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.
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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
This Page Intentionally Left Blank
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
<
<
$3.
21
LINEAR DIVISORS
do not exclude the possibilities k = Q and k = k,(x,, ..., x,) for the time being in order to accentuate the generality of the method. We take the opportunity to introduce a terminology which has become customary. As we saw in the classical case, the prime polynomials p ( x ) and their associated integral domains i, correspond to the points or places x = t, including 5 = 00 of the Riemann number sphere. One now associates places with the i,,, in a formal manner, without necessarily bringing a geometric concept into play. We will never properly speak of the places as such, but rather in connections of the following variety: an element is integral “ a t the place” p if p occurs in its prime decomposition with a degree v(p) 2 0. Basically the places are but elements of an index set. The correct use of the expression and its practicality will appear by itself in what follows. For i = Z or i = k,[x,, ..., x , ] we have the important and self-evident equation (1) i=
nip
pfm
to which we remark that, for the case k = Q, i.e., i = 2, we have not defined i m and that therefore the restriction p # co is superfluous. Let k“ be the n-dimensional vector space over k and w i a basis of k“ which will be fixed from now on. A linear dirisor a of k” will be understood to be a system of i,-modules a, with the following properties: (a) With every place p (including p = co if it is defined) there is associated a finite i,-module a,, in k” of rank n. These a, will be called the pcomponents of a. (b) For almost all p (that is, for all but a finite number of the p ) the w i form a basis of a,, with respect to i,. The basis w i in the definition can be replaced by any arbitrary basis mi’. For then there exists a system of linear equations w i = cijwj’ in k. The denominators of the cij contain but a finite number of primes. Ifp is not a divisor of the denominator of some c i jor of the numerator of the determinant Icijl, then wi and mi’ are bases of the same i,-module. This is almost always the case. The intersection a, = a, (2)
1
n
P+ w
is a finite i-module of rank n.
Proof. For almost all p , w i E a,,. For the finitely many exceptions, let a p i( i = I , 2, ...) be generating systems for a,,. All these finitely many a p i , together with the m i ,generate, on the one hand, finite i,-modules a,’ -C a,,, and on the other a finite i-module a,’. We have a,‘ =
a,,‘ 2 a,. P#m
22
I.
LINEAR ALGEBRA
By the finiteness criterion of §1,4 we thus know that a, is a finite i-module. It remains to show that it is of rank n. As all the a,, are of rank n there exists, for each place p , an exponent h such that phmiE ap. Because of property (b) we can take h = 0 almost everywhere. Letting a be the product of all the ph, we have am, E a, for all i, completing the proof. T In this connection we consider the dual space k"*. The complementary modules a,* to the a, form the components of a linear divisor a* of k"*; a and a* are said to be complementary. Proof. From §1,3 we immediately have property (a). Letting wj* be the basis complementary to oi, the mi* always form a basis of a,* with respect to i, when the w i form a basis of a,, and this is almost always true. 7 Let gp,i = C a p , i j a j (3) J
be bases of a, (p = 00 and p # matrix in k. Then
00).
Bpi
Further, let A = (aij)be some nonsingular
=
c
"p,ij"JPl
(4)
J.1
are bases of the p-components of a linear divisor b. This divisor is uniquely determined by A and the divisor a, for if other bases of the ap are used, the = A, and A,A appearing in (3) and (4) are simply multiplied matrices from the left by the matrices giving rise to the basis transformations. We write b = aA (5) and call a and b equiualent. The following is clear:
(aA)B = a(AB) (6) for two matrices A and B. This shows that our equivalence relationship has the familiar property of transitivity; reflexivity and symmetry are obvious. Therefore equivalent divisors may be grouped into divisor classes. The divisor complementary to ( 5 ) is
b*
= a*A*,
A* = (A')-'.
(7)
For, if wi* is the basis complementary to wi, and one sets
using the matrix A,* = ( aPnIJ * ..) = (Ad)-' contragredient to A, = (ap,ij), one finds that c1Pi a*. PI = wioj* = d i j . Thus the a:i form a basis of a,*, so that Eq. (7) follows from the identity (A&* = A,*A*.
$3.
23
LINEAR DIVISORS
2. NORM AND DEGREE OF A LINEARDIVISOR The matrices in (3) are determined up to a unimodular factor on the left, thus their determinants are fixed up to a unit factor. Let
with a function y of degree 1, and with units up E i,, u, E i, . The exponents h, are invariants of a,. They are h, = 0, because of property (b), for almost all p . Now associate to a the linear divisor a', defined by a,' = p h p i p , a,' = (l/y)h-ia. It is called the linear norm of a, symbolically a' = N(a). This definition immediately yields the equation N(aA) = N(a)lAI.
(9)
In the case of a function3eld one also defines the linear degree of the linear divisor a bv
where G(p) is the degree of the prime polynomial p. Equivalent diuisors hatie the same linear degree. This. implies, incidentally, that the degree remains invariant under a change of basis, while this does not hold for the norm. Proof. Using the notation of (9,let IAl have the prime decomposition 1/41 = c
n
Pf
pvp
00
with a constant c. The degree of IAl is then G(IA1) =
v,G(p). Further,
Pf,
and thus
3. THEDIMENSION OF A LINEARDIVISOR
From now on throughout we will assume k to be a function field. The intersection of all the a, (including a,) is a linear space over the field of constants k , . The dimension of this intersection is called the dimension of a and is denoted /(a). A vector contained in all a, is called a multiple of a. Equicalent dicisors hare the same dimension.
I.
24
LINEAR ALGEBRA
Proof. Let a = C aimi be contained in all a,. Then, because of (3), the system of linear equations
Ci x p , i a p , i I = a, is solvable for all p with xp,iE i, and, conversely, the solvability implies a E a,. Passing now to the divisor b = aA we see that is simultaneously solvable. This means that the vector = C b p i is contained in every b,. The correspondence between a and /Iis unique in both directions and linear. This yields the assertion. 7 The dimension of a linear divisor is finite. Proof. All multiples are contained in the intersection a, defined by (2),which is a finite i-module. Were I(a) = 00, then there would exist multiplesof arbitrarily high degree in a representation with a basis of a,. But, as these multiples also are contained in a, and a, is a finite module, this is impossible. 7
Riemann-Roch Theorem for Linear Divisors. Let k = k,(x). Associate, with the linear divisor a in k", the divisor a' in kn* by setting a;=
Then,
p'
for p #
x-2a,*
00,
for p = 0 0 .
@) - l(a? = -G(a)
+n
holdr. Furthermore, aa' = 0 for multiples a, a' of a, a'. The fact that ( 1 1 ) defines a divisor follows from 43,l. By §1,(7) we have (ap*)* = a, and (x-2a,*)* = x'a, . Thus, the divisor associated with a' is a" = a. Thus (12) implies G(a)
+ G(a') = 2n,
which could also be proved directly by setting y = x in (8). An immediate consequence of (7) is
(aA)' = a'A*, A* = (A')-'. (13) Thus the correspondence a t ) a' between the linear divisors is also an involution of the divisor classes. Correspondingly, Eq. (12) contains only invariants of divisor classes. The exceptional status of p = 00 in (1 1 ) is doubtlessly an esthetic flaw, but it will dissolve of itself in the applications of the theorem below. Furthermore, we can carry out the following argument here. By taking a projective transformation x' = (ax + b)/(cx + d )
$3.
25
LINEAR DIVISORS
of the variables, the places p are caused to permute; in particular, if c # 0 the place p = cx + d passes to p' = a.It is therefore plausible to replace p = 00 in (1 1) by any other place p = cx + d (c # 0) and set apl = p'a, there. In doing this the correspondence between divisors is in fact altered, but not that between the classes, and it is in them that we are really interested. The proof of the Riemann-Roch theorem dates back to Dedekind and Weber.? As am is a finite i,-module, there exists to every a E k" a smallest exponent v(a) such that x-'(")aEa, holds. As the basis theorem of #1,1 assures us that a, has a basis, we can construct a special basis ai of a, as follows. Among all 'the possible bases we choose one such that v(aJ has the least possible value. If there is more than one such basis choose, from among theq, one such that v(aJ is as small as possible, and so forth. One then has, of course, v(al) 5 v(az)5 =< ~(a,,). (14) Now we claim that the vectors x-v("i)a,form a basis of a, with respect to i, . For, if this were not so, there would exist elementsf, E i, not all divisible by l / x such that the vector p = x C f t x - v(a4x, (15) would lie in a,. One could then set
fi =A(')) + ( l / x ) g i
9
ft(0)E ko
3
~t E
im *
Then is also in a, and, in our assumption, not all of thefi(0) are 0. Thus there is no loss of generality in assuming that f i = f , ( O ) E ko in (15). Letting r be the largest index for whichf, # 0, we see by (14) that a', = ~ " ( " ~ ) - ~is/ 3a vector in a,. al, ..., a,-1, a,I would generate the same submodule of a, as al, ..., a,. Thus one could replace a, by a,'. But, since v(a,l) 5 v(a,)- 1, this contradicts our construction of the ai . If one takes the a, for the oi then, in the notation of (9, (up,ij)is the unit matrix for all p # co and the diagonal matrix with entries x-'("~). By $3.2 this yields G(a) =
v&).
(16)
Now, a multiple a of a is represented in two ways: This gives a, = b,x-"("). The conditions ui E i, bi E i,
t Theorie der algebraischen Funktionen einer 181-290 (1882).
cannot be satisfied
Veranderlichen,J. Reine Angew. Math. 92,
I.
26
LINEAR ALGEBRA
when v(ai) > 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
The discriminant of a particular and thus, of course, of every basis of K / k is #O if Klk is separable. In the case of a simple inseparable extension K = k(*) of afield of characteristic p all traces are 0, and therefore so is the discriminant of any basis. By the transitivity formula the same holds true for arbitrary inseparable extensions. This state of affairs renders the trace and the discriminant useless for the study of inseparable extensions. As a replacement we introduce a pseudotrace.? What we need is a function a(a) of the elements a E K with values in k with the properties a(a
+ p) = a(a) + o@),
a(aa) = aa(a)
A(o) = la(wio,)l # 0.
for a E k,
(8)
for a basis w , of K / k .
(9)
The last determinant is called the pseudodiscriminant. Given three fields k c L c K with pseudotraces for KIL and Llk, then
aK/da) = G L / ~ ~ K / L ( ~ )
(10)
is a pseudotrace for K/k. Equations (8) obviously hold. The proof of the transitivity formula (7) of §4,3 carries over word for word, yielding the transitivity formula (1 1) AK/dm x tt) = n,/k(AK/L(~))AL/k(tt)'~:~l for pseudotraces. Thus (9) also holds for the pseudotrace defined by (10). For separable extensions the trace is a pseudotrace. For a purely inseparable extension K = k ( 6 ) every a E K can be represented in the form a = so
+ s1qi+ + sp- l ( q i ) p - '
The last coefficient is a pseudotrace: a(a) = sp-
1.
Equation (8) is immediately evident, and the pseudodiscriminant becomes
A(1,
6 ,...) = (-l)p(p-l)/z.
t The concept is due to E. Lamprecht, Uber s-Differenten und Differentiale algebraischer Funktionenkorper einer Veranderlichen, Arch. Math. 4, 412-424 (1953). The term pseudotrace (Pseudospur) is suggested here for the first time.
32
APPENDIX TO CHAPTER I. THE THETA FUNCTION
Using (10) one can now prove the existence of a pseudotrace for an arbitrary extension Klk. I f 0 and a' are two pseudotraces, then there exists a uniquely determined element 6 # 0 in K such that (12)
u'(a) = a(6a)
holds for all a E K . Conversely, if6 # 0, then a(6a) is always a pseudotrace. Proof. With a basis o1of Klk,set
C coIai,
6 =C old, with as yet undetermined coefficients di . Also put a=
a(6a) = C d,a(o,w,)a, = C a'(w,)a, = a'(a).
This is correct for all a if and only if
Cf d p ( o p , ) = a'(o,)
( j = 1,2,
...)
holds. By our assumptions the determinant of this linear system of equations is not zero, and thus such a 6 exists. If conversely such a 6 is given, then define a' with (12) so that (8) holds. Further,
60 =WD implies Iu'(wfo,)l = la(6oiq)l = lO(Ol0,)l
PI = A(o)n@),
which is ZO. 7
Appendix to Chapter I The Theta Function This function is among the most important tools of analytic number theory. Its theory is built primarily on completely elementary considerations. This justifies its insertion at this point. §1*. The Symplectic Group 1.
THEBASICPROPERTIES
Let A, B, C, D be n-rowmatrices with coefficients in R, E the n-row unit matrix, and 0 the zero matrix. Finally let
J = ( -E
")0
*
33
$1. THE SYMPLECTIC GROUP
The symplectic group En of the nth degree over the field of real numbers consists of all 2n-row matrices M = (g i)which satisfy the equation
")(
M~JM=(~' B' D' or, written otherwise,
)(
O E A B) = J -E 0 C D
Computation yields the three equivalent equations A'C - C'A
= 0,
B'D - D'B = 0,
A'D - C'B
= E.
(3)
The group C' consists of all real two-row matrices with determinant 1. M' is in C" The group property of C" follows from (2), and since J E Zn, whenever M is.Special symplectic matrices are V O
U V = (o V * ) with V * = ( V ' ) - ' ,
J,
T,=
(" ')
with S'= S.
(4) These matrices generate the symplectic group En. Proof. The given symplectic matrix (f i)is first multiplied from the left by a Uv and from the right by a Uv3.This transforms A to VAV'. Suitably chosen V, V' put VAV' into the diagonal form with only ones and zeros as entries: (El 8). Thus it is no loss of generality to assume A has this form from the start. Subdividing C into submatrices of the same dimensions, the first of Eqs. (3) yields
Here IC,,J # 0, for otherwise the first n columns of (g i)would be linearly dependent, which would make the determinant 0, incompatible with (2). Now multiply (2 i)from the left by T A E1, being an as yet undetermined parameter. A transforms into A'
=A
+ 1C =
Choice of a suitable A assures that the determinant IA'I # 0. Thus we may assume in full generality that IAl # 0, and thus even arrive at A = E with our first argument. For A = E, C must be symmetric because of the first of g) so that A and C go Eqs. (3). Multiply from the left with J - ' T J = (-I into E and 0, and only the matrix (f i)remains to be treated. Now the two last equations (3) yield D = E and B' = B, and the matrix is equal to T B .
34
APPENDIX TO CHAPTER I. THE THETA FUNCTION
This completes the proof. 7 As an incidental result we see that all the matrices (4) have determinant 1, and thus : The determinant of a symplectic matric is 1. 2. SYMPLECTIC GEOMETRY In the following sections we will consider the space !$' of n-row complex symmetric matrices T=X+iY, whose imaginary components Y = ( y i j ) are matrices of positive definite quadratic forms in n variables t i . This means that
for all real values of the ti not identically 0. We will call the matrix Y itself positive definite for short. Under linear fractional transformations T + T' = (AT with symplectic matrices M = (i Proof
+ B)(CT + D)-'
= M(T)
(5)
i),the space 8"goes into itself.
If
then
M(M'(T)) = M"(T). Hence it suffices to prove the assertion for the special matrices (4) which, as we know from the last section, generate the entire group Z". For these we have T' = YTV', T' = -T-', T' = T + S (S' = S ) , respectively. In all three cases T' is again a symmetric matrix, and in the first and third it is easy to see that the imaginary component is also positive definite. As one can, with real transformations, make a positive definite quadratic form the sum of the squares of the variables, there always exists a real matrix I/ and symmetric matrix Z such that
+ iY = V(Z + iE)V' = Uv(Z + iE). It remains to show that -(Z + iE)-' has a positive definite imaginary component. E + Z2 = E + ZZ' is positive definite and thus has an inverse. Verifying -(z + i,!?)-' = - Z ( E + z')-' + i(E + z')-' T=X
91. THE SYMPLECTIC GROUP
35
gives the assertion. 7 The space b" is convex and thus simply connected.
+
+ +
Proof. Let TI = XI iY, and T2 = X , iY2 be two matrices in 4j". For all s between 0 and 1 it is seen that sY, (1 - s)Y2 is positive definite, and thus sT1 (1 - s)T, E 5". 7
+
3. THEHYPERBOLIC PLANE AND HYPERBOLIC SPACE
For n = 1, T = z = x + iy is a complex number, and Sjl is the complex upper half plane. The transformations ( 5 ) are t + z' = (az
+ b)/(cz + d ) .
As PoincarC showed, Sj' is a model of the hyperbolic plane, and the fractional linear substitutions represent hyperbolic motions. The lines and circles perpendicular to the real axis play the part of the hyperbolic lines. Angles are given faithfully by the model. The hyperbolic surface element is y - 2 dx dy. Three-dimensional hyperbolic space can be generated by a specialization of symplectic geometry. As is well known, complex numbers u = a , + ia, can be represented by two-row matrices
Set
")
H = ( '0 -1 ' so that HG(u)H = G(u)' = G(E) is a representation of the conjugate complex number. Let a, p, y, 6 be four given complex numbers with determinant a6 - By = 1. It is easy to verify that the four-row matrix
is symplectic. Fractional linear transformations ( 5 ) belonging to symplectic matrices of the form (6) applied to the variable matrix T = ( x,
+ ix, XI
-x,
+ ixz
(7)
with real xo, x,, x2 lead back to matrices of the special form (7). Moreover, the halfplane x2 > 0 goes into itself.
36
APPENDIX TO CHAPTER I. THE THETA FUNCTION
It is easily verified that the matrices (6) form a group isomorphic to the complex two-row matrices (F $) of determinant 1. The latter group is generated by the matrices ("0 (-: i). Thus, to prove the asserted theorem, one need but consider the transformations corresponding to these special matrices. In these cases
A), (A
T' = HG(a)HTG(a-')-'
= G(Z)TG(a),
T' = -HT-'H,
T' = T
+ H,
and the assertion follows from a simple calculation each time. 7 The matrices (6) form a subgroup of Z2, and the space of real dimension 3 spanned by the matrices (7) with x2 > 0 is a subspace of B2,which is of real dimension 6. It is the three-dimensional hyperbolic space, the transformations ( 5 ) with matrices (6) being the hyperbolic motions.? To prove this, one first verifies that the hemispheres 2
(xo - co)
+ (xi - cl), + x22= r 2 ,
x2 > O
(8)
are invariant under the three special transformations. The planes x,co + xlcl = c2 must also be included as a special case. Together they play the part of hyperbolic planes. Each plane can be transformed into one fixed : G ( c o - r + i c l ) ) into xo = plane, for example, the plane (8) under (-: :)(: -(2r)-', or the plane xoco + xlcl = c2 under (: ~ C ( - i c 2 ) ) ( ~ G ( a ) H G ( a - I ) ) with a2(co- ic,) = - i into x1 = 0. The plane x1 = 0 is transformed into itself under the substitutions (6) with real a, /3, y, 6, these having the form xo'
+ ix,'
=
a(xo
y(xo
+ ix,) + /3 + ix,) + 6
*
The geometry induced here is, thus, hyperbolic. This confirms the assertion. 7 4. THESYMPLECTIC MODULARGROUP The symplectic modular group is the subset r"c Z" whose matrices have rational integral coefficients. The group property becomes obvious when one calculates the inverse element using (2). If M = (i i) is symplectic, then by (3) the matrices C'A and B'D are symmetric. Consider the subset 0"c r"for which the diagonal elements of C'A, B'D are even numbers; we will show that it also is a group. A quick
t Hyperbolic geometry of n dimensions and its group of motions are investigated in K. Th. Vahlen, uber Bewegungen und komplexe Zahlen, Math. Ann. 56, 585-593 (1902), using Clifford algebras.
