) cusps. The requirements for regular ramification are automatically satisfied for characteristic 0. T a,t

7+-=

C,T

+ d,

+

+

The divisor a used in the definition of differentials of degree h of the first kind is now set a =(5,

5aN)h-1.

(15)

Then for N > 1, according to IV, $4,6, every differential of the first kind with respect to a can be represented as duh = U ( T ) drh with a cusp form ~ ( tof) dimension -2h, while every such cusp form leads to a first kind differential. For N = 1, though, we must replace our divisor by a = 5h-l

hz

hi

qr,cqr,i

9

h, = C2h/3l, h, = ChP1,

(16)

272

V.

CORRESPONDENCESBETWEEN FIELDS OF ALGEBRAIC ~JNCTIONS

to have cusp forms and differentials of first kind with respect to a coincide. Furthermore, the T. are no longer of bounded ramification, so that the trace formula of §2,3and 4 perhaps no longer holds for N = 1. But this case need not be excluded, as long as the trace formula is not needed. Now let a prime correspondence A R of the type §1,(28) be given. By 52,l it operates on a differential u ( t ) drh as

2)

where (:; = R M , . Note that now in the operator [ R M , ] - h we also have the determinant avd, - b,c,, which would seem to contradict IV, @I,( 14); but there that determinant is 1. As our sole interest is in modular forms, we may write this as

It is clear that this sum depends upon the cosets I‘(N)RM, alone. I f u ( t ) is an integral modular form, or even a cuspform, then so is u(t)AR. Our proof does not need application of the theorem in 52,1,which would not apply for N = 1. Instead we argue as follows. To any G E T ( N ) and index v there exists a G’ E r ( N ) and p, such that RMvG = G R M , , , and if G is held fixed p takes on all possible values along with v. Noting IV, $4,(15) we find that (U(T)AR)[G]-” = U ( T ) [ G ‘ ] - ~ ~ [ R M = , , U] -( T~) A~, . That this modular form is integral is obvious. Furthermore, A, maps cusps into (generally several) cusps. Thus cusp forms map into cusp forms. 7 The theorem just proved can naturally be carried over to composed correspondences A = 1 mpARp. __

4. THEPETERSON- METRIC ~~

Let two integral modular forms, U ( T ) and a cusp form. The integral

U(T),

be given, one of which, say

u(T), is

{u(T), u ( T ) }

= I I u ( r ) v ( r ) y z h - ’ dx d y

(T

=x

+ iy)

(18)

8

taken over a fundamental domain 5 of T ( N ) is called the Petersson scalar product of U(T) and u ( t ) . Its value is finite and independent of the choice of fundamental domain 5.

Proof. By the hypothesis and IV, @I,( 16) the power series expansions at the cusp ioo are

$3. m

U(T) =

273

MODULAR FUNCTIONS Q)

1cn exp[2ni~-’nt],

U(T)

n= 1

=

1c,’

exp[2ni~-’n~],

n =O

and it is obvious that (18) converges at this cusp. The other cusps are transforms of ico under T --* T‘ = (: :)T. We can now apply the easily verifiable identity

which shows that the integrand of (18) is invariant under the substitution T + T’. A suitable :) E r transforms this integral (18) in the neighborhood of such a cusp into an integral in the neighborhood of ico. Thus it converges here as well. Any arbitrary fundamental domain is arrived at from any given one by cutting off some part of it, 9, and supplementing with an equivalent part b’= (: :)9. The equality of the integrals over b and 9’is seen as above. 7 The scalar product (18) clearly defines a definite hermitian metric in the space of cusp form, called the Petersson metric. As an easy corollary we have: there exist bases ui(z)of the cusp forms such that

(z

Another formula generalizing §2,(1 1) is also due to Petersson; it states that {u(T)A, u ( T ) ) = { U ( T ) , v(T)A*},

(21)

where A = m p A R , is a correspondence composed of prime correspondences A R pof type §1,9, and A* is its Rosati conjugate. The proof need only be given for prime correspondences A = A , . We introduce the abbreviations Mv=

(z: ii),

a

=(c

b

d)’ u ( T ) [ R ] - ’ ~= u

bc)’ (-) + db (ad - d)” UT CT

-k

(CT -/-

= UR(T)

to write the left side as

By IV, 9443) then,

S=CMvS V

is a fundamental domain of 6 = r(N)n R-’I-(N)R, as the M, can be interpreted as in Ej1,(26). Because U ( T ) [ M ~ ] -=’ ~u(T), the integral above is equal to

274

V.

CORRESPONDENCESBETWEENFIELDS OF ALGEBRAIC FUNCTIONS

which in turn, because of (19), is equal to

The substitution Rt = t' or t = Bt' = (dt' - b)/(- ct' + a) is now carried out. Renewed application of (19) and insertion of (cz + d)(-ct' + a) =IRI

where we have set

The domain R 5 of integration now is a fundamental domain of the group R(T(N)n R-'r(N)R)R-' = T ( N ) n R-'T(N)R. By noting the relation to §1,(28) and (29), the last integral can be put in the form of the right side of (21) in the same way. 7 By using an orthonormal basis u i ( t ) of the module of cusp forms, (21) leads to the following result. If u i ( ~ ) A= C aijuj(Z), I

ui(t)A* =

C a$uj(t> j

are representations by matrices ( a i j )and (a;), then these are hermitian adjoints: Rosati-adjoint correspondences are represented by hermitian-adjoint matrices. Indeed, {ui(~)A,ULT))=

C1 aij{uj(T), u ~ T ) =) {~i(t),u,(t)A*I = C1 {~i(t), uj(T)}ac

so that

aij = By now applying the theory of hermitian and unitary matrices we have the following theorems. A representation of a ring of commuting Rosati-self-adjoint correspondences in the space of cuspforms can be simultaneously put in principal axis form. A representation of an abelian group of automorphisms of Kr(N, can be simultaneously put into principal axis form.

#3.

MODULAR FUNCTIONS

275

As for the last theorem, it suffices to know that any automorphism A taken as a correspondence of &(N) with itself satisfies the equation A-' = A*. The matrix representing A over a basis (20) is then unitary. 5. FOURIER EXPANSIONS OF MODULAR FORMS

A special abelian group of automorphisms of Kr(N) is formed by the A u n . In accordance with the last theorem let U ~ ( T be ) a basis of the module of cusp forms for which the Aun are represented by diagonal matrices,

u,(r)Ain' = xl(n)ul(+ The xl(n) depend upon the residue class of n mod N only, and are thus characters of the multiplicative group of residue classes modulo N prime to N . Such a basis will be said to be normed. More generally, any modular form U ( T ) will be called normed if it satisfies the equations

4+G: = x(M4 (22) with any character x(n) of this group. Rewording the above, the module of cusp forms is spanned by normed cusp forms. Particular normed integral modular forms of dimension -2h were found in the appendix to Chapter I, §2,3; these are the theta series for integral definite quadratic forms F(x) of level N in 4h variables, the character being the Legendre symbol x(n) = (I W)*

The second of Eqs. (10) testifies that the Tn map normed modular forms of character x(n) into each other. Their action on modular forms, in view of (17) and (9,is

where a, b, dare taken as in (6). From now on let m

u,(t) =

C1ci,,, exp[2aii~-'mt]

m=

be a basis of all normed cusp forms of character x(n) and dimension -2h. Then ui(r)Tn = t i j ( n ) u j ( r ) (24) i

is a representation of the Tn by matrices (tij(n)). More precisely, these matrices generate a ring depending on h and x , and this ring is a homomorphic image of the ring of correspondences of Kr(N) generated by the T.. The implications of (1 1) and (22) apply to the (tl,(n)):

276

V.