01. THE SYMPLECTIC GROUP calculation becomes necessary. Let M , , M , matrix M = M , M , we have
E
37
r"be given. For their product
+ C2'Dl'BlC2+ C2'Dl'A1AZ+ A2'C1'BlC2, B'D = B,'Al'C1B2 + D2'B1'D1D2+ D,'Bl'CIBz + B;Al'D1D2. C'A = A2'Cl'AlA2
Applying (3), this yields
+ Cz'Bl'D1CZ + C,'D,'AIA2 + A;A,'DICz - CZ'AZ, B'D = B2'ClrA1B2 + DZ'Bl'DlD2 + B2'A1'0102 + D2'Dl'AlBZ - BZ'Dz. C'A
= AZrC,'A1A2
In the present connection only the diagonals of these matrices are of interest, and in particular only their residue classes mod2. The third and fourth summands are mirror images and may be omitted. Further, we call to mind the fact that a symmetric matrix M = (mij)with integral coefficients and even diagonal entries mii has the property that, for integral values of the variables, its corresponding quadratic form m i j x i x j gives only even values, and that this property in a quadratic form already implies that all the mij are integral and the mii are even. Applying an integral linear substitution with matrix N , the transformed form will again yield only even values. Thus, the diagonal entries of the matrix N'MN are again even. For this reason one can replace C,'A, and B,'Dl in the above equations by the diagonal matrix formed with their diagonal elements. Using natural abbreviations this yields
1
and another congruence of a similar sort. As u2 = u mod 2 holds for every rational integer, one can rewrite these congruences by making column vectors
tt
= C t l M 2= r
t p 2
+rt,
As 0" is characterized by (L = 0, we can immediately see that 0" is a semigroup. However, we shall hereafter produce a system of generators of 0"whose inverse elements also belong to 0".Together this gives: 0" is a group. Taking this result for granted, (2) or M' = JM-'J-' imply that M'belongs to 0"along with M. We now set tM = tLt,& = qLt ; these are column vectors whose components are the residue classes of the diagonal entries of BA' and CD' mod 2. Further, set C M = (tMqM). With this, the last equation can be expressed as CM = C M , M 2 = C M , W + CMI , (9) in which form we shall apply it later.
38
APPENDIX TO CHAPTER I. THE THETA FUNCTION
The group r"is generated by the matrices U , ,J , Tsof (4), with unimodular V and integral symmetric S. The group 0"is generated by these U,, J, and T,, where the diagonal elements of S are even, in addition to the elements 1
J v = ( E , Ev -E
E -Ev
Ev=
, v = l , 2 ,..., n - 1 .
1
0
E, has v ones in the diagonal, and zeros otherwise. J: = E. Theproof dates back to Witt.7 Say that a given element (i i)of r" or 0"has been brought into such a form by left multiplication with permitted U,, J, and T,, that the product of the g.c.d. of the first columns of A and C is as small as possible while the first is not smaller than the second: (ell, ..., cnl)
g.c.d.(a,,,
..., a,,)-g.c.d.
g.c.d.(a,,,
..., a n l )2 g.c.d.(cll, ..., cnl).
= min,
Then multiply from the left with a suitable unimodular U , , so that the first column of C contains only zeros, with the possible exception of c1 If cI1# 0, continue to multiply from the left, with a T s . If S = ( s i j ) , then a,, goes into a , , cl1sil.Suitable choice of the sil will give laill < lclll for all i and then also g.c.d. ( a r l )< IcI1l,which, however, contradicts the assumption made. When dealing with O", one is limited to an even number for sll.Nevertheless the same argument holds with one exception: = ... = anl = 0, while a, is divisible by cll but not by 2c11.But then the determinant is divisible by cI1,which must therefore be & 1. Further, C'A has the element allcll as entry with subscript 11, and it is odd. But this cannot occur for an element of 0". Consequently, under the assumptions made the first column of C consists of zeros only. By multiplying from the left with a suitable U , , one can bring the first column of A to be 1,0, ..., 0. Applications of the first and last of Eqs. (3) now shows that the first row
,.
+
t Eine Identitat zwischen Modulformen zweiten Grades, Abh. Math. Sem. Univ. Hamburg 14,323-337 (1941).
$1.
39
THE SYMPLECTIC GROUP
of C is 0, ..., 0 and the first row of D is I , 0, ..., 0. Left multiplication by Uv with 1 d , , *.* d,,
! :::
v = [o
0 0
...
pj
brings the first column of D to 1,0, ..., 0, while the already fixed rows and columns remain invariant. Finally multiplication from the left with a suitable T, makes the first column of B into 0, ..., 0, leaving all of C, D and the first column of A unchanged. This works not only for r",but also for 0". For in the latter case, ;( i)E 0"together with the normed form of D implies that b,, is even; b,, is made into 0 be setting s,, = -bll. Applying the second of Eqs. (3) shows that the first row of B consists only of zeros. And by the third of these equations the first row of A is 1,0, ..., 0. If n was 1, then the remaining element M is the unit matrix, and the proof is finished. Assume the assertion to be correct for n - 1 in place of n, and cross out the first and (n + I)th columns and rows of M. The remaining matrix M ( " - ' E ) r"-' or CY"", respectively. It can be represented by the respective U ( " - l ) ,J("-'), T("-'),and By completing these matrices to 2n-row symplectic matrices by annexation of two further columns and rows of the forms I , ..., 0, 0, ..., 0, and 0, ..., 0, 1, ..., 0, one arrives at a representation of M as desired, at least in the case of the group 0". In the case of the group r" one must still demonstrate that the matrix J1, arising from completion of J("-'), can be represented by the given generating elements. This is done by verifying the simple identity J,
E, as in (9) for v
=
1.
(10)
= -(J7--,)3
7 (We shall need (10) later for v > 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
APPENDIX TO CHAPTER I. THE THETA FUNCTION
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,.
APPENDIX TO CHAPTER I. THE THETA FUNCTION
=/.:*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
APPENDIX TO CHAPTER I. THE THETA FUNCTION
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
APPENDIX TO CHAPTER I. THE THETA FUNCTION
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
APPENDIX TO CHAPTER I. THE THETA FUNCTION
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
11. IDEALS AND DIVISORS
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, $131. Berlin-Gottingen-Heidelberg, 1959. 1A local ring as such is defined as an integral domain having a single maximal prime ideal.
64
11. IDEALS AND DIVISORS
both certainly holding if a = 8,we write (a) = (8);(2) if (a) 5 (8)and (8)I( y ) then (a) 6 ( y ) ; (3) if (a) 5 (8) and y # 0 then (ay) S (BY); and, finally, (4) 0 consists of all a with (a) S (1) and a = 0. A finite ideal a has a largest element. Assume this to be false, and there to exist an infinite series a, with (a,) < i.e., (a,) 5 (a,+1) but not (a,). For no subscript n, then, can we write a, as (a,+ 1) a,, = ylal
+ + 9..
yn-lan-l,
7,
E 0,
for this equation would imply
+ + 7.-
1 = yl(al/aA
I(@,tlan).
By hypothesis a,/a, = ( ~ , / a , +m--a,,-l/a,, ~ are contained in 0, but not their inverse elements. Thus, the a v / a n E p ,yet 1 cannot be in p. We have thus shown that a cannot be a finite ideal, which is a contradiction. For a maximal a E a we have (8) 5 (a) for all B E a, which means that 8 E oa. Thus, a = oa, as maintained. 7 The discussion also implies : The finite o-ideals form an ordered group under the definition oa 5 08 if(a) 5 (B). Exercise. The triangle inequality o(a + 8) 5 max (oa, OD) holds; i.e., the mapping a -,oa is a valuation of k. A special local ring o will be called discrete if its prime ideal p is finite and if to every a E k there exists a power of p such that uph E 0 . Taking the least power h, one has ap" = 0 . In fact, let p = on. Were ap" # 0, then ap" G on and an-'ph = a$-' 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
',
= a.
(6) (7)
Because of (3) we have oKIL(D&La-la) = D,; thus DzlLa-' c On the other hand, oK,L(DKa&La) = DL so that agILac DZlL. Together this yields (6), while (7) is proved in the next section. 2. DIFFERENT AND DISCRIMINANT The ideal inverse to the pseudocomplement is called the pseudodifferent;
bKIL= D&'.
(8)
From (4) we carry over the transitivity formula b K / k = bK/LbL/k
The norm of the pseudodifferent is called the pseudodiscriminant:
(9)
I
I I
11. IDEALS AND DIVISORS
76
It is an ideal in L. The transitivity formula
follows from (9) and §3,(4) and (7). It is quite analogous to Eqs. 1,§4,(7) and (I l), and stands in connection with these. In fact, we have the theorem: If DK has a basis with [ K : L ]elements mi with respect to O,, then DK/L
(12)
= DLA(wI),
that is, it is the D,-ideal generated by the pseudodiscriminant of the m i . Proof. As was seen in w,l, the complementary basis generates the pseudocomplement. This basis can be computed as mi* =
cipJ
with (cij)(o(oimj)) = (aij) = unit matrix,
by 1,§1,3. From the definition of the norm in $3, then
as was to be proved. 1 For separable K/L, choose the tracefor the pseudotrace. Then sK/L(DK) c 0 , , * and thus DK/L 2 D K . A s a consequence the diferent and discriminant are integral ideals. As in §2,4, to all prime ideals p of L we form the p-components
ap = DLPa of ideals of K. The following theorem then holds: The pseudocomplement of the p-component of'an ideal a is the p-component of its pseudocomplement : * (a,)&, = ( a K / d p * (13) Proof. The definition of pseudocomplement gives a;/,
2
n (ap);/L. P
Using (g) and (h) of §2,4, the right side is an QK-ideal with p-components (ap)ZL.Taking p-components on the left as well, one has ("&L)p
2 (ap);/,
(14)
*
On the other hand, the definition of p-components shows that (a;/L)pap = a&LaO:p ; taking pseudotraces this is &G,L)pap)
= DLP
*
The definition of (ap):/, therefore assures (ajf./L)pc with (14), gives the desired equation (13). 1
which, together
$4. THE COMPLEMENT, DIFFERENT, AND DISCRIMINANT
77
We can now prove (7) as follows. The equation certainly holds for all ap, as these have bases with respect to DLp. By (13), then, ((at/L)&L)p
= ((at/L)p)&L
= (
which in turn gives (7).
3. THEDEDEKIND DISCRIMINANT THEOREM Let K now be a separable extension of L. Ler the decomposition of the prime ideal p of L in K into prime ideals of residue class degrees fp be given by $343). The discriminant D K / L is divisible by p at least to the power f l ( e l - 1) + ... +f,(e, - 1). Furthermore, B K / L is always divisible by p if one of the residue class fields O,/p is an inseparable extension of the residue classJield DL/p. Proof. From (13) it is known that the p-components of the different b K / L are the differents of DKpwith respect to DLp. This, in turn, implies the corresponding statement concerning discriminants. The assertion can therefore be restated identically for all the discriminants of DKpwith respect to DLp By (12), these can be computed as discriminants of bases of DKpwith respect to DLP' To compute these discriminants a special basis of DKpwith respect t o DLpis constructed for each p. It will be used again in 111,§2,4. Using the decomposition of p mentioned in the theorem, §2,2 gives
.
nKp = n D ~. ~
,
P
Using the elements np given in §2,2, we first form K ~ ,= n;(n;'
. a -
..., ep - 1 = eKIL(p0) - 1,
I I ~ K ~ ~ Q )j ~=, 0,
(16)
where the exponent h > 0 will be fixed later. Then, lec AQi ( i = 1, ...,& = fK/L(pp)) be a system of elements of DKpp, such that their residue classes with respect to its subfield DLp/pp.We mod ppp form a basis of DKpp/ppP maintain that the products (17) A p i ~ o j , e = 1, ..., r ; i = 1, ...,fp; j = 0, ..., ep - 1 form a basis of DKpwith respect t o 0L p . By first choosing h sufficiently large, it can be assured that the elements (17) lie in all the DKp,,and thus, by (1 5), in DKp.Because of $3414) there are exactly [ K : L ] elements in (17), as is required. To demonstrate the basis property, start with an a E DKp. To each index e there are elements apl0E DLpsuch that a=
1apio~,iK,omod I
Ppp
-
11.
78
IDEALS AND DIVISORS
Remembering what the rcQIare, we have ( a that new elements aQilE O,, can be found with
1uQioAQiKQo)K;:
EO
K ~SO ~ ,
The same argument applied ep times yields the congruence u= i.i
apijAQiKQj mod pi;
.
Simultaneously, remembering definition (16), we have for h > e, 0=
c a,ijAQi~Qimod P:;~,
Q
#
e.
iJ
Finally, with §3,(8), both congruences together yield a
=
uQijAQiKQ,mod OKpp. Q.1.j
It is to be noted, in this argument, that the uQijare uniquely determined mod pp . This assures the linear independence of the elements (17) mod pp, and then certainly the linear independence as such. If 01, is a basis of O K p with respect to O,,, then a system of linear equations
holds, where c, cv,pijE O L p ,c being the common denominator. If c were divisible by pp , multiplication would give c v , Q i j A Q i K p j = 0 mod OKppp, contradicting the linear independence mod pp . Thus, indeed, the A Q i K Q j are a basis of OKp. We now arrange these basis elements, according to the index e, into r groups, each containing,f,e, elements. The product of any two, with differing e, is, because of (16), divisible by pph. By increasing h we can increase the powers of pp by which the elements of the discriminant matrix are divisible so long as they are not in the r boxes with fQeQrows along the diagonal. The consequence is that the power of pp by which the discriminant IDK/L,p is divisible is the same as that which goes into the product of the fQeQrow subdeterminants
1
) S K / L ( A Qi K p j A Qi ' K p j ' ) l ,
e=
' 9
" ' 9
"
We now show that sKIL(aKoj) 3 0 mod pp for j > 0 and all a E OKp,This shown, all rows and columns of the above matrix would be divisible by pp , except for the.f, first ones. This yields the first assertion. For this last argument we use the method of 1,§4,1 to compute the traces: CLKpj.AaiKui
=
c As'i'Kb'l'aa'i'l',,iI.
u',i',l'
The he, by fQeQ submatrices of the matrix (a,,j,l,,,il)which stand along the diagonal are almost all = 0 modp,, except those of index Q = e ;
$5.
79
DIVISORS
the other elements are not interesting in connection with the trace. The exceptional submatrices are of the form
1; p 3, 0
...
0
...
0
jjj jij
...
...
where 0 stands forS, row matrices divisible by p. a d * for certai matrices with elements in DKp-, * taking thejth place in the first column. This completes the last argument of the proof. In proving the second assertion of the discriminant theorem only the case where all eQ= 1 need be considered, for otherwise the discriminant is divisible by pp anyway. By the hypothesis, let nC)Kpp/pppbe inseparable over DLp/pp. This makes the residue class of ~ s K , L ( A ~ ~ A ~mod ~ , Kpp ~ ~ the ) ~ discriminant of DKpp/pQp with respect to DLp/pp, that is 0. T Another statement concerning the power of p by which the discriminant B K / L is divisible will be arrived at in $6. It is not difficult to rearrange the above considerations so as to get information concerning the powers of pQ by which the different bKIL is divisible. For the Dedekind diferent theorem which results see Hassef where, however, a different method is used. The theory of the different can be carried over to more general classes of rings, as was done in papers by E. Noether and E. Kaehler, which introduce two independent new notions of the different. In our special case they of course coincide with the Dedekind different. For the latest results see Berger and Kunz.$ $5. Divisors 1. THERATIONAL FUNCTION FIELD
In $5 we collect what has been said in 1,$3 and II,$2 concerning the local point of view, and complete the development of this tool. We first illustrate our procedure using the example of the rational function field k = ko(xi) = k 0 ( X l , ...,x m ) . With each of the places p of the introduction, i.e., the prime polynomials of i and p = 00 a prime divisor p = 3g is formally associated. These prime
t H. Hasse, Zahlentheorie, p. 310 ff., Berlin, 1949. 2 R. Berger, Uber verschiedene Differentenbegriffe, S.-B. Heidelberger Akad. Wiss., Math. Nat. KI.pp. 3-44 (1960). E. Kunz, Die Primidealteiler der Differente in allgemeinen Ringen, J. Reine Angew. Math. 204, 165-182 (1959).
80
11. IDEALS AND DIVISORS
divisors are taken as the generators of a free abelian group f, whose elements are called divisors, and, by definition, are power products
of prime divisors. The exponents or order functions vP(a) are all supposed to be 0, with a finite number of exceptions. The multiplication of divisors implies addition of the exponents. The unit divisor is written a = (1). A divisor is said to be integral if all the v,(a) 2 0, and a is called divisible by b if ab-' is integral. These pure formalities being set, let f = f ( x i ) # 0 be an element of k and f = p;' ... PP its decomposition into prime polynomials p i of i = k,[x,, ..., x,]. From now on G(f) will denote the degree of a rational function in the sense of the variables x i . With f associate the principal divisor
(f)= 3;:
3232
with v, = -G(f).
(2)
In particular, for a prime polynomial p we have
(P)= 3pam- G ( P ) . The exponent v , with which pmoccurs in (2) can be interpreted in yet another way. The principal ideal domain i, has only one prime ideal i,(I/y), where y is any first degree function. All i,-ideals are powers of the ideal, and in particular i , j = im(l/y)-G(f), asfL-G(f)is of degree 0 and must therefore be a unit of i, . This lastequation brings p = co into line with the other places, for they also satisfy The divisor associated with the product of two functionsf, g is clearly the product of their divisors, Us)= (f)(s)(3) Hence the principal divisors form a subgroup lj of f. It is isomorphic to the factor group k x/ k , of the multiplicative groups of k and the constant field. Each divisor is assigned a degreeg(p) by settingg(3,) = I and g(3p) = G(p), the degree of the prime polynomial with which 3p is associated. For a general divisor one can then define s(a) = v,(a)s(p). (4)
c P
These degrees satisfy
d a b ) = s w + db).
(5)
$5.
81
DIVISORS
Principal divisors are always of degree 0, as follows immediately from (2). Conversely, every divisor of degree 0 is a principal divisors. To prove this, let the divisor on the right side of (2) be of degree 0. Then, - v, = vig(3,,), so that the divisor is a product (p:‘) of principal divisors:
1
n
In $3 the concept of divisors will bc carried over to algebraic number and function fields. There, however, the simple distinction between principal divisors and other divisors, expressed by the degree, will not generally hold. 2*. PROJECTIVE. INVARIANCE The question as to the generality of this notion of divisor becomes natural. The generating indeterminates x i of the field k are not distinguished by any inner properties. They might, for example, undergo a projective transformation aio + a i l x l aimxm x.’ = aoo aoIxI aOmxm The inverse transformation is
+ + + +
+
where pij is calculated from the matrix identity (aij)(Bij)
= (aij)*
For later use we point out the special relation (a00
+ ~ O I X +I + a O m X m ) ( B 0 0 + B O ~ X I+’ .**
* * a
+ B o m x m ’ ) = 1.
(8)
We will show that the integral domains i, permute under the projective transformation (6). The divisors are thus projective invariants of the generators x i of the ,field. For the case of functions of only one variable x , it is easy to observe that x’ = ( a l o + allx)/(aoo + ao,x) are the only functions which also generate k. If m > 1 indeterminates are given, there are other “ birational transformations” such as xI’= x l , x2‘ = xlxz. The divisors are thus birationally invariant for m = 1 only. For the one variable case though, with which this book is primarily concerned, an essentially further reaching invariance of the divisor concept will appear in $ 5 3 . Under (7), a polynomial f ( x r )of degree G becomes
f ( x J =f’(x;)(boo
+ B o l x l ’ + + BOmxm’)-G a * *
(9)
82
11.
IDEALS AND DIVISORS
with a polynomial f’(x,’) of degree G’ 5 G. Replacing the xi’ by their values from (6) gives
+
ff(xif)=fff(xi)(aoo aolxl
+ + aOmxm)-G’
(10)
with a polynomial f”(xi) of degree G“ 5 G’. If f(xi) was a prime polynomial, then (8), (9), and (10) show that only two possibilities exist: ( I ) f” =fand G = G’ = G; (2) f” = constant,f= aoo aolxl
+
+ -..+ aomxm,G = 1, G‘ = 0.
In the first casef(x;) is also a prime polynomial in the xi’. For, if
f ’(xi7 =fl’(xil )fi’(xi’1 with two factors of degrees G I f ,G2‘ > 0, and the equations corresponding to (10) were fl’(xif) =f;(xi)(aoo aolxl
+
+ f2’(xi’)=f;(xi)(aoo + aolxl +
.-.)-G2’,
the first of the two possibilities applying in both cases. This would necessitate .f;’f; =f”=f, the degrees of the factors being G’; = GI1,G; = G2‘, contrary to the assumed irreducibility off. These considerations show that every prime polynomial, with the single exception of f(xi) = aoo aolxl ... ao,,,x,,,, corresponds under (9) to another prime polynomialf’(x,‘) of the same degree. The rest is obvious, for if f=f(xi) = (aoo aolxl + aomx,)yop,(xi)vL~~~
+
+ +
+
+
is the decomposition of a function into irreducible factors, and pl’(xi‘), ... are the corresponding prime polynomials in the xi’, then f = pl’(xi’)yI
... (Boo + f l O l X l f + ... + ~OmnXm‘)-vO-vIG(PI)-..’
is the decomposition of the same function into irreducible factors in the xi’. Thus the integral domains i, and ip,, correspond for corresponding p, p ‘ , etc., and further i, and i,, correspond when p = aoo + aolxl + .... Finally, i, and ips also correspond for p‘ =/loo BO1x1’+ .... Hence even the exponents remain invariant, which is clear for all p except 00 and is true also for this exception because the degree of the divisor off is 0. This is exactly the assertion that the divisors are projectively invariant. 7
+
3. DIVISORS IN ALGEBRAIC NWER
AND
FUNCTION FIELDS
Let K be a finite extension of the rational fields k = Q or ko(x,). The i, are the same as in the introduction, and the 3, are the principal orders of K
83
$5. DIVISORS
with respect to the i,. Let the prime ideal ( p ) = i,p of k decompose in K as ( p ) = 3 , p = p;'
*.*
p?
into prime ideals pQ.As before, choose an arbitrary function of degree - 1 for the element p of the place p = co. With every prime ideal pp, for all places p of k, one associates formally a place pp and a prime divisor pQof K . Introduction of new special designations is unnecessary. The places pQof K are said to lie over the place p of k. The geometric import of this convention in the classical case will become apparent in III,§2 and IV,§l. As before, form the free abelian group R of all divisors from the prime divisors. If u # 0 E K, and if for each p 3,a = p;pl(a).
.. p,~~r("),
we associate with the element a the principal divisor
where the product is taken over all prime divisors, or, in other words, all prime ideals pQfor all p . Assurance is still needed that only a finite number of the exponents vp(u) in (1 1) are nonzero. By §2,4,(c) almost all the 3, contain u, so that vp(a) 2 0, almost always. The same argument applies to u-l. The principal divisors form a subgroup 8 c R. By its definition it is a homomorphic image of the multiplicative group K " of K. The importance of divisors is, that they permit the global statement of the divisibility properties for all places p , as they are expressed inideal theory for 3,. Moreover, it will turn out that for the case of fields of functions of a single variable the divisors represent a concept which does not depend upon the particular model of the field K. For the case of algebraic number fields K / Q , though, divisor theory cannot yield more than ideal theory for the principal order 3 with respect to i = 2, since ideal theory of 3 already contains the ideal theories of all the 3,, as 3 = 3,. At times it is practical to split up the product representation of a divisor in the manner
n
i.e., the places pp of K lying over a place p of k are grouped together. This procedure will furnish a bridge to the concept of a linear divisor of 143. Let us now investigate the nature of the homomorphic mapping K" -P jj somewhat more exactly. We have !?j E K"/ci,
84
11. IDEAL3 AND DIVISORS
where (F is the group of those E E K with principal divisor (8) = (1). These e are called the units of K . For the case of a number field, these units will be considered more closely in the appendix to this chapter; they form a finitely generated group. The result for a function field is quite simple: c4 = k O X , as long as we assume that k, is the exact constant field. Proof. That ko E (F is obvious. Now let E E (5 and thus be a unit in all 3,. If k = K , then clearly e E k , . Otherwise, let en c1&"-l c,, = 0 be the equation in k which e satisfies. By §l,l, E E 3, if and only if all the ci E i,. This, as we have just seen, is equivalent to ci E k,, if it is to hold for all p. Thus E belongs to the exact field of constants which, by hypothesis, is ko 7 A consequence of the above is: an element is determined by its divisor, up to a factor from the exact constant jield. In the proof we also showed that for a function field
+
+ +
.
ni,=k,,
n3,=k0.
P
P
The distinction from I§3,( 1) should be noted. The elements of the factor group
a = R/fi
-
are called divisor classes. Divisors of the same class are called equivalent, written aa = b a. Automorphisms S of K / k act on the divisor by their action on the ideals. As (a') = (a)' is obvious, the S are also automorphisms of the group of divisor classes. 4. THE BEHAVIOR OF DIVISORS UNDER FIELD EXTENSIONS Let an intermediate field L between k and K be given. The group 2 of divisors of L is embedded as follows into the divisor group R of K . Say a prime ideal p for 3, = 3L,p of L decomposes in K into Z K , p P = p;'
**'
p:.
(13)
Proceed by identifying the divisor of L with the divisor in K of the same notation, p = p"1
... P?,
(14)
and expand this embedding into an isomorphism of 2 onto a subgroup of R. It is unnecessary to distinguish between a divisor in 2 and its image in 53. This embedding, which is formally possible, carries with it an embedding of the group fir. of principal divisors of L into the corresponding group !ijKof K . It is to be expected that this embedding corresponds with thenatural
$5.
85
DIVISORS
embedding of 4 j L Z L*/kOXinto !jjK z K X / k , ". Indeed, this is so. For, if (a) =
n
pvda)
is a principal divisor in L, and the decompositions (14) are substituted for the p, its image in K is (a) =
nn P
pYp(').
a
Considering (a) as a principal ideal in L and K , however, we also have
3L P a =
n
pVp("),
3K,pa=
nnp : p ~ ~ ( ~ ) , P
P
where here the pP simultaneously denotes the prime factors of (p) in 3',, and these p are decomposed in 3,,, according to (13). Thus, here also, divisor theory can be seen as the formal global summarization of ideal theory of the individual places p . In embedding !jjLin Sj, one must carefully avoid the error of thinking that !jjL = f! n $, . It is well possible for a divisor of L that is not principal to become a principal divisor in K . The residue class degree and ramification index of a prime divisor p E Si with respect to a subfield L are fixed as the same numbers as for the corresponding ideals. Further, the norm nKIL(a)of an a E R of the form (I) is nKIL(p)"P("),and the norms of the p, in turn, by defined as the product setting them equal to the ideal norms of 53; they are taken as divisors in L. The product formula 8344) holds, and in particular,
n
nK/L((a))
= (n,/L(a))
(15)
the norm of a principal divisor (a) is equal to the principal divisor of the element norm. In the case of a function field a degree g(p) of a prime divisor is defined as follows. Say p corresponds to a prime factor in K of the ideal 3 , p of residue class degree fKlk(p), p being an irreducible polynomial in the generating variables. Then, set g(p) =f,,,(p)G(p), where C(p)is the degree as polynomial, for p # 00. The case p = 00 is handled similarly; p corresponds to a prime factor of 3m(l/y),y being some rational functions of the first degree. Thus use G(m) = 1. Define the degrees of composite divisors (I) by again using (4). Equation ( 5 ) still hqlds, and the divisors of degree 0 form a subgroup. The degree of a divisor was here defined with respect to the field K . If one is dealing with a divisor of a subfield L, one must distinguish between the degrees gL(a)and gK(a)of a as a divisor of L and K . Every divisor a of K satisfies gda) = gL(nK/L(a))*
(16)
86
11. I D W AND DIVISORS
Clearly we need only demonstrate this for prime divisors. Let, therefore, a = p K , and pL be the prime divisor in L divisible by pK . On the one hand, §3,(11) gives nKIL(p) = pLJKIL,and on the other, by definition and $3,(12), the degrees of p K ,p L are g K ( P K ) =fK/LfL/kG(P),
gL(PI.1 = f L / k G ( P ) ,
which proves (16). A consequence of (15) and (16) applied to L = ko(xi) = k and the fact that the principal divisors of k have degree 0 is the important theorem: The degree of a principal divisor is 0 ; divisors of the same class have the same degree. Formula $3,(7) carries over word for word to divisors. From it, in connection with (16), we have the important relation: the degree of’a dioisor of L in K is [ K :L] times its degree in L. 5. THEPRIMEDIVISORS
The introduction of divisors concludes the common treatment of algebraic number and function fields, so that it now seems fitting to compare these two sorts of fields and consider them in relation to each other. As mentioned in $53, the group of divisors of an algebraic number field is isomorphic to its group of 3-ideals. The reason for this can be said to be, that the intersection of all the integral domains 3,for all primes p of i is equal to 3. This is no longer so for function fields. There, indeed, we still have
3=
n 3,,
P+ w
but by (12) the intersection of all the 3, is the exact constant field. As a consequence, there is no integral domain whose ideal theory can reflect the properties of all the 3,. Thus the divisors are needed to sum up globally the divisibility properties of all the places. There is another difference between number and function fields. In contrast to number fields, function fields can be generated in various m?nners. Any maximal set of algebraically independent functions x i in K defines a model (cf. §2,3). K is a finite algebraic extension of the rational function field ko(xi). For fields where the number of variables m > 1, it was shown in $5,2 with an example that our notion of divisor depends upon the generation of the field. In other words, the divisors are not invariant under birational transformations. For fields in one variable we prove the following:
Invariance Theorem. The prime divisors p of an algebraic function jield K of one variable with constant field ko correspond, one-to-one, with the special local
55. DIVISORS
87
rings 0, of K (which contain ko)in such a manner that 0, consi5ts of all those functions, u, whose principal divisors do not contain negative powers of p. The residue class rings op/pp are extensions of degree g(p) of the exact constant field. The first assertion follows directly from the characterization of local rings in §2,2 and §2,3. Now let x be some nonconstant element. Then K is a finite extension of k = ko(x). Say the prime divisor p divides the prime divisor po of k, and assume, for the time, that po is the numerator divisor of a prime polynomialp(x) of degree g o . Further, say p divides po with residue class degree f. Then the residue class ring i/p(x) is an extension of ko of degree g o . By §3,3, op/ppis an extension of i/p(x) of degree f, and thus an extension of ko of degree g(p) = gofi For the exception, where po is the denominator divisor of x, replace i by i, in the above argument. 7 In this connection we incidentally mention another possibility of defining divisors, based on §1,3. We limit ourselves to the simple case in which K is generated by simple adjunction to a rational subfield. (This generation is always possible if k , is perfect, as follows from 111,#,l. The generalization possibilities of the following considerations lie close at hand.) Let, therefore, K = k,(x, y ) with an equation f ( x , y ) = 0. Using three " homogeneous" variables t, q, [, set x = t[-', y = q t - ' . A homogeneous equation F(& q, [) = 0 then holds, and it defines a finite extension Kh over kh = ko(t, [). Also, set oh = k o [ ( ,(1. Now adjoin another indeterminate z and define, as divisors of K, the cosets ~ ( Z ) & , ( z ) r where u(z) are functions of K,,(z), homogeneous in t, q, and c, while ah(z) is the group of units of the principal order T ) h ( z ) of Kh(z) with respect to oh(z), homogeneous in q, and [. The divisors in this and the previous sense must be shown to correspond one-to-one. Let n(z) E &(z) be a homogeneous prime element. A local ring On,,, c Kh(z)corresponds to n(z). Certainly On(z)(nz) containsa homogeneous polynomialf(5, [) E k0[& [I. This means that n(z) maps the rational subfield k o ( ( [ - ' ) onto a finite extension of k o , and thus does the same t o K. Hence n(z) belongs to a prime divisor p, with local ring 0,= Dn(z)n K. Conversely, let a prime divisor in the previous sense, p, be given. We may assume that p does not divide the denominator of x, for otherwise we could use x - ' . Now, a prime element no(z), integrally dependent upon i(z) (i = ko[x]), corresponds to p . Choose a power of [ such that n(z) = no(z)t' is integrally dependent upon o,,(z). We will even show that if I is minimal, then this element is also a prime element. Once this is known the first correspondence again gives p, so that we are finished. n(z) is homogeneous. Represent this element as the quotient of polynomials, the denominator being a primitive polynomial of oh[z]. After any possible cancellations, homogeneous functions remain in the denominator and numerator. If the denominator is primitive, the prime decomposition of an element amounts to the prime decomposition of the numerator. Now, it is
r,
88
11. I D W S AND DIVISORS
very easy to see that the decomposition of a polynomial in z homogeneous in <, q, and [ gives homogeneous factors. Then, if contrary to our assertion, ao(z)cl= a(z)/?(z)with functions a(z),/?(z)of degrees a, b, homogeneous in c, q, c, and integrally dependent upon oh(z),we should have a decomposition no(z)= a(z)c-"*/?(z)[-b into factors integral at all places of K not divisors of the denominator of x . But, because of the relation of nO(x)to the prime divisor p, any decomposition of this sort must be trivial. Thus, one of the factors is the product of a unit with a power of c, which is what was to be shown. 6. DIVISORS AND LINEAR DIVISORS
The time has come to clarify the relation to the concept of linear divisor introduced in I,§3. Our starting point is 42 of this chapter, but specialized with k as rational number or function field and op = i, ,0,= 3,. An arbitrary basis oi of K/kis given. By associating with every a = C aioi E K the vector (al),we have a one-to-one mapping K o k". If a pseudotrace r ~ is ~given, , ~ then K is simultaneously mapped onto the dual space k"*, as in §4. We now decompose an arbitrary divisor a to a product
a=na,, P
whose factors a, consist only of power products of the finitely many p p lying over p , in the sense of §5,3. As became clear there, the groups R, of these ap are isomorphic with the groups of 3,-ideals carrying the same notation a,, the latter being exactly all a E K with exponents vp,(a) r v,&) = vp,(a). These 3,-ideals a, are simultaneously finite i,-modules of rank n = [K:k ] . They are the p-components of a linear divisor in the sense of 1,§3. All we need show for this is that the above field basis oi forms a basis of a, with respect to i, for almost all p . We show this first under the assumption that Kis a number field or function field of one variable. By §1,1 the intersection 3 of all 3, (except p = 0 0 ) has a basis i I consisting of n elements. Set o,= C iici,; the matrix (c,,) is unimodular for almost all p . Furthermore, as components of an %deal the a, are almost all 3p (§2,4,(d)). Thus, the w i form a basis of a, for almost d p . 1 Using the basis theorem of 1,§1,5, one can carry this argument over to function fields in two variables. We shall not need this theorem for functions of several variables, and therefore omit the proof of the general case. It can be carried out by use of the methods of §1,1 and §1,2, under the same limiting assumptions made there in the case of inseparable field extensions.
$5.
DIVISORS
89
The linear divisors corresponding to equivalent divisors, a and act, are equivalent in the sense of 1,§3,1. For, if a, has the basis api
=
Ci ap,ijwj
with respect to i,, then apia is a basis of au. Setting mi =
C sip, i
we have apiu =
c
a,,iiQpl*
191
The following theorem is important in later applications. There exists a divisor b K / k , such that the linear divisors corresponding to an arbitrary divisor a and to a* = b - l a - ' Klk
(a** = a)
(18)
are complementary in the sense of 1,53,1. We call b K / k the pseudodirerent divisor with respect to k ; its p-components b K / k , p , taken as 3,-ideals, are the pseudodifferent ideals of 3, in the sense of $4. Our assertion is quite the same as formula $4,(6), applied in the case L = k to the components of the divisors and using the abbreviation of $443). For separable K/k,we can use the trace as pseudotrace, b K / k is then the diferent divisor, and the correspondence a t ) a* is unique. For inseparable extensions K/k,(1 8) does not define a unique pseudocomplementary divisor. A t least, though, the class of a* is uniquely determined by a. The distinction between two pseudotraces 0 , 0' is, in fact, that a'(a) = ~ ( d u ) for all a E K with some fixed 6 # 0 E K, according to 1,$4,4. For pseudocomplements this gives a*' P = a,*d-'.
And, for the divisors defined by (17), we then have a*' = a*d-'.
7. THELINEARDEGREE The Riemann-Roch theorem for linear divisors of I,§3,3 can now easily be formulated as a theorem for divisors. As it will turn out, it is the central theorem for the theory of algebraic functions of one variable, and therefore its formulation will belong to Chapter 111. Nevertheless, it is practical to calculate the linear degree G(a) of a divisor at this point. Let K be a function field in one variable.? Fix a basis w i of the principal
t All we really need know is that 3 has a basis with respect to i.By a remark in the last section this also holds for function fields in two variables.
90
11.
IDEALS AND DIVISORS
order 3 with respect to i, and use this basis for the computation of the linear norm according to the procedure prescribed in 3 §3,2. For all places p # co of k the p-components of N(a) and of n K / k ( a ) are the same, as follows immediately from §3,(6). Further, if om( = c i j o j is a basis of 3mwith respect to i, , the same reasoning shows that, for the a-component,
1
N(a>m = (nK/k(a)lcijl)m holds. It follows from I,§3,2, then, that the linear degree of a is G(a) = g k ( n K / k ( a ) )
+
with some rational integer c, which can best be computed by the use of (18) and 1,§3,(12). With (16) it is also possible to write G(a) = g(a)
+ c.
(19)
Because of (l8), the divisor associated with a by the definition 1,§3,(11), is
where b K / k denotes the pseudodifferent divisor and 3 , is the denominator divisor of (x). The degree of 3m as a divisor of K is n because of the theorem at the end of §5,4. On the one hand, this yields
g(a) + g(a') = 2n - g(bK/k)* On the other hand, we saw in 1,§3,(12) that G(a)
+ G(a') = 2n.
Comparison of these two equationsfinally gives c = hg(br/k)in (19). Altogether we thus have (20) G(a) = g ( a ) + h ( b K / k ) *
#*. Decomposition of Prime Ideals in Galois Extensions In $6 K will always beaseparable Galoisextensionofk, ofdegree [K: k] = n. The behavior of a prime o-ideal po of k is obviously completely determined by the behavior of the corresponding prime o,,-ideal, o,, being the local ring associated with p o by §2,2. As ,o, is a principal ideal domain, there is no loss of generality in assuming that o is a principal ideal domain. In $6 we assume throughout that the residue fields S)/p of the principal order 0 of K with respect to all prime ideals p dividing p o are separable
$6, DECOMPOSITION OF PRIME
IDEALS IN GALOISEXTENSIONS
91
extensions of o/po . This assumption is actually dispensable, but simplifies matters.? The theory which we will now develop is due to Hilbert. Among the most important results to be attained is a strengthening of the Dedekind discriminant theorem. 1. THEDECOMPOSITION GROUPAND INERTIA GROUP The elements of the Galois group 6 of K/k are denoted S, T, ... . Application of S to an element or set of elements is written a+aS. Application of S to an ideal a yields another ideal as. One prime ideal p of K will be fixed throughout. The totality of all S E 6 which map p into itself form a subgroup 3 E 6, the decomposition group of p. If 6 = 3S1 u u 3 S r ( S , E 3) (1) is a decomposition of 6 into left cosets, then p1 = psl = p, ..., p, = psr are the prime ideals conjugate to p. The decomposition groups of the pp are S @ - ' ~ S , . Let po be the prime ideal of k divisible by pl. We maintain that P O = ( ~ 1 Pr)',
e 11.