(tij(nl))oi,(nZN =

C

tln1,nz

t

X(0(tij(n1n2t-2)).

We must study the system of Fourier coefficients c ~ and , ~its behavior under the transformations. By (23) we have

First, sum over b, the result vanishing whenever m is not divisible by d. A simple calculation leads to OD

ui(7)Tn=

C m= 1

aln

exp[Zni~- ma^].

ci,,,-IX(a)n1-ba2h-'

(25)

After the left side has been replaced by the right side of (24), the coefficients of some subscript m prime to n are compared. Only the summands with a = 1 are needed, so that Ci.mn

= nh-'

CJ tij(n)Cj,m,

( m , n ) = 1.

(26)

This system of formulas was given the following surprising formulation by Hecke. With the Fourier series of the normed cusp forms u1(7)associate formally? the Dirichlet series m

CLs) =

C ci,mm-', m= 1

CdS)

=

C Ci,mm-',

mlN...

(27)

where, in the latter, only numbers composed of prime factors of N, the level, are taken for m. Consider also the formal injinite matrix product Z(s) = ( E - (tu(p))ph--'-S + E X ( p ) p Z h - ' - Z y (28)

fl

PXN

taken over all prime numbers p not dividing N;E is the unit matrix. Then,

(29) where (ti@))and (ci0(s)) are column matrices with the components indicated. The product (28) is independent of the order of the factors. For level N = 1 the Cio(s) consist of the numbers ci,l only, so that (29) reduces the Ci(s) and thus the series expansion of the ui(t) entirely to knowledge of the representation (ti,(n)) of T.. (C,(s))

= Z(S)(CiO(S)),

Proof. The commutability of the factors of (28) follows from the commutability of the Tnand thus of the matrices (ti,(n)). The individual factors are the infinite series

t It is not difficult to arrive at these t ( s ) from the u0 (23) for all % not in the extended principal class. K and K' need not be assumed isomorphic. The excepted case clearly has s(%%*) = 0. The assumption that the field be conservative assures the coincidence of differential and canonical classes. Clearly (23) is equivalent to s(AA*)>O if A .1.0 (24) for the nonconstant component A of % in the sense of (4). Correspondences in the extended sense were first introduced here to define the number of fixed points, and will also play a part in this proof. We first take care of an easily demonstrated special case in a lemma: for a nonconstant prime correspondence % of degree g'(%) = 1, we have = 2g'g(%),

s(%%*)

with the genus g' of K'.

Proof. From (4) we see that 2l coincides with its nonconstant component A. By §1,1 A is determined by an isomorphic mapping x' -+ xo of K' onto a subfield KO G K, and g(A) = [K : KO].The ideal AK K , is generated by the differences xo - x', a system of generators of K' being taken for x'. Thus K,, is generated by the differences xo - xg , so that AA* is the ideal an extension of the unit correspondence Dobetween KO and K," to a correspondence between K and K". For the unit correspondence we saw in (7) that =2

+ g(b0) = 2g',

(26) with a divisor bo of the canonical class of KO.Now choose a correspondence Bo between KO and K,", strictly equivalent but prime to Do.Extensions of Bo and Doto correspondences between K and K" are also equivalent. We saw that the extension of Do is AA*, and will denote the extension of B, by B. The definition of divisors of fixed points shows that S(D0)

f(W

= fo(B0).

$4.

293

CASTELNUOVO'S INEQUALITY

Thus the degrees g(B,), g"(Bo),f(Bo)are all multiplied by [ K : KO]= g ( A ) when B, is extended to B. The same is then true of s(B,). As B is equivalent to AA*, (25) is a consequence of (26). 7 The principal theorem itself is easily derived from this lemma for the case of an elliptic function field. The algebraic closure of the field k, of constants is first taken; all the degrees and therefore the trace remain unchanged. There exist constant divisors of any given degree. Thus any given class can be represented by a correspondence 8 of degree g'(23) = 1. Apply the RiemannRoch theorem to KK'IK; there exists an integral divisor 2l equivalent to 8, and of degree g'(2l)= 1. So long as the extended class is not principal, 2l is a nonconstant prime correspondence. But the trace is a function of the extended class, so that (25) implies (24). Thus further consideration is necessary for fields of genus g > 1 only. 7. SECOND PROOFOF THE PRINCIPAL THEOREM : PREPARATIONS

Extension of the field k , of constants leaves the divisor degrees, and therefore the trace, invariant, because K and K' were assumed conservative and separably generated (IIT, §3,4). It is thus no loss of generality to assume ko to be algebraically closed. By a nonspecial divisor system we mean g distinct prime divisors pi of K such that dim( W(p, p,)-') = 0, where g is the genus of Kand Wits canonical class. Such a system can be constructed as follows. Let ui be a basis of the module of integral divisors in W. Choose some prime divisor p , by which not all ui are divisible, and determine constants ei E KO such that ui - ei is divisible by p1 for all i, where v , = u p ; ' . Then, oi' = (ui - Qiu,)p;' (i = 2, ...) are linearly independent integral divisors of the class Wp; No further linearly independent integral divisors can exist in this class, for, with such an a, the ui'pl and ap, would be linearly independent integral divisors of W, so that every integral divisor in W could be expressed as UD, = cap, + c2'02'p1 + . This would contradict our choice of pl. Thus dim(Wp;') = dim( W ) - 1 = g - 1. Continuation of this process finally leads to the system desired. The Riemann-Roch theorem shows that any nonspecial divisor system has djmension dim(p, p,) = 1. Lemma. Let pl, ..., pm (m _I g) be contained in a nonspecial divisor system. There then exist m functions wi E K with common denominator Ij of degree

'.

s(b)_I s + m - 2, (27) prime to an arbitrarily given integral divisor, and satisfying the congruences

294

V.


For each i s m find an integral divisor mi of W divisible by p, =r,. It will be of the form mi =rigi with an integral g, in the class Wti-'. The Riemann-Roch theorem shows that dim(Wr;') = dim(ri) 6 dim(p, pe), which is 1 by hypothesis. But, as trivially dim(r,) >= 1, necessarily dim(Wr;') = 1, so that up to constant multiples only a single integral divisor g, can lie in the class of Wr;'. As dim(Wr;'p;') = 0, then that 9, cannot be divisible by p i . Now set

Proof.

-..

with constants ej # 0 to be determined. Multiplication of the wi by suitable constants leads to (28). A common divisor cancels out of the m i , and contains at least P,,,+~ pe. Then suitable choice of the constants ei assures that the common denominator is prime to any divisor given. l~itself divides the divisor Z.eimi(p,,,+, pe)-', whose degree is on the right in (27). 7 We have tacitly reverted to the multiplicative notation for the divisor group, but now return to the additive notation introduced in this chapter. Choice of a suitable class representative. In §4,3 we showed multiplication of correspondences to be an operation between classes in the extended sense and, in the last section, the trace to be a function of these classes. 2l may be replaced by a divisor equivalent in the extended sense, in the proof of (24). For 91 not in the principal class,

can also be proved, which is more than (23). An extension K,/K multiplies both sides of this equation by [Kl : K] [cf. (17)], so that we may extend K finitely whenever convenient in the proof. Consider % as a divisor of KK'/K, and let 6' be some divisor of K' such that g'(2l+ 6') = g', the genus of K'. The Riemann-Roch theorem then states that in the same class there exists an integral divisor 8. This 23 = 2I + b' + c + (r)with a constant divisor c yet to be determined, and g ' ( 8 ) = 9'. A K x K'-ideal is associated with 8,that is, an integral nonconstant component in the sense of (4). Determine c so that 23 has no constant component in K. Then 23 is integral taken as a divisor of KK'/ko. Replace 2l by 8, that is, assume 2l to be integral and g'(2I) = g'. Now choose an extension Kl/K for which % = + + (pe, with g'((pi) = 1, but still write K. Also, choose some arbitrary divisor p of K, but hold it fixed, and form

v,

p2l = a' = q,'

+ .-.+ qi,,

q,' = p p i ,

& CASTELNUOVO’S I. INEQUALITY

295

setting p’pi = ‘pi = qi’ for any Fpi constant in K’. By $1,(18) and because g’(’pi) = 1 the qi’ are prime divisors of K’, so that g’(a’) = g’. After this preparation, we take some nonspecial divisor system pi’ of K’ and form an integral divisor 23 of KK’/Kequivalent to 2l - a’ pl’ ... +pi,; that is, 23 = % - a’ + p l ’ ... + p i . c +(l-).