(2)
Proof. Let p o = p;' p7py1' p?', where the pb and their conjugates differ from the p p . This decomposition continues to hold under automorphisms in 6. Thus we already have el = = e, = e. By §2,2, there exists an a E K divisible by p1 but by none of the p: or their conjugates. Then the conjugates of c1 are divisible neither by the p i nor by their conjugates, and consequently neither is the product of the conjugates of a which is nKIL(a). On the other hand, nKIL(a)is divisible by the p p , and thus by p. Hence we have el' = = 0. 7 The inertia group 2 of p is the totality of all 2 E 6 for which
uT=umodp
for all W E T )
(3)
holds. Certainly (3) holds for o E p, so that 2 E 3. The subfields K, and K , belonging to 3 and 2 in the sense of Galois theory are called the decompositionfield and inertia field of p, respectively. We will denote the principal orders and prime ideals divisible by p, of these fields, Q 9
Q ?
PZ?
PI.
t The theory is carried out under more general conditions in 0. Zariski and P. Samuel, Commutative Algebra I, p. 290 ff., London, Toronto, New York, 1958. (Tr. note: It has become customary to use the symbols 8 and 5, standing for the German words Zerlegung and Tragheit, in English as well.)
92
11. IDEALS AND DIVISORS
The inertia group is a normal subgroup of the decomposition group. y the residue class field D/p is a separable extension of o/po, then every element of D is congruent mod p to an element of D,,so that D/p D,/p,. Under that assumption, the j e l d D/p is a Galob extension of o/po with the group 31%. Proof. Application of (3) on as in place of w with an arbitrary S E 3 yields dTS-' = o mod p. This is equivalent to the first assertion. Let T run through all the elements of 2.Again because of ( 3 , any element a E 0 satisfies the congruence, f ( x ) = JJ( x - a')
= ( x - a)['] = 0 mod
p,
T
[2]signifying the order of 2 as a group. The coefficients of the polynomial on the left belong to the inertia group; let us consider more closely the polynomialf(x) in K,. The above congruence assures us thatf(x) is a [2]th power in D/p. But, as we assume D/p to be separable over o/po and therefore over D,/p,, f ( x ) must already be a [2]th power in D,/p,. Thus, f ( x ) 3 ( x - a,)['] mod p, with some a, E D,, so that a 3 a, mod p. For the residue class fields this means that DIP Q/P, (4)
=
To prove the last assertion we consider D/p as an extension of D,/p,. For arbitrary S and T in Q and 2,and all a E 0 we have, with (3), aTS =$modp. Hence the cosets of 2 in 3 induce automorphisms of D/p leaving Dz/pzinvariant. For an S E 3, S # 2 there exists, by the definition of 2,an a E D such that a mod p. We have [3:21 of these automorphisms of D/p. In other words [D/p :Dz/pz12 13:21. On the other hand, the symmetric functions of all as, S E 3 for one a E 0 are in D,.Thus, every a E D satisfies an equation in D, of degree at most [3:21. This assures us that
+
fKt/K.(P1) = CD,/P,:Dz/Pzl = CWP D,/P,I = c3:21. Furthermore, D/p z D,/pl is a Galois extension of D,/p, with the 31%.The last assertion therefore depends only upon the proof of DZlPZ
O/PO
(5)
group
*
(6) From the definition of the residue class degree in $3,3, (4) implies f K I K , ( p )= 1. The only prime ideal of K which divides p, is p. For, all the prime divisors of p, in K are generated from p by the application of elements of the Galois Group of KIK, [cf. proof of (2)]. But this group is 2,which leaves p fixed. Now, as seen in §3,(14),f K I K I ( p=) 1 implies
eKIKt(p)= [K:K,] or p, = p[K:Krl. Taking norms of (2) according to §3,(11) yields (p)"= ( p y f ' , so that efr = n,
(7)
(8)
$6.
DECOMPOSITIONOF PRIME IDEALS IN GALOIS EXTENSIONS
93
where r = [8:31 = [K,: k]. But, as in $34 12), e is divisible by eK/K1(p)and f by fKt/K,(p,). By (7) and ( 5 ) these numbers are [K: K,] and [ K , :K,].This shows that the left side of (8) isdivisible by [K: K,][K,:K,][K,: k] = [K:k] = n, which is the right side, so that e=eK/k(p)=
[ K :K t l ,
f =fK/k(P)=fKtlkr(Pt)=
IKt :Kzl*
(9)
Equation (9), with §3,(12), yields fKZlk(pn)= 1, which is equivalent to (6). Finally (7) and the first equation of (9,along with §3,(12), yield eKI/Kz(p,)= 1, which is equivalent to Pt = Pz . (10) This completes the proof. The result can also be described with other words. The prime ideal po of k is divisible by a prime ideal p, of degree & , / k ( P , ) = 1 in K, . The other prime divisors of po in K, can have larger degree. Stepping up to K , , (10) assures us that p, remains prime: it is “inert.” But, its degree increases to f = [K,: K,]. In the extension K/K,, p , becomes the eth power of a prime ideal p in K. KIK, is called ramijied at the place p, with index e, if e > 1. But p retains the degree fK/&) =fKtlk(p,) =f. Incidentally, in any intermediate field K‘ between K, and K the prime ideal p’ there satisfies p, = p’[K’:Krl. This is because the degree fK,/Kt(p’) = 1 and the index eK./&’) = [K‘: K,], then, using §3,(14). If K“ is another intermediate field, between K and K’, we again have p’ = p”[K“:K’l.Thus, ramification does not cease once it has started. The norm introduced in $3 for ideals can be written
so as to correspond with the norm of elements. The proof need only be brought for a prime ideal. But for these it follows immediately from (1) and (2), together with the above considerations concerning the order of 3 and 2 and their connection with the degree and ramification index. Finally, assume that the residue class field o/p, has only finitely many elements. The same then holds for D/p. The numbers q and Q of them are then powers of a prime number, with, in particular, Q = 9’.
The Galois group of a finite field is cyclic, as is well known, and it is generated by exponentiation with q. If o/p, has q < co elements, then 312 is a cyclic group and there exists a generating element S such that ws = w4 mod p
(for all w E 0).
(11)
It is uniquely determined up to a factor in 2,and is called the Frobenius automorphism for p.
94
11.
IDEALS AND DIVISORS
2. THERAMIFICATION GROUPS The first ramification group w"
E
consists of all V E 3 such that (for all w E D)
w mod p2
holds. Beside the group !B1 we are interested in the greatest exponent u, ( 2 2 ) for which (for all V E 23,,w E D). w" = w mod p"' (12) ?B1is an invariant subgroup of 2 and 3. Thefactor group 2/g1is isomorphic to a subgroup of the multiplicative group of the residue class,fieldD/p. For the proof we introduce the principal ideal domain D, of all a E K without p in their denominators (cf. §2,2). All Dp-ideals are of the form OPT?"' An element 7c of this sort is called a prime element. Two prime elements differ but by the factor of a unit of D,. By §2,4(i) the residue class rings D/p and Dp/D,n are isomorphic. Comparison of (3) and (12) yields B1tz 2. If S E 3, the application of (12) to ws gives wsys-' = w mod pz. Thus 23, is an invariant subgroup of 3. If T E 2 and 7c is a prime element, then so is 7 c T , and
nT = &TK
(13)
with a unit E ~ If. further T E 2$, then nT _= n mod p2 SO that tT = 1 mod p. We want to show the converse of this: in (13) let tT 3 1 mod p. By the last section every w E D, is congruent to an w, E D,,, = K, n O,, and thus w
= 0, + w,'x mod p2
with some w,' E D,,, . Now, (3) implies wT = w,
so that, altogether, with (13) and
+ w,'nT mod p z E~
= 1 mod p, we have
oT3 o mod p 2 .
By definition, then, T E 113,. If T I ,T, E 2, then by (13),
so that
ET,T,
=
& F : E ~ and ~ , with (3),
A consequence of (14) is that the mapping of T onto the residue class of eT
$6.
DECOMPOSITION OF PRIME IDEALS IN GALOIS EXTENSIONS
95
is a homomorphism of 2 into the multiplicative group of D/p. The kernel is the ramification group Bl, as was shown. It appears plausible to define further ramification groups by strengthening the congruence (12). Say the ith ramijication group !Bi is defined by the congruence w" = w mod pui (for all w E D) (15 ) where vi is maximal. Then, define the (i of all V E Bi for which w"
+ 1)th ramijication group as the totality
= w mod p U i + l
always holds. Then choose a maximal v i f l for which w" z w mod p u t + '
(for all
I/ E Bi+ , o E D).?
(16)
An argument already used twice shows that: The ramification groups 23 are normalsubgroups of the decomposition group 3. The invariant fields K i belonging to the Bi are called the rrrmijication fields. For the prime ideals pi of these fields which are divisible by p we have, by !$,I, p., = pP"1 ,+1 with e i + l = [ K i + , : K i ]= [23i:Bi+1] (17) (here B,, = T,KO = K,). The total ramification index of p is then e = e,,,(p) = ele2 .-..
(18)
Let us yet consider the factor groups B i / B i + lfor i2 1. We know from (15) and (16) that, for a prime element TI and a V E B i , n"
= n + vynuimod
with a v,, E O,,whose residue class mod p corresponds to the element V (where TI is held fixed). If V , , V2 E B i , then nvlvr = - n + (v;: + vV,)nuimod put+'. That is using (3) and the fact that V2 E Z, vv,v,
= v v , + vy, mod
p.
(19)
This represents a homomorphism of Bi into the additive group of D/p. By definition, the kernel is B i + The factor groups Bi/Bi+lare isomorphic to subgroups of the additive groups of O/p.
,.
___-
t The U I lie below a bound that depends only upon n, cf. the book quoted on page 91 (Zariski-Samuel).
11. IDEALS AND DIVISORS
96
One speaks of regular or irregular ramijication, depending upon whether 'Dl consists only of the unity or not. K/k is regularly ramified at the place p ifand only if the ramification index e is divisible neither by p nor by the characteristic of the field. Proof. Let K/k be irregularly ramified. Then, e, = [Bl:B,] > 1. But B,/B, is isomorphic to a subgroup of the additive group of 0 / p , so that there w exists a residue class w f 0 mod p and a divisor e,' of e, such that w = e,'w = 0 mod p. Then e,' = e2 = 0 mod p, so that with (18) either p or the characteristic divides e. Let K/k be regularly ramified, i.e., Bl consist of the unity element only, and, contrary to assertion, let e be divisible by some prime q = 0 mod p. t Now e is the order of 2,and 2 is isomorphic to a subgroup of the multiplicative group of D/p. The residue classes mod p form a field, and the residue classes of 0, I, ..., q - 1 form a finite subfield f.2 because of q = 0 mod p. As the multiplicative group of D/p has a subgroup of order e and e is divisible by q, there must exist a residue class e f 1 mod p with eq = 1 mod p. Adjoining e would yield an extension of f.2 of the qth degree. An extension of this sort over a field of q elements, defined by eq = 1, is well known not to exist. 7
+ +
3. THEDISCRIMINANT We first compute the discriminants BK,/BKr and BK/BK,, where K, denotes the first ramification field. As we saw in $4, we need only the pcomponents. = K, n 0,.Thus Every w E 0,is congruent mod p to some w , E D,,, 0
= 0, -k w'R
with some w' E 0,and a prime element n in 0,. Following the same procedure with a',etc., we find a congruence w
= w, + w,'n +
+ wie-')ne-' mod pe
(20)
with w , , mi', E D,,,. We now maintain that 1, n, ..., ne-' are a basis of 0,with respect to D,,, . In fact, if w, is a basis and
one can substitute the representations (20) for the w, and arrive at a system of congruences np = c p v c ~ p xmod p pe, cip E O,,, . v,o
t This includes the case where 9 is the characteristic.
$6.
DECOMPOSITIONOF PRIME IDEALS IN GALOIS EXTENSIONS
97
The elements 1, n, ..., n e - l are linearly independent mod pe, so that we have the matrix congruence (c,,J(civ) E E mod p , . But this is only possible if the matrix (c,,") is unimodular in D1,,. Similarly we find that if n1 is a prime element of O,,, = Kl n 0,, then I , nl,..., nfl-l is a basis ofD,,, with respect to D1,,. Further, 1, R , ..., nel'-l (el' = elel) is a basis of D, with respect to D1,,. By §4,2, the p,-Fmponent of I D K I I K t is the principal ideal generated by
=
&
n
( R p
-Rp)
Q+O
where TQruns through a system of representatives for the cosets of B1in Z. By (1 3) we have here nTQ I - 1) = n & 1 - nTu 1 = np(&pTR e I T , (&pTP O I - I),
and because we assume TQT;' # Bl we know &zTg1f 1 mod p. The ptcomponent of IDKllKIis thus equal to p;l(el-l) = pF1-l. The pl-component of a,,,, is similarly found to be the principal ideal generated by (R", - n"-), (21)
n
Q+U
where V Q , V, traverse the entire group 23,. The V Q , V, are represented as products = * ' * vzQzVIQl , V# = " ' V 2 U z v l U , 3 the V i Q i ,Vial traversing a system of representatives of the cosets of S i + ] in B i . We have V,V"= v2Q*v1QlV;;V~; **' . 'Q
1.3
We here assume that VIP,= V,,, , ..., V i - l , Q i - ,- Vi-l,or-l, Viol# Vi,, . The VQV,-' then lie in 'Bi but not in B i + , ,all the subgroups occurring being normal. Then §6,2 give us; n"R - n"'J = ( x " R " U - ' - n)", ( v ~ p ~ u - I ~ u ~mod ) " Op U i + ' ; i.e., every term is divisible by p exactly vi times. Thus in the product (21), holding the V,fixed, there areexactly [Si]- [23i+l]= (ei+]- l)(e/el . . ~ e ~ + ~ ) factors equal to (7~'")"~ up to units, this holding for i = 1,2, .. . . [V,] denotes the order of S i . Remembering still that R",) = p"" = pl, it has been
(n
n
98
APPENDIX TO CHAPTER 11. ALGEBRAIC NUMBER FIELDS
shown that the p,-components of Vi(CWI
aK/K,
- [%I) -I-V 2 ( " 1 2 I - W = u,(e2
3 I )
is pI to the power
+
***
- l)(e/ele2) + v2(e3- l)(e/e,e2e3)+
.
Finally we apply the transitivity formula $4,(1I), noting that, by the Dedekind discriminant theorem of §4,3, JDKtIk is relatively prime to p whenever the residue class field O/p is separable over ~ / p This . gives the final result: Let the residue class field O/p be separable over o/p,. a K / k is then divisible where by
pt,
6 =f(k - l)(e/el) + v1(e2- l)(e/ele2)+ 0 2 ( e 3 - l)(e/ele2e3)+ .-->; (22) (18) should be taken into account. For regular ramification at p this exponent is 6 =f ( e - 1); otherwise it is larger. Appendix to Chapter 11*
Topics from the Theory of Algebraic Number Fields The theorem of the unique decomposition into prime ideals gives us a basis for the theory of algebraic numbers. The main task remains to find as close as possible a connection between elements and ideals, and this is carried out primarily in two theorems ($1): first, that the set of ideals decomposes into a finite number of classes (11,$1,4), and second, an explanation of the structure of the group of units (11,$5,3), so that the connection between elements and principal ideals becomes clear. The results of these two theorems can be interpreted as saying that the concept of an ideal leads only a finite step beyond the concept of an element. In $2 the information thus obtained is briefly illustrated on two types of fields, as important as they are simple. The space here does not permit a deeper penetration of algebraic number fields. The interested reader will find it in the following texts: H. HASSE,Vorlesungen iiber Zahlentheorie, 2nd ed. Springer Verlag, BerlinGottingen-Heidelberg, 1964. H. HASSE,Zahlentheorie. Akademie Verlag, Berlin, 1949; rev. ed., East Berlin, 1963. E. HECKE,Vorlesungen uber die Theorie der algebraischen Zahlen. Leipzig, 1923, reprinted, Chelsea Press, New York, 1948. $1. The Finiteness Theorems
1. THEFINITENESS OF THE IDEAL CLASS NUMBER
As was discussed in 11,$5,3, the divisor concept can offer nothing more for the study of algebraic number fields than the ideal concept of 11,$1,4, where o is taken to be the principal order i = Z of the rational base field k = Q. We will stay with these assumptions throughout.
81.
THE FINITENESS THEOREMS
99
Let the degree [ K :Q ] be n. By the fundamental theorem of algebra, K can be considered as a subfield of the field C of complex numbers. Let 9 be a primitive element of K and 9, = 9, 9,,..., 9, be the algebraic conjugate elements to 9. The nonreal elements among these occur in pairs, for if 9, is conjugate to 9, so is 9,. The number of real conjugates will be denoted rl throughout, the number of pairs of complex conjugates with r 2 . Then, n = rl 2 r 2 . (1)
+
These elements 9, generate n number fields K , = k(9,) conjugate to K which, however, need not be distinct. The number of ideal classes is finite. In many cases, in fact, this number is one. This means that then every ideal is a principal ideal, and every element can be written uniquely, except for units, as a product of prime elements. The calculation of the number of ideal classes or, what is more, the determination of the structure of the group of ideal classes is a difficult problem, insofar as one desires to make general statements.? The proof of the theorem consists of finding a certain finite set of ideals containing a representative of every ideal class. Let 1, be a basis of the principal order 3: of K, and A some ideal class; let it contain the ideal a. Furthermore, let @, be a basis of the inverse ideal a - l . Then there is a linear substitution I
with rational mij. By 11,§3,2, its determinant has absolute value
llmijll = n(a)-'. By 1,§2,1 there exist rational integers x i , not all vanishing, such that the ineaualities (2) hold for allj. We now set
and majorize the norm of p. As defined in 1,$4,1 it is In(P)I= If(yj)l with a homogeneous polynomial in the y j of degree n. If 2-"7 denotes the maximum of If(zj)l for lzjl 5 1, then by (2), In(P)I c cn(a)-'. Now, the ideal ab belongs to the class A , is integral, and has norm n(aB) = n(a)ln(P)l< c.
t H. Hasse, Uber die Klassenzahl abelscher Zahlkorper, Akademie Verlag, East Berlin, 1952. C. Meyer, Die Berechnung der Klassenzahl abelscher Korper iiber quadratischen Zahlkorpern, Akademie Verlag, East Berlin, 1957.
100
APPENDIX TO CHAPTER 11. ALGEBRAIC NUMBER FIELDS
There can exist but finitely many integral ideals whose norm is thus bounded, namely, those occurring in the prime decomposition of natural numbers < c. 2. THEDISCRIMINANT The discriminant BK,pof the field K is, according to 11,§4,(12), equal to the principal ideal iD(ii),where i i is, again, a basis of 3.It can be minorized ID(ii)l > (n/2)'"
2 1.