+

+

+

+

As before, c is determined as a constant divisor in K making 23 integral. We are interested in the product p23 = p1’

+ + pi. + (y’),

y’ # 0 E K’,

*.*

which we compute by the second rule of #1,5. Residue class formation mod p reduces the function field KK’/K modulo the prime divisor p of the constant field, a summand like c having no effect. (Incidentally, this coincides with the definition (6) of the product of generalized correspondences.) As 23 is integral, so is p b , which is equivalent to pl’ ... . But, the pl’ are a nonspecial system of divisors, so that dim(p,‘ + = 1. This implies that p23 = pl’ + ... + p i . (in other words, the principal divisor (7‘) above is the unit divisor). Again write 91 in place of 23 and extend K again, so that BI still decomposes to prime divisors ‘!Qi of degree g’(Cpi) = 1. Suitable indexing gives p’!Qi = pi’. Some constant prime divisors may occur among the ‘pi. They coincide with the pi’ and can be dropped, as they contribute nothing to s(%BC*).The remaining nonconstant divisors will be denoted by roman letters. The configuration at hand has finally been reduced to the following. Every class in the extended sense contains a representative of the form

+

..a)

A

+ + P,,

=P ,

g’(Pi) = 1,

m 5 9’.

We have p’p, = pi‘ with some prime divisor p of K, the pi’ being contained in a nonspecial divisor system of K’.Clearly no two ‘pi can coincide, as this is excluded for the pi‘. In case m > 0, we will now prove (29). 8.

SECOND PROOF OF THE PRINCIPAL

THEOREM: CONCLUSION

Using the class representatives determined we have S(AA*) =

c S(PiPi*) + 2 1 S(PiPj*). i

i
0. We group the a, over which (3) sums, into classes A, and these into degrees. We separate the A of degree g ( A ) 5 2g - 2 from the rest, for which dim(A) = n - g 1, by the Riemann-Roch theorem. Such a class thus contains (q"-'+' - l)(q - l)-' integral divisors, and there are h classes per degree. Thus the second sum is

+

h =-

q-1

h

q1-g+(2e-2+r)(1-S)

--

1 - g(1-s)

C

4- 1 n > Z g - 2

q-.

In these sums n runs through all integers >2g - 2 which occur as degrees of integral divisors or, as we have seen, over all multiples of r which are > 29 - 2. This equation assures the finiteness of the class number. The first partial sum is

where n runs through all multiples of r between 0 and 2g - 2.We combine the last sums of cl(s)and CZ(s) and see that 1

c ( ~)

2g-2

1 C

q - 1n=O

g(A)=n

qdim(A)-ns

+"(q

i-g+(zg-z+r)(i-s)

q -1

1 - q'(l-s)

+L) 1 - q'" '

which again has the form of two sums c3(s) and C4(s). Obviously [4(s)q(8-')s remains invariant when s and 1 - s are interchanged. We see the same for [3(s)q(B-')s:

Write A' = WA-' for the canonical conjugate class of A, whose degree g(A') = n' = 2g - 2 - n must also lie between 0 and 2g - 2. Its dimension is related to that of A by dim(A) = n - g 1 dim(". Hence

+ +

Because A' traverse all these classes along with A the asserted invariance holds.

$5, APPLICATIONS IN NUMBER THEORY

303

We finally also see that

p-

1)s

[(s) =

p - 1"

--I)

5(1 - s),

(10)

where the case g = 0 need not be excluded due to (9). This relationship leads further. There exists a polynomial L(q-") = L(u) of degree 2g such that if r = 1 (see next section), then

L(u) has rational integers as coefficients, and L(u) = 1

+ ( N 1 - q + l)u + + ***

q8u28,

(12)

where N , is the number of prime divisors of first degree of K. We also have the functional equation q u s L ( q - s ) = q e ( l -"'L(q"- 1). (1 3)

Proofs. Equation (13) is a simple consequence of (10) and (11). If r = 1, the above computations show that L(u) is a polynomial L(u) = Zu,u" with rational integers a,. Because of (13),

C a,u" = C

~,,qg-~u'g-~.

Thus the highest power possible is uzg.As clearly a, = 1, (13) yields aZg= $. The coefficient of u is found by comparing (3) and (1 1). T As long as we do not yet know that r = 1 we can only conclude, however, that, instead of (1 l),

with a polynomial L(q-"). [(s) has first-order poles at s = 1 and s = 0. The residue at s = 0 coincides with that of the partial sum C2(s), and is thus -1 L(1) ---

log qr 1 - qr

1 1 - q'log q"

__--

h

so that

h = L(1).

(14)

3. EXTENSION OF THE FIELD OF CONSTANTS

If k , is replaced by the unique extension ko, of degree n, a new field

K, = Kk,, of functions is found. A simple relation holds between its zeta function ~Js) and [(s). Let some prime divisor p of K be given. The residue

304


class field modulo p of the integral domain 3, of functions of K integral at p is an extension of ko of degree g(p). Choose a 9 E 3+, whose residue class generates this field. There then exists an irreducible polynomialf(9) of degree g(p) over k, such that j ( 3 ) = 0 mod p. Now, f(9) is separable because ko is perfect, so that suitable choice of 9 within its residue class modulo p assures that the numerator off(9) is divisible by p only once. In the extension of ko to konwe know, from the theory of Galois fields, that f(9) decomposes to t = g.c.d.(n, g(p)) distinct irreducible factorsf@). This extension of constants therefore also decomposes p to

P =PI

**.

f

Pt,

&I).

= g.c.d.(n,

The irreducibility offr(9) makes the pI prime divisors. Divisor degrees remain invariant under extensions of constants, so that = 1,

A P T ) = g(P)t-',

***,

t,

There are Q elements in kon,so that q must be replaced by q" to form the zeta function of K,, . The individual prime divisors of K,, arise from those of K as described. Thus r.(s) is found by replacing each factor of the Euler product (4) by (1 - q - e ( + , ) n r - l s ) - r

n- 1

= fl(l

- q-e(p)(s-Zniv/nloe4))-1.

v=o

This means that the zeta function of K, is

Writing this equation for the function (6) we have n- 1

n

z,(u") = v = oZ(U eZniv/" 1.

(16)

Applying (15) with n = r to (lla) we find

which exhibits poles of order r ; so r = 1 because of @,2. 7 Certainly the denominator of (1 1) satisfies the functional equation (15). This means that (15) also holds for the numerator L(q-") and (16) for the polynomial L(u). Decomposing it to 2#

L(u) = n ( 1 - 0 , u ) v=l

(17)

$5.