(3)
Hence, using the Dedekind discriminant theorem of 11,§4,3, we know that there are always rational prime numbers which ramify in K. The proof depends upon refining the argument of the last section. Let iiv ( v = 1, ..., n) be the conjugates of the zi. Then
Assume, contrary to the assertion, that (writing the determinant)
I/ ..-11 for the absolute value of
5 (n/2P.
(4)
IIiivlI
If the sign < holds we can choose positive numbers c, with the product
Then, by 1,§2,1 there exist rational integers xi, not all vanishing such that
holds. The norm of
< = 1xiii is then, taken absolutely,
If (4) is an equality let c, = cvo+ t with positive t and positive cvowhose product is 1. ( 5 ) defines a convex body changing continuously with t, and for all t > 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
APPENDIX TO CHAPTER 11. ALGEBRAIC NUMBER FIELDS
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
APPENDIX TO CHAPTER 11. ALGEBRAIC NUMBER FIELDS
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
APPENDIX TO CHAPTER 11. ALGEBRAIC NUMBER FIELDS
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
POWER SERIES EXPANSIONS OF ALGEBRAIC FUNCTIONS
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
111. ALGEBRAIC FUNCTIONS AND
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.
POWER SERIES EXPANSIONS OF ALGEBRAIC FUNCTIONS
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 $1 ,I assures the same for all rational functions in x and y , that is for all functions of K. In a field extension Kl of k the polynomial (1 2) decomposes to factors of the first degree n
f(y, X> =
JJ (Y - wi)
i= 1
whose zeros, by our assumption, do not coincide. The discriminant icj
can, as is familiar, be calculated as a polynomial in the ci(x), so that it is itself a polynomial in x not identically 0. This polynomial has, at most, a finite number of zeros x = t e k , , called the critical places? of Eq. (12), other E k, being called regular places. At a regular place the equation f ( y , 5 ) = 0 has exactly n distinct zeros y = q i in an extension of k , , and we maintain : For a regular place 5 there exists, to euery zero y = q i o f f ( y , c), exactly one power series
<
t Tr. note: The term "place" here, for the German "Stelle," is not the plaCe defined in Chapter 11, but rather simply means point. As the development of the subject proceeds it will turn out, though, that the two senses of " place" coincide in effect and require the same word, thus justifying the terminology of Chapter I1 anew.
116
111.
ALGEBRAIC FUNCTIONS AND DIFFERENTIALS m
Y=
C qip(x - 5)”
wit11 qio = q i ,
p=O
satisjying (12) (identically in x). The coeficients q i p are elements of thefield
kohi). The respective radii of convergence of these series are not zero. Proof. To (12) apply the variable substitution x + x‘ = x
- 5,
y - r y’ = y - qi
(14)
leading to the equation f’(y’, x‘) = 0 which has the simple zero y’ = 0 for x‘ = 0. This means that the last two coefficients off ’(y‘, x’) have the properties cn’(0)= 0, cL-,(O) # 0. To be proved is the existence of a power series y’(x’) with first exponent v > 0 which solvesf ’(y’, x‘) = 0. To simplify the notation we omit the primes and consider Eq. (12) under the assumption en-
,(O) # 0,
C,(O)
= 0.
It can now be written as
which is of the form (3). Thus the solvability now follows from the lemma in IlI,$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
POWER SERIES EXPANSIONS OF ALGEBRAIC FUNCTIONS
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(<) n natural numbers eQ= e,(t) whose sum is
<
e,
+ + e, = n ;
(19)
similar numbers e,(oo) are associated with the symbol t = co. Save for afinite number of exceptions, r ( t ) = n and eQ(t)= 1 always hold.
$1.
POWER SERIES EXPANSIONS OF ALGEBRAIC FUNCTIONS
119
These numbers hatie the following meaning : Setting z< = x
-
r,
z,
= l/x,
(20)
the irreducible equation (12) satisfied by an arbitrary function y of K over k has for solutions the r = r(<)power series
c m
y, =
aQ,,(ea(cfijp,
z 0,
apv,
e = 1,2, ..., r(t).
(21)
p=va
With a primitive e,th root of unity form yQi =
1 aQ,,Pe(cJiJJ,
i
= 0,
..., eQ(<)- 1 ;
(22)
P
then the left side of (12) is identical with
The coeficients a,,, are elements of a finite field extension k , of ko , and their images under isomorphisms of k , give permutations of the yQiin (23). The power series (21) have respective radii of convergence # 0. There is not much left to prove. By $l,4 every y E K is of the form y = h(x)-'y,, where h(x) is a polynomial and y , satisfies an equation (12) with polynomials ci(x). But, for such yl we have shown the existence of solutions (21) in the last two sections, and this then also holds for y by what was shown in $1 , l . As for the symbol = 00, the variable x is replaced there by z, = x - l . The place z, = 0 can turn out to be regular or critical, leading to power series (21) with e,(.o) = 1 or e,(co) > 1. It has already been shown that the aQpbelong to a finite extension k , of ko . Now, the fact that (21) solves (12) is expressed in algebraic equations in the a,,, with coefficients in ko ,which continue to hold under an automorphism. This means that automorphisms of k,/ko transform solutions of (12) into solutions of (12). Only Eqs. (19) and (23) remain to be proved, the first of these being an immediate consequence of the second. At regular places we found n differing solutions in $1,4, and, since there cannot be more, (23) is obvious for that case. Now let 5 be a critical place. The partial products
<
of (23) are polynomials in y of degree e, . The coefficients are power series in but because of the symmetry of their construction they remain invariant under multiplication of the variable by an e,th root of unity. Hence, the coefficients are power series in z c . The same is true of the right side of
'd&,
111.
120
ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
(23), as well as for the quotient of both sides of (23). If, further, the quotient still depends on y , the process of the last two sections can be used to find further solutions which will turn out to be of the sort (21). (Note that in the last section we explicitly permitted (16) to be an infinite power series.) But, under the assumption that (21) and (22) are all the possible solutions, (23) must hold. 7
Corollary. If the coeflcients of (1 2) can be expanded in integral power series at x = t, then all the power series (21) turn out integral. The proof is contained in §1,5. 52. Algebraic Function Fields 1. DIVISORS IN RATIONAL FUNCTION FIELDS
In §2,1-6 the constant field k , will be assumed to be algebraically closed and of characteristic 0. As is familiar, every rational function w # 0 E k = k,(x) can be decomposed to a product of powers of first degree factors
r
where the are a finite number of elements of k, . With every element 5 E k, we formally associate a place, without considering these to be in a space, the geometry of which we want to consider. It is true that in the classical case ko = C the t are visualized to be in the complex number plane, which is even closed by a point 00. This visualization is without consequence in the algebraic theory, though. The exponent v&w) of ( I ) is called the order of w at ?he place Now, w has an order at every place t, but as only a finite number of factors occur in (1) all but a finite number of these orders are 0. Finally, formally introduce the place = co where the order is defined to be
r.
Note, incidentally, that -v,(w) is the degree of the rational function w. With the places we further associate prime dioisors p = 3 < . Let these generate the free abelian group f , the elements of which are called diuisors. By definition these are thus products
with but a finite number of orders v,(a) # 0. Add orders to multiply divisors;
$2. ALGEBRAIC FUNCTION
FIELDS
121
the unit element of the group is then the unit divisor, written (I), whose order is vp(l) = 0 for all p. A divisor a is called integral if all its orders vp(a) 2 0. a is said to be divisible by b if ab-' is integral, or, what is the same, vp(a) 2 vp(b) for all p. With an element w # 0 E k we can associate a special divisor by setting
with vc(w) as in (1) and v,(w) from (2). Note that now, as opposed to (l), the place = co occurs in the product. These divisors are called principal divisors. Comparison of (1) and (4) shows (WlW2) = (wJw2); (5) that is, the product of two principal divisors is the principal divisor associated with the product of the two elements. Hence the principal divisors form a subgroup $ c t. The elements of the factor group €/$ are called divisor classes. Two divisors a, b are called equioalent, written a b, if their quotient is a principal divisor. There is no danger of ambiguity in writing b = wa for b = (w)a. A principal divisor (w)is the unit divisor (1) if and only if w E k , , as is seen with a glance at (4). This implies that the group $ of principal divisors is isomorphic to the factor group k x / k o of the multiplicative groups of k and k, . The divisor a is said to have the degree
-
and then Our definition (2) assures us of the important fact the degree of a principal divisor is 0. For rational functions, no more is to be learned from divisors as can be seen from formal consideration of (1). The importance lies in the fact that divisors can be carried over to the theory of algebraic functions, where the decomposition (1) is no longer possible. Exercise. All a E k with v&a) 2 ve(a) form the <-component of a linear divisor in the space k', in the sense of 1,$3. Its linear degree is equal to g(a). 2. DIVISORS IN ALGEBRAIC FUNCTION FIELDS Now let K be a finite extension of k = k,(x) generated, say, by adjoining a root y of Eq. $1,(12). The contents of the last section can be carried over
122
111.
ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
literally as soon as we have defined places, their prime divisors, and orders in K. First, with each of the r(5) power series of §1,(21) associate a place t4 of K(p = 1, ..., r(5)). The places 5, are said to lie ouer the place 5 of k. This is to be taken as a convention of speech, the application of which is yet to be explained. The e,-fold ambiguous root ‘ h o f 91,(21) is to be fixed arbitrarily. Any element w E K can be written as a polynomial w = P(Y, X I
in y whose coefficients are rational functions in x. Inserting the power series 51,(21) for y and expanding the coefficients of P(y, x) in power series in zy we obtain power series
Evidently (9) depends only on w, not on the particular polynomial PCy, x) representing w ; the only ambiguity is the e,th root of unity with which may be multiplied. Equation (9) is called the power series expansion of w at the place t4. Equation (9) can be understood as an isomorphic mapping of K into the field of power series in a variable which varies with the point t4. We do not, however, consider w as a continuous function in x or 5, even if the power series occurring have positive radii of convergence. It is only after the question as to the correspondence between the 5, and the series $1,(21) has been settled that continuous functions are found. This matter will come up in IV,§l. By §1,(23)the yQiare conjugate among themselves and with y, with respect to k. The conjugates of w = PCy, x ) are thus P(y,,, x), and w satisfies the equation
‘d<
‘fi
d w , x) =
n ad
(w
- P(Y,i 4 )= 0 Y
in k. The second and last coefficients of this polynomial in w are called the trace and norm of w :
Remembering that for a primitive eth root of unity
s= 1
holds, (9) and (10) give
for r = O m o d e for r f O m o d e
$2.
ALGEBRAIC FUNCTION FIELDS
123
The arbitrariness left in our choice of an e,th root of zc has no effect on the first exponent v,(w) = vCP(w)of series (9). It is called the order of w at the place t,. We now assert that the order of a function w # 0 is 0 except at a finite number of places. Proof. Let w = P(y, x) = a,(x)
+ a l ( x ) y + + a,-l(x)yn-l
r
with ai(x)E k. Except for a finite number of places of k the ai(x) can be expanded as integral power series in zc = x - 5. In the proof of Puiseux's theorem in $1 it was further seen that the series §1,(21) contains no negative exponents if y is taken as a solution of an equation of the type §1,(12) whose coefficients are polynomials. (It was shown there that this last assumption is no loss of generality.) Thus integral power series (9) are found at almost all places for w. The same is also true of w - l , which gives the assertion. 7 The reader can verify that our definition of orders based on power series corresponds, in the case K = k, to the orders defined in the last section. The definition (10) of the norm yields r(<) V&K,k(W))
1 VCe(W).
=
Q=
Together with (2) we have
1
ce, vse(w)
= 0.
(12)
(13)
The orders also behave as expected under multiplication : VCP(W1W2)
= v&1)
+ v&*).
(14)
With every place (, we now associate a prime divisor p = 3ce,proceeding as above to define the group R of all divisors and its subgroup $3 c R of principal divisors ( w ) = ~ V P ( W )= 3vCe(w), CP
I7 P
Ce
whose orders we have just defined. The group property of $ and its embedding in R stem from Eq. (14). The adaptation of the concepts of integral divisors, divisibility and equivalence of divisors, and divisor classes is obvious. A proof is needed, though, for the following theorem: A principal divisor ( w ) is the unit divisor (1) if and only if w E k , . Thus, !ijz K " / k o x .
ProoJ If w E k, ,clearly ( w ) = (1). Now, conversely, let ( w ) = (I), that is, let all power series (9) for w start with the constant term. If y E ko , then all the orders of y - w must be 2 0, and if further y is equal to the constant term of one of the series (9),then at least one is even > 0. But this would contradict (13) were not y - w = 0
111.
124
ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
The divisor a again written in the form (3) is said to be of degree =
c v,(a). P
As before Eq. (7) holds, and (13) can be restated as (8): every principal divisor has degree 0. This further implies that equivalent divisors have the same degree, so that the degree is a function of the divisor class.
3. DECOMPOSITION OF RATIONAL DIVISORS The concepts of divisors in different fields are logically independent. We are, nevertheless, free to embed the divisor group f of k into the divisor group R of K. Care must be taken, though, to be sure that such an embedding coincides with the natural embedding of k"/k," into K X / k o xFor . every place of k we set ( r = r(5)), (16) 3< = 3;; ... 3;:
<
where the ee and 5, are as in the last section. To prove the consistency, let w # 0 be an element of k and
be its decomposition into prime divisors of k and K; we must show the latter to result from the former by the substitution (16). Now, the rational function w is divisible exactly vt(w) times by the functionz< = x - <.But in K the order of zt at is exactly e Q .Thus we arrive at the exponent ve,(w) = eQvt(w)of 3t, for w, as was to be shown. We call 3< the norm of the prime divisors 35,, written 3< = nK/k(&e). In general, this norm is defined as
rQ
Then, holds, giving a homomorphic mapping of A into f . Furthermore, nK/k((w))
= (nK/k(w));
(18)
that is, the norm of a principal divisor is the principal divisor of the element norm. This follows immediately from (12). The degrees of a divisor are defined differently in k and K, so that they must be distinguished by subscripts. We have gK(a) = gk(nK/k(a))*
(19)
$2.
ALGEBRAIC FUNCTION FIELDS
125
This is obvious for prime divisors, and carries over to all divisors by (17). A special case of (19) is gK(a)= [K : k ] g k ( a ) which will be needed below.
if a E f,
(20)
4. THEPRINCIPAL ORDERS
From now on we must try to translate statements concerning power series into the language of linear algebra. The knowledge to be gained is already contained in II,§2 and 5, but here we follow a different path. As it turns out it is advantageous to make as little use as possible of the special properties of the power series. Those functions of k which are integral (i.e., can be expanded in an integral power series) at a place (including 5 = co) form an integral domain i,. Those functions y of K, which are integrally dependent on i, in the sense of II,§l,l, or in other words which satisfy an equation §1,(12) with cI(x)in i , , form an integral domain 3, in K , the principal order of K with respect t o i, .3,is of rank n = [K : k] over k. For, if y E K and h is the smallest exponent such that, for all i, zFci(x)E i , , then y' = z,"y satisfies the equation y'" + + z ~ c ~ ( x ) ~+" ... ' -= ~ 0, so that z,"y E 3 , . Now, if wi are any n elements of K linearly independent with respect to k we can thus find an h such that all the z,"wi E 3,. These are of course still linearly independent. 3, is a principal ideal domain. Proof. Let a E 3,be an 3,-ideal. Find an element a, of a whose norm with respect to k is divisible by a minimal power of z c . Such an a, exists, as the norms of all elements of 3, lie in i, . Now let a E a and a E k, be arbitrarily given. Then
is a polynomial in a. Using the assumed property of a, and the fact that aa, - a E a, we know thatf(a) lies in i , for all a. We assert that even all the bi(x) are in i , . Say this were not so, and the largest negative power z i h occurred in bi,(x),.. ., bi,(x). These negative powers would have to cancel out when any a E ko is inserted intof(a). In other words, if we set z , = 0 in
+ + z~bi,(x)an-im,
z ~ b i l ( x ) a " - i l ...
then this expression vanishes for all c1 E k o . Hence the coefficients of this polynomial in a must all vanish when set z , = 0, which in turn contradicts the assumption that all the bi,(x) have denominator 2:. Now, sincef(a/a,) = 0, and the coefficients of this polynomial are integral at 5, we have a/a, E 3,. Thus a = 3
111. ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
126
As is well known, the theorem of unique decomposition into prime elements holds in a principal ideal domain, or, more precisely, two given decompositions of an element into prime elements can differ at most by factors of units. From 11,#1,2 we see: 3, is aJinite ic-module of rank n. 3, has only a Jinite number of prime elements q l , ..., qr which all occur in the decomposition z , = uqe,' ...q:, (21) u being a unit. Proof. Let q be a prime element. As we saw above, we can find an h such that z,b-' E 3,. Thus q is a divisor of z , . 7 In (21) r = r(5) and a suitable indexing gives
the
E~
of (21) being equal to the e p I
Proof. By the corollary of #1,6, the elements of 3, have nonnegative orders, this being also true, in particular, of the unit u and u - ' . Thus, vcu(u) = 0 for all u. Similarly, vtQ(qQ)2 0. Taking orders of (21) leads to the equation
C EQvCU(qQ) = eu
for u = 1, ..., r(5).
Q
As the qQare mutually prime there must exist, to any pair of indices el # e2, elements a, b E 3, with aqQl
+ bqQ,
=
But this implies that vcu(qQ,)> Oand vru(qQ2)> Ocannot hold simultaneously. Hence the sums (23) are reduced to one term each, with e = eu , and we have eQev,u(qQ$= eU 3 v,u(qQ) = for @ # @O ' Furthermore, holding one e fixed, v,,(qQ) = 0 cannot hold for all 6. For then qQand qQ-' would each satisfy an equation with coefficients in i, , so that qQwould be a unit of 3,. Thus r 5 r(<),and we can renumber the qQor the tQ so that 'QVrp(qQ) =eQr @= (24) holds. The rest of the proof depends upon the additional assertion: A basis of 3, with respect to i , is given by qpi = qy q:'qi-rQ(e= 1, ..., r ; i = 0, .. ., E~ - l), and we have ' 9
. " ) '9
ci = n = [ K : k ] .
For, assuming (25) to hold, then (24) and #1,(19) yield
(25)
$2.