APPLICATIONS IN NUMBER THEORY

305

we have, for the corresponding polynomial of K,, ,

n 29

L,(u) =

(1 - o v n U ) .

v= 1

(18)

In particular this yields the equation 28

W,n v= 1

= 9"

+ 1 - N,,,

where N,, is the number of first degree prime divisors of K,,. This equation takes on importance later. In the decomposition (17) we should note that the o,,which are the reciprocals of the zeros of L(u), can be indexed so as to make v = 1,

o,wv+,= 4,

..., 9 ,

(20)

valid, because of the functional equation (13). 4. RIEMANN'S CONJECTURE

Riemann's conjecture is literally identical to the conjecture for the Riemann zeta function. It states that the real component of any zero of the function (3) is equal to 4. Here no "trivial zeros" occur, as for the Riemann zeta function. Combined with (1 1) and the decomposition (17) of the polynomial L(u) this conjecture requires that all the w , have the absolute value Iwvl

=

J4.

(21)

To prove this we make use of the Frobenius correspondence F between K and an independent isomorphic field K', as considered in $I$. First, the trace of F must be found. The automorphism of ko denoted K in §1,8 is the identity under our present hypotheses, and F maps any a E K onto a'" E K'. As a traverses an integral domain 3 of K, like that of $4,4, the 3-ideal found as a representation of f ( F ) contains all the differences a - a". (Of course, this ideal only represents those local components f,(F) of the divisors f ( F ) for which 3 c 3, .) Let p be a first degree prime ideal not contained in the denominator of any function of 3 and a E ko be the residue class of a mod p. Then a - a" = a c1 - ( a - u)" is divisible by p. Choose a divisible by p to exactly the first power; the same is then true of a - a", and then p also goes into f(F)exactly once. Suitable choice of 3 then assures that all fixed points are included. Now let p be a prime divisor of higher degree that does not occur in the denominator of a function of 3. Choose some a E 3 not congruent to any a E k , modulo p. Then a - a" f 0 mod p, so that p does not divide f(F).

306

V.

CORRESPONDENCESBETWEEN FIELDS OF ALGEBRAIC FUNCTIONS

We have shown that f(F) is the product of prime divisors of the first degree of K,and therefore f(F) = Nl. Thus s(F) = s(F*) = 4 + 1 - N1. (22) By Castelnuovo's inequality we know that s((mD + F)(mD + F)*) 2 0 for all rational integers ni. Further, $1422) states that FF* = qD and Eqs. (2) and (7) in §4 state that s(D) = 2g. Altogether 2g(m2

+ q ) + 2m(q + 1 - N,) 1 0

must always hold. But this can only happen if 14

+ 1 - N11 I 2 g J q .

The same argument applied to the extension K,,yields (4"

+ 1 - N,,I I 2 g J 4 "

which, because of (19), means that

We can now extract the result. Use (17) to form m

log L(u) = -

Cn n= 1

This series converges absolutely for 1u1 < q - l l 2 because of (24). This means that all the w , 5 (q)'/'. But, (20) shows that qw;' occur among the w , , so that (21) must hold. 7 Another result is also contained in our considerations and it serves to place Riemann's conjecture into the right perspective : The class of the Frobenius correspondence F satisfies the equation L'(F) = 0, where L'(u) = uZBL(u-').(Set Fo = D.) The proof requires Castelnuovo's inequality in the strong form of the principal theorem of $4. If F is taken as a correspondence of the extended field K,, , then its nth power F" = F,, is the Frobenius correspondence of K,,. By replacing F with F, in (22) we find that s(F") = q" + 1 - N , . Because of (22) we thus have 28

s(F") =

for n

=

C w," v= 1

1,... . This equation is trivial for n = 0.

$5,


307

The F*" also have the same traces as the F",so that §1,(22) implies that

s(F-") = s(q-"F*") = (4"+ 1 - N")q-" =

c 0,"4-". 28

v=

1

The fact that the 40;' also take on the values of all 0,[which is a consequence of the functional equation (13) and of (17)] shows that (25) holds for negative n as well. We know that L'(0,) = 0, so that (17) and (25) imply that

s(L'(F)F") =

c JJ V

(0, - w,>w;

=0

P

for all n. Renewed application of §1,(22) gives

S(L'(F)L'(F)*)= s(L'(F)L'(qF-'))= 0 so that L'(F) is the zero class by the principal theorem of §4. 7 The inequality (23), which is essentially equivalent to the Riemann conjecture, can be interpreted elementarily. Generate the field in question as K = k,(x, y ) with f ( x , y ) = 0. Prime divisors of the first degrees not contained in the denominators of x and y map x and y onto elements ( and q of k , , and these satisfy the equation f((,q ) = 0. Conversely, let (, q be a pair with this property. Then x + ( , y + q defines a homomorphism of the integral domain generated by x and y over k,, whose kernel is a prime ideal corresponding to a prime divisor of the first degree. This shows that N , is essentially the number of solutions of f ( x , y ) = 0 in k, , and only "infinitely distant" solutions would have to be defined and adjoined. Then (23) states that this number of solutions differs from q + 1 by at most 2gq'". For the congruence f ( x , y ) = xzy2 xz y2 - 1 = 0 mod p this was conjectured by Gauss.?

+ +

5. MODULAR FUNCTIONS The generalization of Ramanujan's conjecture mentioned in §3,6 can be derived from the Riemann conjecture and thus proved by the last section, insofar as it applies to modular forms u(z) of dimension -2, that is more correctly to modular forms corresponding to first kind differentials u(z) dz. Certain preparation is necessary. First we remark on the model of the field Kr") of modular forms of level N. Kr(N) is a Galois extension of the field K = C(J(7)) of modular functions of level 1, and its group is defined in IV,§4,(10) as !UI(N). It was shown there

t G. HERGLOTZ first proved this conjecture in: Zur letzten Eintragung im Cuupschen Tugebuch, Ber. Verh. SLhs. Akad. Wiss. Leipzig Math.-Phys. KI. 73,271-276 (1921).

308

V.


that III1(N) is faithfully represented by the permutations suffered by the classes T(N)al/a2 of cusps. The following section showed the same permutations to act on the Eisenstein series, Gk(r;al, a 2 ,N). Thus the second degree differentials G4(r; a,, a2 ,N)dz2 of when transformed by substitutions of r, undergo a group of permutations which faithfully represents the Galois group. Therefore the quotients W,(T)

N4 G4(r; Q1, a29 N) 360 G ~ ( z ;0, 0, 1)

=-

generate the field Kr(N)over Kr . From IV,&l,(23) and (24) we see that the w,(z) can be expanded into Fourier series Cc,, exp(2ninN-'z) whose coefficients c,, are numbers in the Nth cyclotomic field QN = Q(exp[2niN-']).t Such a Fourier series, with coefficients even in Q, exists for the modular function j ( r ) which generates Kr . Thus &(N) is generated by functions having Fourier series expansions with coefficients in QN. These functions generate a subfield with constant field QN which shall again be denoted Kr(N)in the sequel and be the object of our study. Any function q ( z ) with Fourier coefficients in QN and invariant under T ( N ) belongs to this field Kr(N).For, let q ( ~=fg-' ) with polynomialsf, g in the w,(r). Let the coefficients off and g belong to the field L over QN, let w , be a basis of L/QN and f = Co,f,,g = Co,g, , where f,,g, are polynomials in the w,(z) with coefficients in QN . Substitute the Fourier series for the w,,(z) into

CPb)

c

OVQ"

=C o v f v

and then compare coefficients, to find that Wgv

=fv

which proves the assertion. Now let x = j ( z ) , y , be a system of generators of Kr(N,; a system of polynomial equations f J x , y,) = 0 must hold. By comparing coefficients after substitution of the Fourier expansions a system of linear equations with coefficients in QN is found, and the coefficientsf, can be computed from these.

t Clearly. we must also show that the constant terms of the expansion, (2ni)-4 X' m-4, 9ElZt(N) belong to the field QN , But this follows from the Fourier expansion

after setting (hi)-4

X

m sa t ( N )

1 N-1 m-4=-

24N

c

(;)('

-

1--

exp [-2niaavN-1]

(as f: 0 mod N).