127
ALGEBRAIC FUNCTION FIELDS
If r(5) > r we would have > on the left, and > would also have to hold on the right if only one vte(qQ)> 1. Thus the proof of the former assertion depends upon (25). To prove the last assertion and (25), consider some qQand some residue class ii of 3, mod qQ. It is not difficult to see that it satisfies an equationf(ii) = 0 whose coefficients are residue classes of i, mod z c . As k, is algebraically closed, these residue classes can all be represented by elements of k, . Now, the residue class ring of 3, mod qr can have no divisors of zero, so that ii must even satisfy an irreducible equation of this sort, and, k, being algebraically closed, this equation will be of degree 1. Hence each a E 3, is congruent mod qQto some a, E i, . The elements qQoare $0 mod qe . Now let some a E 3, be given, and aq;; = up, E i, mod qQ. Further, let (a
- aQoqQO)q; ‘4;;
= (a
- ‘QoqQO)qill
mod q Q
9
etc. This yields elements aQiE i, satisfying the congruences a
+
+
+
mod 4: * The construction of the qQi entails the right side of this congruence to be = 0 mod q: for e # n. Thus ‘QOqQo
‘QlqQl
’*’
aQ,sg-lqQ.Ep-l
Now let w , be a basis of 3, with respect to i,, which we know to exist from I,§l, 1, and let CQi,” E i, . qpi = cpi.vwv
c
1
V
On the other hand, apply (26) to the w y ,so that w,
= 1ciVQiqQi mod 3,zc. Q.i
and then The w , must also be linearly independent mod zc , so that
must hold. Hence (cYi,J is a unimodular matrix over i,, and the qQialso form a basis of 3,. As 3, has the rank n with respect to k, Eq. (25) is simply a matter of counting. 7 This has brought us to our goal. The basis qQiwill soon play an important role. Noting that these calculations made no use of the power series (9), one
111. ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
128
might conjecture that they could be dispensed with completely. This will be confirmed in 111,§2,7. We also point out the fact that a basis similar to q Q i ,but even more complicated, was used in the proof of the Dedekind discriminant theorem in 11,§4,3. One last theorem serves to bridge the gap between the older concepts, based on power series, and the newer, more abstract, algebraic ones: 3, coincides with that integral domain of functions f E K whose power series are integral at allplaces rQlying over c. To show this one need only choose the representation f = C qQplfQiwith E k,(x), and prove that all the power series for f at the 5, are integral if and only if the power series of all the&, are integral at <.This is not difficult, when the construction of the qQiand (22) are used (exercise). As was already predicted in the introduction to this chapter, the power series now fade into the background of the development of the algebraic theory. 5. DIVISORS AND LINEAR DIVISORS
Our path meets here with that of Chapter I1 at 11,§5,6. As we do not want to presuppose that chapter now, we must consider some material anew. A divisor a is written as
<
with traversing all the places of k. The <-componenta, of a thus defined can be associated with the set of all a E K with vcQ(a)2 v,,(a), Q = 1, ..., r(<).This is clearly an 3,-ideal, and no misunderstanding can occur if we denote it also by a,. Choose an h 2 max(0, - vtn(a)),then z?aQ E 3,. The finiteness criterion of I,§l,l together with the fact that 3, is a finite ic-module implies that ac is finite. Given an arbitrary b E K, we can clearly also require z"b E at ;certainly then a( has rank n with respect to k. Finally, as almost all v,,(a) = 0 we have at = 3, almost always. A basis representation a = C a , o , with a, E k of the elements a E K is now interpreted as an identification a tt (a,) of these elements with the vectors of an n-dimensional space over k. We can then assert that the at are the components, in the sense of 1,§3, of a linear divisor. To show this we form the intersection 3= q. (28) c+ 03 It is an i-module, where i= ( i,l = k,,[x].
n
,+a
By definition 3 consists of all elements of K integrally dependent upon i and by II,§l,l (where now o = i and D = 3) we see that 3 is a finite i-module of
$2.
129
ALGEBRAIC FUNCTION FIELDS
rank n, and therefore has a basis o,with respect to i(I)$l)l).The o,are also a basis of 3, with respect to i, for all # 03. For, were a = o v a , E 3,with a, E k but not all a, E i,, we could find a common denominator h = hlztt of the a,, with h, prime to z , . Then ah, = o,a,h, E 3,,while not all the a,h, E i, . On the other hand, avhl E it, for all t’ # t, so that by our hypothesis, ah, E 3. But some of the avhl are nonintegral, which contradicts the basis property of the o,. As the o,now form a basis of 3,with respect to i, forevery 5, and as a, = 3, almost always, the w, are almost always a basis of a,. Thus all the properties of a linear divisor have been confirmed. The linear divisors corresponding to equivalent divisors a and aa are equivalent in the sense ofI,$3. The proof can be taken exactly from 11,$5,6, if only the symbol p there is replaced by t. The field K hasbeen associated in the natural manner with the n-dimensional vector space k”. Simultaneously we may associate K with the dual space k”* if we can find a bilinear form aa* of K with values in k which possesses the property (3) in the list of 1,$1,3. We choose the trace: aa* = sKIk(aa*). The property ( 3 ) required is then the fact that the discriminant D(w,) = ~ S ~ / ~ ( C O , , C O , ) ~ of a basis o,of K/kis not 0. Let complementary modules and complementary linear divisors be defined now in this sense. There exists a divisor b K / k such that an arbitrary divisor a and its “cornplement ” a* = b- l a - 1 (29) Klk
<
1
correspond to linear divisors which are complementary in the sense of 1)$3. Proof. As opposed to 11,$5,6, we can now explicitly describe b K / k . In the last section it was seen that 3, is a principal ideal domain, so that for every there exists an a , E K such that
<
a, = 3p,.
Define the module 3,*by sK/k(wo*) E
i,
for all w E 3,, w*
E
3,*;
then clearly a,* = 3,*a;
(31)
Using the fact that 3, is an integral domain and applying obvious notation we have S K l k ( 3 , ’ 3<*) = s K / k ( 3 < 3 < * 3<*) = S K / k ( 3 { * 3,3,*)* This means that 3,* is an 3,-ideal, and (31) becomes
130
111. ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
This is Eq. (29) if the local components of b K l k are given by We can determine these ideals explicitly : 3,*= 3 , q p ... qf-e’,
(33) with the q, and eP of the last section. From (22) it is not difficult to derive the power series expansions
for all e. The elements of 3, have integral power series at all places t P Then . (1 1) and (34) show that the elements of 3,q all have traces in i, , This means that 3,q E 3,*.But, if 3,q c 3,*,say 3,* = 3,qrf1 .--,where some& >= e,, then choose some a E 3, whose series expansion at (, starts with (e$$@-e~, e) starts with(e&)fu-eu+8,e > 0. BY (11), s K / k ( a q ; ” while that at (,(a would not be in i, . This contradiction proves (33). According to Puiseux’s theorem almost all the e, = 1, so that the product
+
is finite and defines a divisor of K, whose (-components, understood as 3,-ideals, are (bK l k )< --3 tq1e l - 1 ... 9p-l. These are just the inverses of the ideals (33). Thus (32) gives the local components of a divisor. 7 Simultaneously, the formula g(bK/k)
=
t.P
(e
- 1)
(36)
has been found. We call (36) the ramification index of K with respect to k. (The arithmetic justification of this terminology is to be found in 11,§3, while its geometric justification will arise in IV, $1 .) b K l k is called the different divisor of K with respect to k. We must finally calculate the linear degree of a divisor, in the sense of I, 93. This is done exactly as in II,§5,6 and 7. The assertion made there, that with a basis ovof 3 the (-components of N(a) and n K / f ( a ) coincide for all ( # 00, follows easily from (30). For W,q
=
1apvw,
9
apv E
k,
V
is a basis of a,, and the determinant (apVJ is the (-component of N(a). But this is also true to n K / k ( a ) by the definition in 1,w.Altogether, by II,§5,(20)
$2.
ALGEBRAIC FUNCTION FIELDS
131
6. THEINVARIANCE OF THE CONCEPT OF DIVISORS We introduced the concept of divisors with the help of a specific model of the field K as a finite extension k(y) of k = k,(x). The element x was thus distinguished. K could just as well be considered a finite extension of k' = k,(y); this gives K = k'(x) where x satisfies the same equation 51412). Clearly there are even infinitely many possible models. However, each prime divisor p can be associated with the integral domain 3,of those functions w with v,(w) 2 0. Now, such an 3, is a discrete local ring in the sense of 11,§2,2, and in II,§2,3 it was shown that there exist no other discrete local rings in K than the 3,,where the p are the 3,@ defined in §2,2. Hence the concept of the discrete local ring is independent of the specific model of K, and then the same holds true for prime divisors and, finally, for the concept of divisors. 7*. EXTENSION TO MOREGENERAL CONSTANT FIELDS
From now on let ko be an arbitrary algebraically closed field. Further, let there exist what is called a separable model of K, that is, let K be a separable extension of k = k,(x) by suitable choice of x . In $4,1 it will be shown that this is certainly possible whenever k, is algebraically closed. The definition of the i, remains unchanged, and the prime divisors 3, of k, and then the general divisors of k, can be introduced as in §2,1. To introduce the divisors of K, though, we can now no longer use the power series (9); they need not exist. Instead, we use the method of $2,4 to form the principal orders 3,.There is no change in the proof that these are principal ideal domains. Equation (21) also holds. The exponents E , and the number r occurring there must be left in place of the e4 and r(C), for there is now no other way of defining these. Prime divisors 3,, are associated with the q, . Everything else in §2,2, §2,3, §2,5, and $2,6 may be taken over verbatim, with the sole exception of the computation of the complement (33). For here we no longer can use formula (1 1) and, in fact, (33) is no longer correct if any one of the e, is divisible by the characteristic. Nevertheless, a complementary ideal 3,*to 3, exists, for the separability of K/k assures us that the discriminant of K with respect to k does not vanish. Set
34 * = 3,q;*'
... q;6'.
then 6, = 0 for almost all (, as here again 3,is the (-component of a linear divisor. In place of (35) define b , k =
so that Eq. (32) is restored.
n 5
3;:
..*
3;;9
(38)
132
111.
ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
All we now have for the relation between linear degree and divisor degree is G(a) = g(a) + j g @ K / k ) ,
(39)
and the ramification index g(bK/k) is, in general, larger than the sum (36). It can be computed with the help of (20), for g@K/k)
= gk(aK/k),
IDK/k
= nK/k(bK/k)*
The latter is the discriminant divisor. By 11,§4,(12)its t;-components are the discriminants of bases of 3, with respect to i, . A similar procedure is difficult when k, is not algebraically closed. It is possible, though, by a trick used in algebraic geometry. K is extended to R by adjoining the algebraic closure I;, of the exact constant field ko , and then the above arguments hold in R. An integral divisor a of R is said to be a divisor of K(or ais said to be defined for k,) whenever a is the g.c.d. of numerators of functions a, of K. In a fully cancelled representation one then has (ai) = abicT1, with integral b, , ti, the b, being mutually prime. To these divisors of K we annex their quotients, so that the divisors of K form a subgroup R c 6. Whenever two integral divisors of A have a common divisor in R it is, by definition, also in A. Thus every divisor of A can be represented uniquely as a power product of indecomposable divisors of R. These are then called the prime divisors of K. The elements w E K generate the group $j of principal divisors of K, and the factor group A/$ consists of the divisor classes of K . It is finally easy to see that the complement (29) of a divisor a also belongs to R.
93. The Riemann-Roch Theorem 1. DIMENSION OF A DIVISOR CLASS No assumptions are made concerning the exact field of constants ko . Whenever K is a separable extension of the rational subfield ko(x), and thus in particular whenever the characteristic is 0, the trace is taken for the pseudotrace, so that the prefix “pseudo” also drops out for the different and discriminant. Let a be an arbitrary divisor. The dimension of the linear divisor corresponding to a-’, all in the sense of 1,§3, is called the dimension of a, written
dim(a) = I(a-’).
(1)
The linear divisor to a-’ is a system of finite i,-modules a;’ or ic-modules a;’ in K. Thus there are exactly dim(a) elements bi E K linearly independent
63.
THE RIEMANN-ROCH THEOREM
133
with respect to k, and lying in all the a;' or a;', respectively. For all p(or l ) this means that a,b, E 3, (or aebi E 3<). This can also be expressed by saying that the divisors abi = bi are integral. Then (bi) = bia-
with integral divisors b i , and it is plausible to call the bi multiples of a-'. Thus dim(a) can also be defined as the number of multiples of a-' linearly independent with respect to k , . As we are assuming that ko is the exact constant field of K, the formula !jj 2 K"/k," of II,§5,3 (or, respectively, 111,§2,2) states that the biare uniquely determined up to a nonvanishing factor in k , by the (bi) and therefore by the b i . Changing these factors does not affect the linear independence, though, and it becomes clear that we can speak of linearly independent (or linearly dependent) divisors bi of a divisor class. In this connection it is also customary to multiply linearly independent divisors of a class with constants eiE k , and to add them. An expression of the sort b = eibi is then the numerator divisor of b = eibi.This notation has a unique meaning only after the bi for which (bi) = bia-' have been fixed, and then (b) = ba-I. By I, §3,3 the dimension as well as the degree is a function of the divisor class. Classes will always be denoted by upper-case roman letters from now on. The unit class of a = (1) is usually called the principal class, as it consists of the principal divisors.
2. THERIEMANN-ROCH THEOREM There is a distinguished divisor class W and a rational number g that the formula dim(A) = g(A)
+ dim(W/A) + 1 - g
2 0 such (2)
holdsfor every divisor class A. I f g ( A ) < 0 then dim(A) = 0. The principal class A = (1) has the dimension dim(1) = 1 , while all other classes A of degree 0 have dim(A) = 0. One calls g the genus and W the canonical class of K . The term differential class used in the classical case is, in general, misunderstandable and we will not use it. In this regard cf. §4,8. 1
Proof. Let a be a divisor of the class A. Denote the linear divisor associated with a - ' by 6. The divisor b' associated with b by I,§3(11) is, according to II,§5,6 and 111,§2,5 (or §2,7), b ' = b - ' b - 'Klkjm 2 = a b i / h m 2 , (3) where b,,, is the pseudodifferent and
ja
the denominator divisor of (x). We
111.
134
ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
maintain the canonical class is that of the divisor of bK,k3;2 :
w =( b K / k 3 3 . By 1,§3,(12) we have
4 6 ) - l(6') = n
(4)
- G(6),
where G(b) denotes the linear degree. Compiling (3), (4), and II,$5,(20) or 111,§2,(37) or (39), respectively, we have dim A
- dim(W/A) = g ( A ) + n - +g(bK/k),
n = [K:k ] ,
which, using the abbreviation = b@K/k) - C K :
k1 +
(5)
'9
is identical to (2). The degree of an integral divisor is nonnegative, so that a class of negative degree can contain no integral divisors. This means dim(A) = 0 for g ( A ) c 0. Further, clearly (1) is the only integral divisor in the principal class, so that dim( 1) = 1. If g(A) = 0 and dim@) c 0 there is an integral divisor a in A, and it must be a = (1). When A is the principal class (2) goes into dim W = g.
(6)
This assures that g 2 0, completing the proof. 7 Setting A = W in (2) and using (6), we have g(W) = 29
- 2.
(7)
An expression in terms of the degree of the pseudodiscriminant in k for the genus is found by combining ( 5 ) with 11,§5,(16) for L = k or 111,§2,(20): = +gk(aK/k)
- [ K : k ]+ 1 -
(8)
If L is an intermediate field between k and K the transitivity formula II,&l,(ll) gives gk(DK/k)
= gk(nL/k(DK/L))
+ CK
Llgk(BL/k)
= gL(DK/L)
+ [ K :L l g L ( b L / k ) ,
where g L denotes the divisor degree in L. Substitution into (8) yields 9 = fgL(IDK/L)
- CK:LI
1
CK: L I ( j g ~ ( b ~ / k) CL:k]
+ 1).
We introduce the relative genus g(K/L) = f g L ( I D K / L )
-CK:
L1 + 1 = b ( b K / L )
- [ K :L] + 1
(9)
and thus can finally write 9 = g(K/k) = g(K/L)
+ CK: LIg(L/k).
(10)
$3.
THE RIEMANN-ROCH THEOREM
135
The pair of equations (9) and (10) is called the Hurwitz genus formula. Note that for the relative genus g(K/L)2 0 need not necessarily hold.
3. QUESTIONS OF INVARIANCE As already mentioned in I1,§5,5 and 111,§2,6, the element x , for which the field K is considered a finite extension of k,(x), is actually arbitrary. The concept of divisor and, in particular, of prime divisor is invariant under a change of this element. This immediately gives the invariance of the divisor degree under the special assumptions of $2 from its definition in connection with prime divisor decomposition §2,(3). Under the more general assumptions of Chapter I1 the invariance theorem of II,§5,5 states that the degree of a prime divisor is independent of x , and then so is the degree of every divisor. Finally, the concepts of divisor class, integral divisor, and dimension are invariant of the choice of the model of K . With these considerations (2) implies the inoariance of the genus g , as for a class A of sufficient degree we can write (2) as g = g ( A ) - dim(A)
+1
for g ( A ) > 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.
ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
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
111. ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
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.
THE RIEMANN-ROCH THEOREM
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.
ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
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.
THE RIEMANN-ROCH THEOREM
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.
ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
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 for u, the solutions u = q being regular places for the first equation due to our assumptions. Thus the two power series u=
C a,(u - q),,
u
- q = 1b,(x - t)”
solve these equations, and the chain rule holds for these power series. Surely it also holds when the differential quotient is computed by the rule (1). 2. THEDIFFERENTIAL CALCULUS WITH CHARACTERISTICp Several remarks are in order for the case of a field of characteristic p > 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
111. ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
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
111. ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
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
<
<,
where 3, is the denominator divisor of x. Further, the divisors of all differentials belong to the canonical class. The proof here will assume the characteristic to be 0; in §4,8 the general case will be considered. By §2,4 there exists, to every place 5, lying over a place t, a function q, with Puiseux expansion qa
Jc + ...,
= ee
w. DIFFERENTIALS
149
the dots indicating higher powers of zC= x - t or x - I . These qp are prime elements for the places I&. For the orders this means
Using $2,(35) this is equivalent to (16). In $3,(4) we saw that bK,ko(x,j;2lies in the canonical class W, and thus with (1 5 ) all divisors of differentials dw lie in W. 7
4*.
CONTINUATION ; SEPARABLE A N D INSEPARABLE PRIME
DIVISORS
The residue class field of 3, modulo a prime divisor p of degree g(p) = r was seen in II,$5,5 to be an extension of ko of degree r. The prime divisor p is called separable or inseparable, depending upon whether this extension is separable or inseparable. We now want to associate a divisor with any differential in a separably generated function field over an arbitrary field of constants ko . Let kg be the algebraic closure of ko and K" = Kk; (cf. $3,4). We define (dw) as the divisor in K of the last section, but show: There exists a purely inseparable finite extension of constants ko'lkoin which (dw) is defined.