55. APPLICATIONS IN NUMBER THEORY

309

These coefficients may thus be assumed to be elements of QN . The equations fp(x,y,) = 0 define a finite extension of Q,(x) which becomes the field of all modular functions of T ( N ) under the extension of constants Q + C. The exact constant field of Kq,) is Q , . Second we note that the elements of r permute the functions wa(r), according to IV, §4,4. They thus define automorphisms of Kr"). The form §3,(5) of the modular correspondences T(n) shows that they are correspondences of Kr(,) with itself (or, in other words, that they are defined over Q,). For the have Fourier symmetric functions of cp(U; '(t f;)(r)) for any q ( r ) E coefficients in Q, and are invariant under T(N). For a number r prime to N , IV, §4,(23) and the footnote on page 308 show

to have Fourier coefficients derived from those of w,(T) by the automorphism a,; exp[2niN-'] + exp[2nirN-'] of Q , . Now let 1be an element of Q , such that the ,l'r (r varying mod N ) form what is cal!ed a normal basis of Q,/Q. By setting with Fourier series cp,(z)

having rational coefficients, wra(r)

=

C

lugur~a,s(~).

S

As the determinant IAuSurl # 0, the cp,Jr) can be expressed linearly in the wra(r), making them modular functions of level N. Thus Kr(,, can already be generated by modular functions with rational Fourier coefficients. The previous argument shows that these form a function field I@(,, = Q(x, yl, ...) with generating equations f l ( x ,yl) = = 0 having coefficients in Q . Adjoining the Nth roots of unity we get Kr(,,:

-

Kr(N) = KF(N)QN We consider this field as an extension of Q(j(r)).The elements of the Galois group are generated by two types of automorphisms: the automorphisms cr, of Q,/Q, operating on the Fourier coefficients of the functions, and the substitutions of r on r. The effect of the former on the functions w,(r) is seen by comparison of (26) and (27): w,(ry. = wr,(r).

From this equation we deduce that the automorphisms Usdefined by $3,(2) commute with the 0 , . Therefore the Usare automorphisms of thefield Kf(,). This latter fact is quite important in what follows. The first consequence we derive from it is: The T,,, Tn*,as defined by §3,(5) and (8), are correspondences of the jield Kf(,). For the proof we have to form the ideal §1,(27)

310

V.

CORRESPONDENCES

BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS

and to show that its Fourier expansions with respect to z have rational coefficients. In the case of T p ,9 a prime, we have (writing z' instead of z'/p does not matter in this connection)

where x ( t ) and y ( t ) generate the field KP,,, . Because Upis an automorphism, x(U;'(z)) and y(U;'(z)) have rational Fourier expansions in z, as does the rest of our product. Because the Tnare products of the Tp they are also correspondences of KP,,,,and the Tn*are derived from them by the Rosati antiautomorphism. Incidentally, it is easy to show (cf. G. Shimura, §3,7,[9]) that is normal over Q(j(z)) and that the Galois group is isomorphic to GL(2,Z/N)/{&(: y)}. The subgroup belonging to is that generated by a, = ('0 y). Third we reduce the I$,,, modulo a regular prime number p o to a field RP,,, of functions over the finite field k, = Q/po of constants. The theory developed in 111,§6 must be used. A homomorphism a +i of the divisor group of K?,,, into that of K&,, was defined there for each regular p o . It maps classes into classes of the same degree, and also maps correspondences and their classes. The divisor of fixed points of A is mapped onto the divisor of fixed points of A. Finally, for the trace defined in terms of divisors degrees, s( 3) = s(A).

(28)

We can now assert, in full generality: afield K of functions over afinite algebraic number field ko reduces modulo a regular p o in such a manner that its ring of correspondence classes is mapped isomorphically onto a subring of the ring of correspondence classes of R (in the extended sense). Proof. The mapping is certainly a homomorphism. Let Al ,... be correspon.. their images. Let a polynomial relationf(A) = 0, dence classes of K and Al,. with rational integers as coefficients, hold. Then, s( f (A)f(A)*)= 0. By 111,§6,6 residue formation modulo ( a prime divisor of) p o and extension of constants commute. Multiplying correspondences amounts to the latter. Thus, A 3 = All (29) for two correspondences A, B. By (28), then,

s(f(A)f(;I)*)= s(f(A)f (A)*)= s(f(A)f(A) *) = 0. This meansf(A) is the principal class, as was to be proved. 7

(30)

$5. APPLICATIONS IN NUMBER THEORY

311

6. THEEIGENVALUES OF MODULAR CORRESPONDENCES Our further considerations depend upon the important equation

-

Tp* = F*

+ CpF

(31)

which is valid for the reduction of KtN)modulo the same prime number p , with the sole assumption that p is regular in the sense of III,§6. F is the Frobenius correspondence and Uptheautomorphismof K#N)defined in §3,1. (We now write p instead of p o .) Proof. Let K = KF,,) be the field of modular functions in z, and let K’be a duplicate but independent field of modular functions in another variable 7’.By §3,1 and Eq. §1,(27) the ideal ( T p * ) K xis K generated , as the product of the ideals which contain all the differences

where a, 6, d take on the values of §3,(6); that is, the matrices

occur. Only such functions rp need be considered whose Fourier coefficients do not contain p in their denominator. The Fourier series immediately show that rp(PT)

= cp(7)” mod P

or

cp(r’) - cp(pt) = cp(z’) - ~ p ( z )mod ~ p.

(32)

We can then set v(ud(t)) = $(r) = Zc, exp[2ninN-’z] and form

with a primitive pth root of unity 5. The residue class of this expression modulo p is the same as that modulo the prime n = 1 - 5 of Q(C). But, Yb3 1 mod n. Thus t+Nb n (p(r’) - $ ( =y) -c ) exp[ZniiiN -‘TI b

cpp(~’)~

= cp(r’)P

cn

- $(z)

mod

R.

312

V.

CORRESPONDENCESBETWEEN mms OF ALGEBRAIC FUNCTIONS

As both sides of this congruence have only rational coefficients, it must also hold mod p, so that

Because of (32) and (33) the residue class asserted in (31). 7 From (31) we can yet derive

TP*is the sum of F* and

U,,F as

Tp= F + Ui'F* (34) by using 53,(5) and (8). We saw in §4,9that the correspondenceclasses of K&!) form a semisimple ring. The elements U p , F, and I;* commute, so there exlsts a faithful representation of this ring in which those elements have diagonal form:

Moreover, by $1,(22), cp,cp,* = p. The C, are roots of unity of an order q ( N ) dependent upon N and p. The cp, are from among the o,of $5,4. By (34) then, Tpsatisfies the equation

where the product is taken over all hth roots of unity and all zeros of the polynomial L'(u) of 554. The coefficients of the polynomialf( Tp)are rational integers. The last theorem of the last section shows that Tpalso satisfies this equation. Hence the eigenvalues z,(p) of Tpmust occur among the w, + [p/w, in every representation. By (21) these eigenvalues can be majorized by l%(P)l 4 2 J i .

(35)

Comparison of these results with those of $3,6 leads to the conjecture that the roots of unity [, that occur here in the eigenvalues ~ ( pof) Tpfor some fixed character ~ ( n are ) equal to the ~ ( p )and , that the roots o,= w,(p) have the same meaning here as in $3,6. This is, in fact, a consequence of $3,(34), as the o,(p) here are uniquely determined by the left side up to the possibility of interchanging o,(p) and ~(p)po,(p)-'.The question as to how the roots w, of L'(u)= 0 are each associated with an eigenvalue z,(p) for modular

$5.


313

forms of a given character is left open here, though. We must refer to the literature for the answer.?

7. MODULAR FUNCTIONS TO THE PRINCIPAL CHARACTER A simplification of the problem is found by limiting it to the subfield KTQfN),O of functions invariant under all automorphisms U p . Its differentials dui = ui(r) dz yield modular forms ui(z) of the principal character, using the language of 53. In place of (34) we then have

T, = F + F*,

(36)

and the eigenvalues of any representation of T p , and in particular of the representation (ti,@)) of $3, are 7 d P ) = 0, + P/%:,

(37)

where the w, are certain zeros of the polynomial L’(u). This is precisely the conjecture of $3,6. We now assert, moreover, that L’(u) is the g-row determinant I(u2

+ PIE - u(tij
)I = L’(u),

(38)

Because of (37) the zeros on the left occur on the right, as well, and the degrees and constant terms on both sides coincide. We must know, though, that even the multiplicities of the zeros are correct. This is found by considering the trace. The trace of the matrix (ti,(p)) is that of the representation 52,(8) by first kind differentials. It is real, as (37) and p / w , = 3,show the eigenvalues to be real; thus it is one-half the trace s(T,,) in §4,(2).The same holds for traces of powers of (tij(p)). By (26) and (36) we thus have trace(tij(p)”) = +((F

+ F*)”),

n = 0,1 ,... .

(39)

By numbering the w, as in (20), and using (25) and FF* = p D , we see that

Then (39) implies that the w, + p / o , are the eigenvalues of the (ti/@)) with the right multiplicities, proving (38). We conclude with a remarkable consequence of (38). By $53, K&N),o is a field of functions over the exact constant field Q. The field KP(N),O can be reduced modulo all regular prime numbers p, the zeta functions C,(s) of all

t G. SHIMURA, Correspondences modulaires et les fonctions J. Math. SOC. Japan 10, 1-28 (1958).

5

des courbes algkbriques,

314

V.


these residue class fields R&N),O are formed, and their product is as follows:

The absolute convergence of this infinite product for Re(s) > 3/2 results from (21). The factors of the first and second kinds essentially give the Riemann zeta functions [(s) and ((s - 1) if the fact that finitely many irregular prime numbers have been omitted from the product is ignored. The product of the L(p-’) is, according to (38), the reciprocal of the determinant of the matrix zeta functions §3,(28), a finite number of factors again being left out. This reduces the product (40) to known functions satisfying functional equations of the Riemann type. But the Riemann conjecture proved in $5,4 for cp(s) is not related to the classical Riemann conjecture for c(s).

NOTES As already mentioned in #,4, the history of the problems delved into here goes back to Gauss. After the proof by Herglotz of Gauss’s conjecture, Artin [l] took up the study of the zeta function of quadratically generated fields. Schmidt [9] generalized the theory to arbitrary fields of functions. Our presentation in &1-3 follows Hasse [5]. Hasse [6] was first able to prove the Riemann conjecture for elliptic fields. In addition, Davenport and Hasse [2] could prove the conjecture in certain fields permitting explicit computation. The zeros of c(s) could be determined as special types of Gauss sums. The turning point came when Weil (cf. the paper cited on page 298) recognized the connection of this problem with Castelnuovo’s inequality. The papers which followed all use this idea, differing but in their proof of that inequality. This literature was discussed in $4,10. The product (40) of zeta functions of fields K arising from reduction of a field K of functions over a finite algebraic number field modulo the regular po was first brought up by Hasse. Weil [l 11 and Hasse [7] investigated these products for fields K = ko with ax‘“ + By” + y = 0, showing them to be meromorphic functions and to satisfy functional equations of familiar types. Deuring [3] achieved similar results for certain (so-called singular) elliptic fields. The work of Hasse and Deuring pays particular attention to the finitely many irregular p. Their contributions c,(s) can be defined meaningfully, and they make the product (40) into a function of particularly simple type. A third class of function fields for which the product (40) is essentially known is that of the modular functions considered in 455-7. The present author [4] attained the first of these results. They were soon generalized by Shimura (cf. footnote, page 313). Igusa (cf. III,§6,7,[3]) could show that only

$6. ELLIPTIC FUNCTION FIELDS

315

the prime divisors of the level N are exceptional irregular primes. Rangachari [8] studied the products of certain L series corresponding to (40) in the same manner for these fields. A zeta function can also be defined for an algebraic variety of several dimensions (Weil [ 121). In spite of the expected complications its construction is similar to (1 l), in that it is a quotient of certain polynomials in q-’ which are associated with the dimensions 0, 1,..., n, where n is the dimension of the variety. Weil presents a conjecture similar to Riemann’s for the zeros of these polynomials, but to date only special cases have been verified. Due to Taniyama [lo] certain classes of abelian varieties have been mastered, and their products of type (40) are known. REFERENCES

[l] E. ARTIN, Quadratische Korper im Gebiet der hoheren Kongruenzen 11, Math. 2. 19, 153-246 (1924). [2] H. DAVENPORT und H. HAWE,Die Nullstellen der Kongruenzzetafunktion in gewissen zyklischen Fallen, J. Reine Angew. Math. 172, 151-182 (1934) [3] M. DELJRING,Die Zetafunktion einer algebraischen Kurve vom Geschlecht Eins, Nachr. Akad. Wiss. Gottingen, Math.-Phys. K1. I, 84-94 (1953); 11, 13-42 (1955); I l l , 37-76 (1956); ZV, 55-80 (1957). Quarternare quadratische Formen und die Riemannsche Vermutung fur [4] M. EICHLER, die Kongruenzzetafunktion,Arch, Math. 5 , 355-366 (1954). [5] H. HASSE,Uber die Kongruenzzetafunktion,S.-B.Preuss. Akad. Wiss.Berlin, Math.Phys. K1. XVII, (1934). [6] H. HASSE, Zur Theorie der abstrakten elliptischen Funktwnenkorper, J. Reine Angew. Math. 175 11: 69-88, 111: 193-208 (1936). [7] H. HASSE, Zetafunktion und L-Funktionen zu einem arithmetischen Funktionenkorper vom Fermatschen Typus, Abh. Deutsch. Akad. Wiss. Berlin K1. Math. Allg. Naturwiss. (1954). Modulare Korrespondenzen undlReihen, J . Reine Angew. Math. [8] S. S. RANGACHARI, 205, 119-155 (1961). [9] F. K. SCHMIDT, Analytische Zahlentheorie in Korpern der Charakteristik p , Math, 2. 33, 1-32 (1931). L-functions of number fields and zeta-functions of abelian varieties, [lo] Y. TANNAMA, J. Math. SOC. Japan 9, 330-366 (1957). [ l l ] A. WEIL,On Jacobi sums as tGrossencharakterer, Trans. Amer. Math. SOC,73, 487-495 (1952). [12] A. WEIL,Numbers of solutions of equations infinitefields, Bull. Amer. Math. SOC.55, 497-508 (1949).

§6. Elliptic Function Fields

We conclude with a report on the correspondences of elliptic function fields. Although we are dealing with the oldest and most established part of the theory we cannot carry out proofs here. A complete presentation would

316


require the presupposition of a broader basis of algebraic number theory than has been possible to include in the scope of this book. Another part of the theory, and in particular the more recent results, build upon the study of the group Q, mentioned in $4,9. A decisive reason for waiving a complete presentation is seen in the following. The special importance attached to elliptic function fields within algebraic function theory is due to the fact that they are one-dimensionalabelian varieties. Their special treatment not including a general study of abelian varieties would hardly seem appropriate today. 1.