Proof. First let p be a separable prime divisor and 9 a primitive element of the residue class field 3,/p over k,. Let it satisfy the irreducible equation f(S) = 9' + ... = 0, f ( S ) being a separable polynomial of degree r = g(p). Hence there must exist a separable extension k , / k o in whichf(9) decomposes to r mutually prime factors 9 - cli . For an element 9 in the residueclass 9,f(9) is an element of K whose numerator is divisible by p. In K, = Kkl we have the decomposition f(9)=(9-cl,).**(9-cc,) into r factors. The difference between any two of them is prime to p. This leads to the decomposition of p to p = p1 ... P,
(18)
in K,. Comparison of degrees shows that the p4 are all of degree 1, and must therefore be prime divisors. A prime element q for p in K is a prime element for every ppin Kl ,as is seen
150
111. ALGEBRAIC FUNCTIONS AND
DIFFERENTIALS
by (18). For a differential (12) in K and the orders defined by (13) this means v,,(dw) = * * * = v,r(dw) = v,(dw), so that
This shows that the contributions of the individual pp to the divisor (dw) altogether yield a power of p, which is, of course, a divisor of K. Thus the p-component of (dw)for separable prime divisors p can be computed as in the last section. It is well known that, for an inseparable prime divisor p, a finite purely inseparable extension ko' of ko can be found such that the field composed of ko' and 3,/p is separable over ko'. Then p is decomposed into separable prime divisors over the extension K' = Kko', and the above argument can be applied. This need only be done for finitely many p because, as it was seen in the last section, only finitely many p contribute to (dw). The assertion is proved. 7 Simultaneously we have proved the Lemma. A finite extension of constants K = Kk,' always exists in which a given prime divisor decomposes into prime divisors of the first degree. For separable p this extension can even be chosen separable. By definition we have: The divisor (dw) of a digerential is invariant under extension of constants. Formula (15) is true in the general case, and therefore divisors of all diferentials belong to the same class. Formula (16), on the other hand, is no longer generally true, and the divisors of differentials need not lie in the canonical class. Consider, for example, the field K = k o ( x , , / x p - a) mentioned in §3,4. Now, as the divisors of differentialsare defined by closure of the constant field, they are invariant under extension of constants. But in the previous mention of this field we saw that its genus changes under extension of constants, and therefore also its canonical class. W,8 will again deal with this question. 5. CARTIER'S OPERATOR We undertake some considerations in preparation of what follows, assuming the characteristic to be p > 0. A diferential of K,
+
+ +
u dx/x = ( u O U1X uP-1xp-') dX/X, ui E KO, (19) is in unique correspondence with the differential uo dxpIxP of KO; that is, the correspondence does not depend upon the element x used. By dxp we mean the differential of the element x p in KO.A linear map S of the differentials of K onto those of KO is clearly defined by this theorem; it satisfies S(al d w ,
+ a, dw,) = alS(dwl) + azS(dw2),
a l , a2 E K O . (20)
$4. DIFFERENTJALS
151
The proof uses the following not inessential concept. A differential dw is called exact if it is the differential of an element of K. For example, x i - dx = d x ' / i for i 0 mod p. Similarly to all differentials, the exact differentials form a KO-module,denoted by E, so that we can write
+
dw,
E
dw, mod E
whenever dw, - dw, is exact. Clearly uo dxp/xp is a function of the class of u dxlx mod E. Now let y be another p-variable, and
dx
u-=u X
0
dx -=u X
0
Y dx dY dY ---=uogo-modE. XdY Y Y
But, by the lemma in §4,2, this can be expressed as dx
yPdxPdy ---modE. X OxPdyP y Therefore, if y is used as the variable instead of x , the differential S(u dx/x) corresponding to u d x / x in KObecomes u-=u
y p dxP dyP xPdyP y p
uo---
If instead, uo d x p / x pis immediately calculated out for the variable y p the result is the same, completing the proof. 7 The following statements are obvious. A place p o f K lies over exactly one place po of KO.I f ( d w ) is integral at p then S(dw) is integral at po . The operator S was discovered by P. Cartier?; it can be defined in the same sense for functions of several variables. 6 . RESIDUES OF DIFFERENTIALS$
First consider a place p of the first degree. In §1,2 we saw that all functions can be expanded in power series with coefficients in k, in terms of powers of a
t Une nouvelle opdration sur les formes diffdrentielles, Compt. Rend. Paris 244, 426-428 (1957) and Questions de rationalitd des diviseurs en gdomdtrie algdbrique, Bull. SOC.Math. France 86 177-251 (1958). 1Another approach to this question is found in H. Hasse, Theorie der Differentiale in algebraischen Funktionenkarpern mit vollkommenem Konstantenkorper, J. Reine Angew. Math. 172, 55-64 (19341, as well as in the book [31 in §3,7. In the latter a concept of the differential is used which need only coincide with ours for perfect fields of constants.
152
111. ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
prime element q. Let a differential dv dw = u dv = u dq dq =
(F
c,, E ko ,
c,,q”):,
be given; define the residue of dw at the place p as the coefficient res,(dw) = c,
(22)
of the power series expansion (21). For two differentials dw, and dw, , and two constants a,, a2 E k, this implies resp(al dw,
+ a2 dw,) = a, res,(dw,) + a2 res,(dw,).
(23)
It must be shown that the definition (22) does not depend upon the choice of prime element q. But, using (23), we need only prove invariance for differentials Q dq/q. Let q‘ be another prime element, and q = a,q’
+ a,q’, + ..’,
a, #
0, a i ~ k o .
(24)
Then
This shows the invariance for p = 0. For p not divisible by the characteristic we have
Substituting (24) on the right we get the differential of a power series, which cannot contain a term in dq’lq’. All that remains is a case where the characteristicp > 0. Here we can apply Cartier’s operator S. Clearly res,(dw) = res,,(S(dw)), (25) where po is the place of KO over which p lies. Repeated application of S eventually leads to a differential, integral at the respective place, or with a pole of order 1 there. In both cases the residue is proved independent of the prime element. 7 Now we can let p be a separable prime divisor of a higher degree. By the lemma of #4,4 there exists a separable extension ko‘/k, such that p decomposes into prime ideals pl‘, ...,p: of the first degree in K‘ = Kk,‘. No generality is lost in assuming the extension to be normal; let the Galois group be 0. By 43,4 k,‘ is then the exact constant field of K’, and K has the same Galois group 0 over K. The residue at the place p is defined by r
res,(dw) =
C res,,; Q= 1
(dw)
$4.
DIFFERENTIALS
153
and it will be proved : the residue of a separable prime divisor lies in the constant field. For, let ql’ be a prime element for p,’ in the decomposition field of pl‘ (cf. 11,§6,1). Its behavior under automorphisms of 6 will be that ofp,’. The same holds true for the power series P
as well as for their coefficients c I P . Applying the substitutions S, of 11,§6,(1) yields r such power series. Their coefficients and all the elements 2 of the decomposition group of pl’ satisfy
Now let q be a “ purely inseparable ” prime divisor, that is, let it decompose to q = q‘phwith q’ of degree I in a purely inseparable extension k,’ of k, .Then define the residue of dw at q to be equal to the residue of dw at the place q’ of K’. This residue also lies in the constant field ko , without extension. Proof. Apply the operator S h times, remembering (25), and considering q = qlphas a separable divisor of the field k,’KPh.On the other hand, Sh(dw) is a differential of koKPh.Thus the residue lies in ko . 7 Exercise. Combining the last two steps define the residue of a differential at an arbitrary place and show that it lies in the field k, . Finally we remark that as the orders of a differential are almost always 0 the residues of a differential are 0 at almost every place.
7 . THERESIDUE THEOREM The sum of the residues of a differential in a separably generatedfield is 0. The proof, which is an easy corollary of the Cauchy integral theorem in the classical case, takes considerably more effort in general. We may assume we have an algebraically closed constant field, for (26) shows that the sum of residues is unaffected by a separable extension of k, and it does not change under an inseparable extension as a matter of definition. Let us first take care of the rational function field K = k,(x). Decompose u of the differential dw = u dx to the familiar partial sums
We can now prove the theorem for the individual summands
154
111.
ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
All residues of the first of these vanish except for i form all vanish. A quick computation shows rest
dx
-= 1,
1; those of the second
=
dx res, -- - 1
X - t
X - 5
completing the proof for the rational case. Now let K be a separable extension of k = ko(x). The residue theorem is an immediate consequence of our special case once we have the formula
which we proceed to prove. The sum is, of course, taken over all [,lying over (. There is no loss of generality in choosing ( = 0. The numerator divisor 30 of x decomposes in K,say to 30 = p;'
...
PF7
so that we can rewrite (27) correspondingly as r
1resp,(u d x ) = res0(s,,,(u)
dx).
(28)
Q=l
First, simplify the matter. Choose elements ql, that
..., qr according to $2,4, so
These q, can be viewed as primitive elements of K with respect to k = ko(x). For if they were not they could be replaced by q,(l + SX") with a primitive 9 and sufficiently large h. At the place p, we can expand u dx as u dx =
1cp,,q'lp-l d q , ,
e = 1, ..., r,
cqP E ko
.
I(
It suffices to prove (28) for the special differentials qF-' dq,, because of the additivity (23) of residues. Our choice of the qQ assures res,*(q;-' dq,) = 0
for
0
#
e,
so that the left side of (28) reduces to a single term for this differential. Only =, reSo(SK/dq;-l ) dq,/dx) d x ) resp(qg-' &I
remains to be shown. It clearly suffices to do this for Let q1 satisfy the irreducible equation f(41,X ) = 41" - cl(x)q;-'
+ - + (-
(29)
e = 1.
n n
l>"Cn(x) =
v=l
(41 - ~
v )
#4. DIFFERENTIALS
155
whose zeros K , we consider to lie in an extension of k having only the elements of k in common with K. Now, c,(x) is the norm of q1 with respect to k, and it is divisible by 30 exactly f times, f being the residue class degree of p1 in K/k.But the degree of p1 is 1 by our assumption that ko be algebraically closed, so that certainly f = 1. Thus
n n
K,
= c,(x)
and
vo(cn(x))= 1.
v= 1
Differentiating this equation logarithmically yields
This implies
which is the asserted equation (29) for the case p = 0. If p is not divisible by the characteristic p , then
As the power series on the right can contain no expression in x - l dx, (29) is also proved for this case. The differential qlp dq,/qlwith exponent p # 0 divisible by the characteristic p remains. We now apply Cartier’s operator. Let
Then
E ko(xp).Thus we have and then ql”giE KO implies sKlk(qlPgi)
(30) This means that the operations S and taking of traces commute. Repeated application of S finally leads either to a differential integral at the place in question, making (29) trivial, or to a differential with a pole of first order which has already been treated. This completes the proof. 7 At a later point we shall need the following generalization of (27). Let L be afield between K and k = ko(x), and p a divisor of L whose decomposition
156
111. ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
in K is p = pyp',L
. For any u E K, then res,(sK,,(u) d x ) =
1resp,(u dx).
(31)
i
Proof. It again suffices to treat first degree prime divisors only, Let the divisor of k divisible by p be po , and let p', p", ... be the other divisors of L which divide p o . It is no loss of generality to assume po not to be the denominator of x. Let 3, be the principal order of L with respect to ko[x]. The divisors p, p', ... correspond to 3,-ideals p, ,p,', ... . Using an argument of 11,$2,2 there exists a w E 3, such that w = 0 mod p L , w = 1 mod pL'pL" ... . The element v = (1 - w ) ~ satisfies the congruences u = 1 mod p L , u 3 0 mod (pL'pL"...)h. Sufficiently large choice of h yields residues res,,(vu dx) = resp,(udx). But also, resP,(sK/,(uu)dx) = res,,,(s,,,(u)u dx) = 0, etc. We now apply (28) twice: i
resp,(uu d x ) = respo(sK/k(vu)d x ) = respo(sL/k(sK/L(uu))d x ) = res,(sK/,(uu) d x )
+ res,,(s,,,(uu)
dx)
+
= res,(s,,,(vu)
dx)
which, by the above, gives the assertion. 7 8*. THEDIFFERENTIAL CLASS
In §4,4 we saw that all differentials lie in the same class, the differential class, which we shall now consider more closely. We first observe: bK/kjz2 is always divisible by the divisor (dx). Thus, (16) is equivalent to the statement that (dx) is in the canonical class. Proof. Let q be a prime element in ko[x] and pp (e = 1, ..., r) the prime divisors of K by which its numerator jqis divisible. By the procedure of II,§5,6 associate with the divisor (dx)-' the 3,-ideal (dx)-' of all u E K satisfying vpQ(u)&vPp((dx);'). For all u ~ ( d x ) ; 'and all v O ~ 3 we , then have uu,dx/dq~3, and thus resp,(uvodx)= 0. By (28) then, resaq(sK/k(uuo)dx) = 0. On the other hand, we can say that resaq(sK/L(UvO) dx) = 0 if and only if u belongs to the complement 3,* = (bK/k)q' of J,, for this is simply another form of the definition of this complement. Thus, (dx);' E (bK&'. At the place 00 we must replace x by x-', arriving at (dx-I);' = x2(dx),' E (bK/k);'. This verifies the assertion: bK/k3G2 is divisible by (dx). 7 Equation (16) holds generally for separable extensions K of k = ko(x) if the different bKlkis divisible by separable prime divisors only. The proof is first produced under the additional assumption that all prime divisors p4 of bK/k are of degree 1; the exact powers of pp which go into (dx)
w. DIFFERENTIALS
157
and b K / k are determined. Our new assumption assures (by 51,2) that every element u E K can be expanded in a power series in a prime element q, with coefficients in k, . As above we assume the q, not to be divisible by the prime divisors p, # p,; and also that the q, are primitive elements. From the previous proof we can now extract the fact that the exact power 6, of pp which divides (dx) is also the largest integer such that resp,(uq;'" dx) = 0 for all u E K integral at p,. We now determine this power by setting an arbitrary integral power series in q, for u ; then 6, is the largest integer such that respp(q;'P d x ) = respp(qp'R+' d x ) = ... = 0. In terms of (28) then, 6, is the largest integer for which we have SK/k(q,-"), SK,k(q;"+ I), ... E i,. But then, from the definition of the different in 11,$4,2, the p,-component of (dx) and b K / k is the same. As the same argument can be applied with x - ' in place of x for the place 00, this case is proved. If bKIkis divisible by separable prime divisors of higher degrees, the lemma of §4,4 states that these decompose to first degree factors in a separable extension of constants. According to #,3 and §3,4 the divisor (dx) and the genus remain invariant under such extensions. The different divisor can, at most, pass to one of its proper divisors. But Eq. $3,(8) implies, using the invariance of the genus, that its degree remains invariant. Then so does b K / kNow, . (16) holds after this extension of constants; therefore it must have held before. 7 A field K is called conservative if its genus remains invariant under all finite extensions of constants. Equation (16) holds for, and only for, conservativefields.
Proof. The above argument shows (16) to hold for conservative fields. Conversely, let (16) hold and k,' be a finite extension of ko . For the extension K' = Kk,' we can deduce as follows: first, (dx) remains unchanged, second, as above, it is certainly a divisor of b K ' / k o ' ( x ) j m 2 , and third, b K ' / k o ' ( x ) j i 2 divides b,/,jZ2 = (dx). This implies that b K ' / k o ' ( x ) j Z 2 = b K / k & ) j m 2 , SO that the invariance of the genus follows from §3,(8). 7 Combining the last two theorems we have: a separably generated field K/k,(x) is conservative if bK/ko(x)is divisible only by separable prime divisors. In dosing we remark that recently a new notion of differentials, whose divisors always lie in the canonical class, was suggested.? It has the same
t H. J. Nastold, Zum Dualitatssatz in inseparablen Funktionenkorpern der Dimension 1, Math. Z. 76, 75-84 (1961). This paper, in turn, is based on E. Kunz, Differentialformen inseparabler algebraischer Funkrionenkorper, ibid., pp. 5 6 7 4 . A similar treatment is given by 0. Zariski and P. Falb, On differentials in function fields, Amer. J. Math. 83, 542-556 (1961).
158
111. ALGEBRAIC
FUNCTIONS AND DIFFERENTIALS
range of applicability as the divisor concept of the book [3] mentioned in $3,7. It fails to coincide with our divisors only in the case that the constant field ko is itself a function field in one or more variables, and then it uses differentiation in k,,, too. Thus this notion properly belongs to n-dimensional algebraic geometry.
$5. Differentials and Principal Part Systems
I . DIFFERENTIALS OF HIGHER DEGREES This paragraph is preparatory to Chapter V. Its subject matter can be considered to be an analogy, in an abstract algebraic sense, to integral calculus. We assume throughout that K is separably generated and that the divisors of differentials lie in the canonical class. We saw in $43that this is always true for conservative fields. Along with the previous differentials, the differentials of degree h, where h may be any rational integer, play a part. By such a differential we mean a formal expression dwh = u doh, with u and v in K and v a separating element. Set u duh = u’ dvIh
whenever u’
= u(dv/dv’)h.
For h = 1 this gives the ordinary differentials introduced in $4. With such a differential of degree h we associate an order v , ( d d ) = V,(U dd‘) = v,(u(do/dq)h)
for every place p, where q is a prime element there. This can again, of course, only be done for separable p, but a suitable extension of the constant field, as in $4,4, serves to generalize; a finite purely inseparable extension suffices in each case. Set (dWh) =
n
p y ~ ( d ~ h )
P
to define the divisor of a differential. By definition it is invariant under extension of constants. Then the divisors of two differentials as in $4,(15) satisfy dVIh) - ( ul ( dul)h) (Uzdu:) u.2 do, (u1
*
Thus it is seen that the divisors of all d d lie in the hth power of the canonical class, provided the field K satisfies the above assumptions. Among the differentials of some arbitrary degree h we now distinguish a certain set, whose importance will not become apparent until later. To this
55.
DIFFERENTIALS AND PRINCIPAL PART SYSTEMS
159
end we first fix a divisor a once and for all which we stipulate to satisfy one of the following assumptions: ( I ) either g(a) - 2(g - 1)(1 - h) 2 0, and a does not lie in the (1 - h)th power of the canonical class, (2) or a lies in the (1 - h)th power of the canonical class. We now say that the differential dw" is of the first kind with respect to a if a(dw")is an integral divisor. For h = 1 and a = (1) we also define differentials of the second and third kinds. In particular, 6w is said to be of the second kind if to every place p there exists a function wp E K such that vp(dw - dw,) 2 0 at that place. All other differentials are said to be of the third kind. For example, x - ' dx, where x is a separating element, falls in the last class. The differentials of the first kind with respect to a form a k,-module of rank (da) gh,n
= g(a)
+ ( 2 h - l)(g - 1)
+ (2h - l)(g - 1) + 1
in case (l), in case (2).
Proof. All (dw") lie in the hth power of the canonical class W,so that the integral divisor a(dw")lies in the class a Wh.The number of linearly independent a(6wh) can be determined from the Riemann-Roch theorem together with Eq. §3,(7). 7
2 . PRINCIPAL PARTSYSTEMS Hold the divisor a fixed throughout $5. By a principal part system dhn,= (dw,") of degree h for the divisor a we mean a set of differentials dwph of degree h over K, associated in a one-to-one manner with the places p of K , and such that the orders vp(a- w,") 2 0 for almost all p. We call two principal part systems (dw,h) and (du,") the same if vp(a- '(du," - dw,")) 2 0 for all p without exception. For instance, in the case a = (1) we are concerned only with the principal parts of the poles of the dw,h at p. Thus such a system is given by a finite number of data. Furthermore, the terminology is thereby explained. Under the extension of constants k,'/k,, the principal part system (dw,") of K passes to a principal part system (dw:.) of K' = Kk,' defined as follows. Say the places pQ'of K' lie over the place p of K . Then set dw&. = dwPh
(e = 1, ..., r).
This convention permits us to identify the principal part systems (dwph)and (dw:,) of K and K'. Principal part systems can be multiplied by elements u, p E k,, and added; u(dw,h)
They form a k,-module.
+ p(du,h) = (a d ~ , +h p do,").
111. ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
160
If to two principal part systems (do;) and (dw,h) there exists a differential dw" in K such that v,(a-'(dw,h - d o t - dw")) 2 0
-
holds for all p, then (du,h) and (dw,h) are called equivalent, written (dw,h) (do;). That this is really an equivalence relationship is clear. Principal part system classes form a k,-module. We call a principal part system d b exact if dhn, (0), that is, if a differential (dw") exists in K such that v,(a-'(dw,h - dwh)) >= 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
ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
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.
ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
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
$5. DIFFERENTIALS AND PRINCIPAL PART SYSTEMS
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.
111. ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
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=,
$5. DIFFERENTIALS AND PRINCIPAL PART SYSTEMS
167
The prime divisor ID of K corresponding to the ideal I D K x K , is called the diagonal divisor, or simply the diagonal of K K . If KK‘ is taken as a field of functions of one variable over the constant field K’, then 9 becomes a prime divisor of the first degree, for the residue class field of Z$, 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
111. ALGEBRAIC FUNCTIONS AND
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
111. ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
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.
ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
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.
REDUCTION OF A FUNCTION FIELD
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.
ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
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.
REDUCTION OF A FUNCTION FIELD
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
111. ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
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.
ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
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
111. ALGEBRAIC FUNCTIONS AND
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.
REDUCTION OF A FUNCTION FIELD
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
111. ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
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.