THERINGOF CORRESPONDENCE CLASSES In the sequel K and K' will be two isomorphic but algebraically inde-

pendent function fields of genus g = 1 over an algebraically closed field ko of constants. Every class of correspondences in the extended sense between Kand K' other than the principal class contains a correspondenceM of degree g'(M) = 1 (cf. end of $43).M maps K' isomorphically onto a subfield KO c K of index [K : KO]= g((M). Now, K and K' and thus K and KO are isomorphic. Thus an isomorphism of K onto its subfield KO is given by M. Such a mapping is called a meromorphism. Meromorphisms can exist for rational and elliptic function fields only, incidentally. For, by 111,§3,2, the genera g and go of K and KO and the different bK/Kosatisfy the equation g

-

=

CK: K O l ( g O

-

-k

b@K/Ko)*

If Kand KOwere isomorphic and g > 1, then g = go would follow, so that the degree of the different would be negative, impossible for an integral divisor, We also see that in the elliptic case &,(/KO) = 0, so that K is an unramified extension of KO. We must take note of another important detail. The addition theorem of IV,§2,2 showed that the prime divisors of K of the first degree form an abelian group, and a meromorphism M must map this group onto a subgroup of the corresponding group of KO.The factor group turns out to be of order g(M), and also to be the Galois group of K / K o ,which is, as is seen, a Galois extension. Composition can be used in the natural manner to define multiplication of meromorphisms. It corresponds to the multiplication of the correspondence classes involved. The addition of meromorphisms is defined by the addition of their correspondence classes, which in turn is nothing but the addition of the first degree prime divisors of KK'/K in the sense of the addition theorem of IV, §2,2. By excluding the " constant" prime divisors of KK'IK, that is, those of K', a ring of meromorphisms coinciding with the ring of correspondence classes is attained.

§6.

ELLIPTIC FUNCTION FIELDS

317

The theory of meromorphisms can be carried over to arbitrary abelian varieties, no distinguished position being held by the elliptic, that is, onedimensional varieties. A very comprehensible presentation of the theory of meromorphisms for the elliptic case is given by Hasse in §5,8,[6]. The structure of the ring of meromorphisms or of correspondence classes is closely linked with the formula MM*

N

g(M)D.

(1)

In the proof of the lemma of $43 we saw that MM* is the extension of the unit correspondence Do between the fields KO and Kg to a correspondence between K and K”. Thus, only

Do

9W)D

need be proved. Do,as a prime divisor of KoK,”,decomposes to Do = D,

+

* - a

in KK”. We know that K/Ko is unramified and Galois, and that its group is the additive group of the prime divisors of K. Thus this group consists of translation automorphisms as considered in IV, §2,3. This means that the Di are all equivalent in the extended sense, proving (1). If only correspondence classes are to be considered,

MM* = g ( M ) D

(2)

may be written in place of (1). MM* is called the norm of M:

It is not difficult to see that no divisors of zero can occur. For, were M , M , = 0, then either n(MJ or n(M2) would vanish, which would imply M , or M2 = 0 by the principal theorem of $4. A ring with a norm such as (3), associating a natural number with each element must belong to one of the following types: (a) it is isomorphic to Z;

(b) it is isomorphic to an integral domain of integers of an imaginary quadratic number field (singular case); or (c) it is isomorphic to an integral domain of integers of a definite quaternion algebra (supersingular case).

In fact, all these three types do occur among the rings of correspondence classes. The third case is excluded, though, if the characteristic is 0. For then

318

V. CORRESPONDENCESBETWEEN

FIELDS OF ALGEBRAIC FUNCTIONS

the correspondence classes are faithfully represented by the only differential of the first kind:

du M = p du. By constructing elliptic function fields to given periods it is easy to generate fields whose correspondence classes are given integral domains of imaginary quadratic number fields. The third type of ring occurs only for fields of characteristic p > 0. In particular, Deuringf showed that only maximal integral domains of those quaternion algebras over Q ramified nowhere except 00 and the prime p can occur. Conversely, any maximal integral domain of such an algebra really occurs as a ring of correspondence classes. The supersingular case is characterized by the first kind differential being mapped onto 0 by the Cartier operator (cf. 111,54,4). 2. COMPLEX MULTIPLICATION

Let the constant field be ko = C and the ring of correspondence classes be the principal order of an imaginary quadratic number field A. Iff is an ideal of A with the basis K,, rc2 over 2,then the elliptic functions with base periods K,, x2 generate a field K with the given ring of correspondence classes. Kdepends only upon the class of I.The invariant j ( K ) which uniquely characterized the field K in IV,§2,6can thus be taken to be a function of the class of €, and written j ( K ) = j ( € ) .To the h different classes of ideals of A there are h different fields K and invariantsj ( € ) . It turns out that these j ( € ) are algebraically conjugate algebraic integers with the following remarkable properties: (a) they generate a Galois extension Al of degree h over A, whose group is isomorphic to that of the ideal classes of A; (b) a prime ideal p of A decomposes in A, to prime ideals of residue class degreef, where f is the smallest exponent such that pf lies in the principal class; (c) Al/A is unramified; (d) every ideal of A becomes a principal ideal in A, ;and (e) any prime ideal p of A satisfies the congruences j(p €)=j(€)”(p)mod p

(4) in A,. These congruences are the key to the theory and lead rather quickly to properties (a) through (c), property (d) lying deeper. Equation (4) has the same roots, incidentally, as does §5,(34). Along with the invariant j(t), the so-called partial values of elliptic functions play an equally outstanding role. They are the constants onto which

t Die Typen der Multiplikatorenringe elliptischer Funktionenkorper, Abh. Math. Sem. Univ. Hamburg 14, 197-272 (1941). ‘‘Multiplikatoren’’ are correspondence classes.

96.

ELLIPTIC FUNCTION FIELDS

319

the functions x, y are mapped by a first degree prime divisor p of K, and between which the Weierstrass normal equation IVY42,(21) is valid, insofar as o can be represented by an integral multiple of p, in the sense of the addition theorem. These partial values also generate Galois extensions of A with abelian groups permitting individual description. It can even be shown (Kronecker’s “Jugendtraum”) that any extension of A with an abelian Galois group can be generated by partial values. A presentation of this theory is found in M. Deuring, Die KIassenkcYrper der komplexen Multiplikation, Enzycloptidie d. math. Wiss., new ed., Vol. I(2), Part 10,,(23). Stuttgart, 1958. As mentioned above, this theory permits an extensive analog for abelian varieties, which can be found in the monograph G. Shimura und Y .Taniyama, Complex multiplication of abelian varieties, Publ. Math. SOC. Japan 6 (1961).


Author Index Numbers in parentheses are reference numbers and indicate that an author’s work is referred to although his name is not cited in the text. Numbers in italics show the page on which the complete reference is listed. Nastold, H. J., 157 Nering, E. D., 184 Ntron, A., 213 (2). 214

Artin, E.,314, 315 Behnke, H., 141 (2), 142 Berger, R., 79

Peterson, H., 171, 231 (4, 6), 232,278,281 Pfetzer, W.,232

Cartier, P., 151 Cassels, J. W. S., 204 Chevalley, C., 135, 142, 151 ( 3 ) Conforto, C., 204 Courant, R., 202 Davenport, H., 314,315 Deuring, M., 114, 174, 184, 200, 202, 244, 266,314,315,319 Eichler, M., 19, 231 (2), 232, 265, 278 (3), 280 (4,4a), 281, 314 (4), 315 Falb, P., 157 Fueter, R., 228 Grothendieck, A., 298,299 Hasse, H., 79, 98, 99, 101, 151, 184, 203, 204,314, 315 Hecke, E.,98,231,278 (6),279,280,281 Herglotz, G., 307 Hofmann, J. E., 204 Hua, L. K., 41 Hurwitz, A., 202, 242,266 Igusa, J. I., 184, 203, 204, 278 Kappus, H., 171,265 Klinger, H., 41 Knopp, K., 202 Kuga, M., 214 Kunz, E., 79, 157 Lamprecht, E., 31,183, 184 Lang, S., 204,213 ( I , 2), 214 Mattuck, A., 183 (7), 184, 298, 299 Meyer, C., 99 Mordell, L. J., 203, 204

Rangachari, S. S., 315 Reiner, J., 41 Roquette, P., 141, 142, 184, 298, 299 Rosenlicht, M., 142 SafereviE, I. R., 190 Samuel, P., 91, 95 Schiffer, M., 141 (2), 142 Schmidt, F. K., 142,314, 315 Schoeneberg, B., 232 Serre, J.-P., 298 Shimura, G., 214, 278, 281, 313,319 Siegel, C. L., 41,43,206 Sommer, F., 141 (2), 142 Spencer, D. C., 141 (I), 142 Tamagawa, T., 142 Taniyama, Y.,315, 319 Tate, J . , 135, 298, 299 Teichmuller, O., 171 Tricomi, F., 203, 204 Vahlen, K. Th., 36 Van der Blij, 278, 281 Van der Waerden, B. L., 63, 142 Weil, A., 141 ( l l ) , 142, 204, 298, 299, 314, 31s Weyl, H., 141 (12). 142 Witt, E., 38, 142 Wohlfahrt, K.,280, 281 Zariski, 0..91, 95, 157

321

Subject Index Abel's theorem, 199, 209 Abelian integral, 213 Abelian variety, see Variety Addition theorem, 192, 198, 21 1 Antiautomorphism, see Rosati adjoint Associative law, 240

Degree, of a correspondence, 235 of a divisor, 80, 85,89, 120, 124, 130 of a linear divisor, 23,89,130 Diagonal, 167 Different, 75ff, see also Divisor Differential, 147ff, 158, 223, 252 class, see Class exact, 151 kind, 155 quotient, 143 Dimension, 5 , 23, 132 Dirichlet, see Unit theorem Discriminant, 29,75ff, 96ff, 228,278 theorem, 77 Distortion ratio, 263 Divisor, 79ff, 12W, 128ff, see also Class diagonal, 167 different, 89, 128 of a differential, 147, 158ff of fixed points, 283, 286ff integral, 121 Kronecker, 57, 87 linear, 22, 88, 128 principal, 57, 80, 82, 120ff system, nonspecialized, 294 unit, 79, 120 Dual space, 10, 128

Base units, see Unit Basis, 5 Complementary, 10 theorem, 5 , 13 Bounded ramification, see Ramification Canonical class, see Class Canonical system of incisions, 189 Cartier operator, 15W Character, 219, 275 Class, canonical, I33ff, 156ff differential, 156ff divisor, 22ff, 80, 12W, 179 ideal, 54,98 principal, 133 Complement, 73ff Complementary basis, see Basis Complementary module, see Module Component, 21, 67ff, 88ff, 128ff Congruence subgroup, 217ff Conservative, 157, 174 Content, 60 Convergence radius, 110 Correspondence, 233ff in extended sense, 285ff Frobenius, 249, 306 inseparable, 246ff modular, 266ff, 308ff prime, 245 Cusp, 217 form, 228,271ff Decomposition group, 104

Eisenstein series, 220, 225 Elementary divisor theorem, 7 Equivalence, of correspondences, 235,244,285 of divisors, 84, 88, 120, 129 of principal part systems, 160 quasi, 62 Field, of constants, I extension of, 135ff cyclotomic, 106 322

SUBJECT INDEX

decomposition, 106 of definition, 133 of elliptic functions, 190ff, 315ff inertial, 91 quadratic number, 104 Finiteness criteria, 7, 12 Fixed points, 260ff, 282ff number of, 289 theorem, 264 Frobenius, see Correspondence Fundamental domain, 39 Gauss sums, 44ff Generators, system of, 6 Genus, 133ff, 188 relative, 134 topological, 188 Green’s function, I65ff Holomorph, 220 Hurwitz genus formula, 135 Hyperbolic plane, 36 Hyperbolic space, 36 Ideal, 5ff, 53ff, see also Class, Norm extension of, 70 integral, 54 prime, 58,72 conjugate, 91 regular, 174 Inertia theorem, 173 Inertial group, 91 Inseparable extension, 30, 54, see also Correspondence Integral(s), dependence, 53 of first kind, 196ff, 204ff of third kind, 208ff Invariance, 86, 135 projective, 81 Jacobian variety, see Variety Lattice point theorem, 14ff Legendre relation, 206 symbol, 44ff Lemniscatic, 204 Level, 217 Linear divisor, see Divisor Local ring, 63ff

323

Local theory, 3 Local uniformizers, 187, 214ff Manifold, 186ff Meromorph, 186,220 Meromorphism, 316 Model, 83,86, 131 Modular correspondence, see Correspondence Modular form, 219ff, 275ff Modular function, 43, 220ff, 266ff Modular surface, 214 Modular triangle, 40, 214ff Module, complementary, 10, 22 Multiple, 23, 232 Multiplier, 43, 132 Noetherian ring, llff, 61 Norm, absolute, 300 of divisors, 85, 124 of ideals, 70,93 of linear divisors, 23 Normal form, see Weierstrass Normed, 275 Order of a fixed point, 265 Order function, 79ff, 120ff, 147ff, 158 p-component, 21,67 p-constant, 146 p-variable, 146 Period, 197, 204ff, 257 matrix, 204 relation, 204 Place, 20ff, 64, 79ff, 120ff critical, 114ff regular, 114ff Pole, 110, 148 Power series, IlOff, 133 Prime correspondence, see Correspondence Prime element, 64, 112, 124, 147 Primitive, 221 Principal part system, 160 Product, of correspondences, 236ff of rings, 234 scalar, Peterson, 273ff scalar, of principal part systems, 160 Pseudocomplement, 73ff, 89

324

SUBJECT INDEX

Pseudodifferent, 75 divisor, 89 Pseudodiscriminant, 30, 75 Pseudotrace, 30 Ramification, 93,216ff bounded, 253,271 field, 95 group, 94ff index, 72,84,95 irregular, 95 number, 72,84,95 regular, 95 Reciprocity law, 47, 139 Reflection automorphism, 193ff Regulator, 104 Residue, 151ff class degree, 72 theorem, 153 Riemann-Roch theorem, 26,133ff, 141 Riemann surface, 185 Rosati adjoint, 241, 259 Separable, 30, 131, 143, 149ff Separating element, 143 Subdegree, 117

Symplectic, group, 32ff, 205 matrix, 32,205 modular form, 42 modular group, 36 System of generators, 6 Theta function, 41ff, 44ff, 232 Trace, 27ff, l22,259ff, 283,291 formula, 263 Transitivity formula, 27, 72 Translation automorphism, 193ff Unimodular, 7 Unit, 5 , 83 base, 101, 105 divisor, see Divisor theorem, 101 Variety, Abelian, 214 Jacobian, 21 Iff Weierstrass normal form, 200 Zero, 110, 148 Zeta function, 299ff

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