REDUCTION OF A FUNCTION FIELD
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
111. ALGEBRAIC FUNCTIONS AND DIFFERENTIALS
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(<), ..., e,(5). The radii of the U ( t J are the same as those of the U ( 0 , and all lie under the limits R(5). The U(r,) are associated with the r separate places tQ defined in 111, §2,2 to lie over the place of So,and are neighborhoods of them. Still lacking is a determination of the connectedness of this structure, which is now given as follows: a point (6. over belongs to a neighborhood U(t,) if, first, t’ belongs to the corresand, second, the power series 111,§1,(21) ponding neighborhood U ( t ) of corresponding to coincides as an analytic function in a sufficiently small circle about with the power series associated with t,, under a suitable choice for the e,th root ?J<.The neighborhoods of the places t plying over the points 4 of sothus form a Hausdorff space %; it is a covering space of So with n sheets. % also can be covered by a finite number of the given neighborhoods. % is called the Riernann surface of the function y. % is connected, and y is univalent everywhere on % and, except for poles, continuous everywhere. The same is naturally true for all functions of the field K . Conversely, every function univalent on % which permits an expansion 111,§1,(21) with nonvanishing convergence radius at every point of % belongs to the field K. (These functions are said to be meromorphic.) This last statement does not belong to algebra in the strict sense, as the notion of the Riemann surface is of essentially analytic nature.
r
r
r’
r’
2. THERIEMANN SURFACE AS
A
COMPLEX MANIFOLD
Just as a distinction is made between the field K and its particular model K = ko(x,y ) with f(y, x ) = 0, we also distinguish an (abstract) Riemann surface % from its particular description as a covering space of !No. A Hausdorff space is called a (topological) manifold if the neighborhoods U ( r ) of the points 5 are homeomorphic to an open Euclidean circular disk.
187
$1. RIEMANN SURFACES
The Riemann surfaces defined in the last section are then manifolds. A Hausdorff space is called a complex manifold if to every neighborhood U(5) there is given a homeomorphism q,,(<,which maps it onto the interior of a circle, this system of mappings having the following property : whenever two neighborhoods U(5) and V(q) have a nonvanishing intersection, the ~ ) V(q)) is a conmapping qv(o)q&defined in the region W = q o c c , ( U ( n formal, orientation-preserving mapping of W onto another region of the circle ( P Y ( r I ) ( W ) . the Riemann surface given in By choosing the mapping qutr,(x)= the last section is made into a complex manifold. The complex structure of a Riemann surface carries with it the concept of a conformal mapping of one Riemann surface onto another. A local homeomorphism f ( x ’ ) = x of %’ onto % is said to be conformal if in sufficiently small neighborhoods U ( t ) c % and U’(t’) c %‘ the mappings q~,,(~,fq‘;!(~., are conformal in their regions of definition [that is, in the regions c P u c e , ( f ( W ’ ) ) n U<5>)3. Two fields offunctions, K = k,(x, y ) with f ( x , y ) = 0 and K‘ = ko(x’,y ’ ) with f ’(x‘, y’), are isomorphic if and only if their Riemann surfaces % and %‘ are conformally equivalent.
‘dz;,
Proof. If 9’ 3 and 3‘ are conformally equivalent, then every univalent analytic function on % can also be considered as a univalent analytic function on %’, and conversely. Thus the univalent analytic functions of % and %’ coincide, so that the fields which these functions form, according to Ql,l coincide. If, conversely, K and K‘ are isomorphic, it is no loss of generality to assume K = K‘. Then x and y are rational functions of x’ and y’, and conversely. Each point of % is given by a value x = 5 and a power series y = But this also defines a value x‘ and a power series y’ = x a B t ( e ’ ~ z ~ that , ) P is, , a univalent mapping of % onto %’. The inverse mapping is clearly also univalent. It remains to be shown that this mapping is conformal. A prime element for a place 5, (cf. 111,§1,2) is called a local uniformizing variable here. It is a function q E K , vanishing at t,, with respect to which any function in K can be expanded in a power series. A prime element is a convergent power series
xaflcd<)”.
c m
q =
c”(eq<)”,
c1 # 0.
p= 1
The lemma of III,§l,3 showed that it has an inverse power series
‘q<= x c”q’, m
I(=
1
c1’ # 0,
188
IV.
ALGEBRAIC FUNCTIONS OVER THE COMPLEX NUMBER FIELD
which is also convergent. Equations of the same form can be written for and then substitution yields the convergent power series
"d;/,;,
'fie''6
and are the mappings of neighborhoods of tRand ti. onto Now, the interior of a circle, introduced above. We thus see that the last equation, taken in the above context, yields a conformal mapping of % onto 3' in a neighborhood of t Rand ti,. 7 The theorem just proved shows that a Riemann surface is associated with each function field invariantly, that is, independently of the particular model of the field. By the definition in 111,§2,2 the prime divisors jr, of K correspond one-to-one with the points of %. The invariance of % thus implies the invariance of the prime divisors.
3. THERIEMANN SURFACE AS A TOPOLOGICAL MANIFOLD It was seen in §1,1 that a finite number of the disks V(5)suffice to cover the Riemann surface %. The bounding circles themselves, any two of which have at most two points in common, divide % into a finite number e2 of simply connected open surface sections, a finite number e1 of curve segments, and a finite number eo of points. The alternating sum -eO
+ e' - e2 =I(%)
is called the Euler characteristic of %, According to the Euler polyhedron theorem it is the same for all subdivisions of % and thus a topological invariant of %. It is always even, and 9 = 9(W = +I(%)
+1
(1)
is called the topological genus of %. The genus g of a functionjeld K defined in III,53 coincides with the topological genus of the Riemann surface % of K . To prove this, consider % as the n sheet covering space of the sphere So, as in §l,l, and use a special subdivision. Say the ramification points lie over the points x = t(")(a = 1, ..., h) of the sphere illo. From some other point ((O) make nonintersecting cuts to all the points t("),cutting through all the sheets. % decomposes into n = e2 copies of a cutup sphere; these are simply connected. Altogether, hn = e1 cuts were made. % contains n points over t('), but only r ( t @ ) )
$1. RIEMANN SURFACES
189
For the ramification indices e,(((")), e = 1, ..., r(<(')), 111,§1,(19) states Cpe,(<(u))= n. We can also write r
C (e,(t('))
- I)
r = r(t(')).
= n - r,
Q=l
This is then used in the computation of (1): h
g(%)=t.(-eo+e'-e2)+1
=fC(n-r(((")>)-n+~, u= 1
In 111, $2,(36) the sum ~ ( e , ( < ( " ) ) 1) taken over o and e was seen to be equal to the degree of the different b K l k , and then, by 111,§3,(8),g(%) turns out to be equal to the genus of K . v It is a theorem of the topology of two-dimensional manifolds that, if its genus is g > 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
&<.>
+ + ,/<
p" = 0' - p
and, in particular, p" = - p for
CT
To,,,
for
CT
(9)
= CT,,~.
= CT o , o . Setting = =o,oCTo,o'
(10)
7
this yields, for every prim.: divisor of degree 1, p' = p
Because of (1 1)
t,.,~is
+ 0'
for
.
(11)
t = to,,.
called the translation automorphism to
0, 0'.
$2. FIELDS OF ELLIPTIC FUNCTIONS
195
If 'c' = T,,,.. is another translation automorphism, then by (1 1) we havet prr'
=p
+ 0' +
0".
But the translation automorphism 'co,of+o,, has the same effect on the prime divisors of first degree. This fact will enable us to prove ~o,o,'co,o,,
= 'c0.0' +o"
(12)
*
Indeed, in the algebraic closure of k , the reflection and translation automorphisms remain such, but all prime divisors are of the first degree. We have seen, though, that the automorphisms T,,,. 'c0,,,, and T ~ , ~ , have + ~ ~ the , same effect on these divisors, and this remains valid if k , is left unextended. By the lemma above, then, they are identical. The purpose of Eqs. (11) and (12) is the following theorem, which they clearly imply. The translation automorphisms form a group isomorphic to the additive group of prime divisors of the first degree and to the multiplicative group of divisor classes of degree 0. For the full group of automorphisms we have: the translations are a normal subgroup of the group of all automorphisms, and the factor group is finite. Proof. Let 0 be any automorphism and 'c = c',~ be a translation. Because of (1 l), application of 'c to a class of degree 0 amounts to multiplication by the class of o'/o. Hence ara-1
(9 - ( 7 0 ) paor
a-1-
P era-' Doa-'
p 0''
N - -
(13)
D O
with a certain o", and then the same argument that verified (12) now assures that 0'' = D'a-l - O a - ' . 0 T O , , , 0 - 1 = 'co,o,, , (14) This is exactly the first assertion. Now let oa = o', where CJ is still an arbitrary automorphism, so that CJ and 'co,oIboth map o onto 0 ' . Thus q , , , ~= - ~~- l ' c o , o - leaves o fixed. We see that the cosets with respect to the translation group each have a single representatice which leaves o Bxed All that remains to be shown is that the group of automorphisms leaving o fixed is of finite order. As this proof is more closely connected with the subject of tj2,6, we postpone it until then. No further use shall be made of this result. 7
+
+ +
t One might now incorrectly argue that prr' = (p 0')T' = p 0' 20". Instead, it should be noted that, in the sense of (4), v + 0' must be represented by a prime divisor q of the first degree. The formula is then clear.
196
IV. ALGEBRAIC FUNCTIONS OVER THE COMPLEX NUMBER FIELD
4. THEINTEGRAL OF THE FIRSTKIND From now on the field C of all complex numbers is taken for the constant field k,,. By 111,g the divisor of every differential then (and even more generally, if the constant field is perfect) belongs to the canonical class. According to the Riemann-Roch theorem there exists, up to multiples with constants, a single differential du with an integral divisor, and that divisor is then the unit divisor. We will now denote the points of the Riemann surface and the corresponding prime divisors with the same symbol. In particular, o will be the infinitely distant point over the x-plane. We introduce the integral oftheJirstkind u = U ( O , p) =
s.'
du,
(15)
and first compute it in a canonically incised copy 8 of the Riemann surface % (cf. $1,3). 8 is a direct connected surface section, and the integral (15) depends only upon the starting and end points. The fact that the divisor of the differential du is the unit divisor has the effect that, for a local uniformizing variable p at the place p, the differential quotient du/dp # 0 and # m. This is so for any place p. The image of 5 under the function u in a plane (the u-plane) is nonoverlapping and finite, that is, it is a finite, simp!y connected surface section 8,. 3. is a quadrangle bounded by four curves (a, b, a', b') which are, in general, curved. Now let p be a point on the side a of 8, and p' the other copy of the point p lying on a'. We compute the integral Jdufrom p to p' in 8. We may choose to do so by first following a path of integration along a until we meet the curve b. We then integrate along 6, and finally along a' until we reach p'. The integral consists of three parts, the first and last of which are equal but opposite, because du is the same on a and a'. Hence, J:'dU = I d "
=@b,
independent of the location of p on a. This equation shows that the images of p and p' in the u-plane differ by the translations of a fixed complex number q,whenever p and p' are corresponding points on a and a'. Of course the analog holds for p and p' on the boundaries b and b' of 8. Translations of the amounts
thus displace the image 3. of 8 in the u-plane in such a manner that 5. and the new images meet exactly along the boundaries b and a'. The translations by PO,, + VO, with arbitrary rational integers p and v
$2. FIELDS OF ELLIPTIC
FUNCTIONS
197
fully pave (i.e., cover without omissions or overlappings) the u-plane with copies of 8,. Proof. We give a concrete realization of the paving. The basic figure 3" is found by computing (15) for all points of 5.To the " basic path" from o to p lying inside &,, now add a closed path from p to p, homotopic to the path a. This yields the translated figure 8, w, . According to §l,3, every closed path on % is homotopic to p traversals of a followed by v traversals of b. Thus the addition of an arbitrary closed path to the path of integration increases the integral (1 5 ) by the number pw, + vwb. Now, say the regions 8, translated by all pw, + vwb cover some part U of the u-plane. There exists a bound E such that all u of distance not more than E from some arbitrary uo E 8. must lie in 3, or one of the eight neighboring regions &, pwa vwb with p, v = 0, f 1. As the translations leave the metric invariant, the fact that some uo E U implies that all u not further than E from uo also lie in U. This entails that U is the entire u-plane. We must yet eliminate the possibility that our regions (partially) cover the u-plane multiply. This would imply the existence of two different points p and q and some integration path in % such that u(o, p) = u(o, q) or u ( p , x ) = 0. Let the point x traverse this path W from p to q. Then u(p, x) follows a closed path W, in the u-plane. Such a path can be continuously deformed into a point. But then W is also deformed into a point which is impossible, because p # q. The theorem is proved. 7 These pw, + vwb which we have found are the periods of the integral u. They form an abelian group generated by the elements w, and wb , called the period module o = [w, , wb]. The points p of R ! and the residue classes of the complex numbers mod o are placed in one-to-one correspondence by (I 5). This is already proved once it is shown that one point p is associated with every residue class u mod o. Choose some representative of this class in 3,.Then, as 8 is mapped onto 8, homeomorphically, exactly one p corresponds to that representative. Note the incidental result: the ratio
+
+
+
(16)
7 = wa/wb
is not real. For otherwise, the translations by and w, would be parallel, and then, 8, being a proper surface section, only a strip of the u-plane could be paved. 5. THEADDITION THEOREM AND THE ABELTHEOREM
As the differential du is determined up to a constant factor by its divisor, we have, for any automorphism 6, du" = pa du,
pa # 0 E k,
.
(17)
198
IV.
ALGEBRAIC FUNCTIONSOVER THE COMPLEX NUMBER FIELD
For a reflection automorphism u we have p,, = - 1, while if r is a translation automorphisrn then pc = 1. Proof. Let 0 , 0‘ be two prime divisors and z an element with the divisor (z) = j(oo’)-l. Our differential of the first kind is put in the form du = dz/w, so that, by 111,§4, its divisor takes the form
with the different divisor bK/ko(r).This gives
From 111,§2,(35) and el = = 2 we see that b&ko(r)is a divisor in k,(z). All divisors of degree 0 in k,(z) are principal, so that w z E k,(z). Now, w itself cannot lie in k,(z), as this would make du a differential of k,(z) so that the corresponding divisors in the two fields would satisfy (dU)K
= (dU)ko(z)
bK/ko(z)
But b K / k o ( ras ) , the product of ramified divisors, could not compensate for the poles of (du)ko,,,. According to §2,3, K is an extension of degree 2 of k,(z). So we see that K is obtained from k,(z) by the adjunction of the square root of wz. Furthermore, the reflection automorphism u = uo,o,is that of K/k,(z), and thus it replaces w by - w . This means that du‘ = -du, as asserted. Definition (10) then immediately gives the result for the translation automorphism. T Application of the translation r = to,,.leads to the formula Jop
du
=Jop
du*-l =
jo: = ~ o ~ + o ’ du
du
mod w
(18)
in which p + 0‘ must be considered a point (or prime divisor) in the sense of $2,2. A result is the following Addition theorem for integrals of the first kind: joPI
du+
s.”
fI+PZ
du=
du
modw,
(19)
where the coordinates xp, y p of the points p = p1 , p2, pi + p2 necessary to compute the integrals are found by the process of §2,2. Indeed, because of (18) with p = p 2 , 0’ = pl, the left side is = ~ o p ’ d u + [ ~ ~ + p z d um o d o . T
$2. FIELDS OF ELLIPTIC FUNCTIONS
199
This surprising interplay between the results of algebra and analysis undoubtedly represents a highlight of the theory of algebraic functions. More surprising, though, is the fact that it was discovered by Euler long before any systematic theory of algebraic functions existed. A special case (the socalled case of the lemniscate, with p1 = p , ) was even known earlier to Fagnano. Because the additive group of prime divisors is isomorphic to the multiplicative group of divisor classes of degree 0, one can express (19) as: By representing the divisor classes of degree 0 as quotietits PO-' of prime divisors and assigning to them the integrals (15), the multiplicative group of divisor classes of degree 0 is mapped homomorphically onto the additive group of residue classes of complex numbers with respect to the period module w . Abel Theorem. This mapping is even an isomorphism. In other words, a divisor of degree 0 and with the prime decomposition p1 ph(q, ..-q h ) - l is a principal divisor if and only is the congruence
f: j : du = 0
mod w
i= 1
holds. The irrelevance of the numbering of the pi and qi can be seen, for example, from the difference of the integrals
J::
du
+ J:
du,
J::
du
+
1;:
du
being the same integral over the closed path q,, pl, q,, p , , q, which is G 0 mod w , as all closed paths can be combined from the paths a and b. Proof. Equivalent to the assertion is the Statement: ... qho-h if and only if
q,
p,
... p h o - h
-
holds. By introducing two further points in the sense of (4),
and remembering (19) we arrive at another equivalent form of the assertion: PO-' q0-l if and only if
-
Jop
holds. But by $2,l, PO-'
N
du
=
j:
du
mod w
qo-' is equivalent to p = q. Now, if the path of
200
Iv. ALGEBRAIC FUNCTIONS OVER THE COMPLEX NUMBER FIELD
integration is chosen completely within a copy of % with canonical incisions, then the last congruence is equivalent to the equation
1: =IOq du
du.
As the mapping of % onto 5. in the u-plane given by u is one to one, this equation can only hold if p = q, so that the theorem is proved. 7 At this point we break off our line of thought. In $3, we will carry these theorems proved for elliptic functions over to fields of higher genus. To close $2 we undertake some considerations, a part of whose importance lies in their applications in &I. 6. THEWEIERSTRASS NORMAL FORM
In this section we again permit arbitrary ko . For simplicity’s sake, though, we assume the characteristic not to be 2 or 3.t Let x and y be the elements of $2,1 which generate K, and consider the divisor class of 0 6 . Its dimension, by the Riemann-Roch theorem, is 6. But, 1, x, x2, x3, y, y2, xy are seven multiples of o - ~ ,which means that there must be a linear relation in ko between them. The element yz must occur in it, for otherwise K = ko(x);and x 3 must also occur, if K is not to have genus 0 by 111,§3,5. Hence, K is defined by an equation y2
+ (ax + b)y + cx3 + d x z + ex +f = 0, a ,...,f e k , ,
c#O.
For characteristic unequal to 2 and 3 we can set x
+ ( d - az/4)/3c = cx’,
2y + ax
+ b = c’y’
and then again write x and y for x’ and y‘. The equation becomes y z = 4x3 - gzx - 8 3 ,
(21) gz 9 3 E ko which is called the Weierstrass normal form of the generating equation of an elliptic field. Of course the polynomial on the right cannot have multiple zeros, that is, its discriminant does not vanish. U p to the factor 16 it is given by gZ3- 2 7 d . t 1
9
t Cf. M. DEURING, Invarianten und Normalformen elliptischer Funktionenkorper, Math. 2. 41, 47-56 (1941), continued and corrected in M. DEURING, Zur Theorie der elliptischen Funktionenkorper, Abh. Math. Sem. Univ. Hamburg 15,211-261 (1945).The cases excluded here are investigated there. 2 B. L. VAN DER WABRDEN,Algebra, 2nd ed., Vol. I, $58. Berlin, 1937. (This edition reprinted New York, 1943.)
$2. FIELDS OF
ELLIPTIC FUNCTIONS
20 1
The substitution x-,
t2X,
y-,
r3y,
t # 0,
leaves (21) in the same form, but with the coefficient change g2
-+
tP4g2,
(23;
g 3 -,t d 6 g 3 .
Let the characteristic be # 2 , 3. The Weierstrass normal form then corresponds uniquely, up to the substitution (22), (23), to the field K . The inLwiunt j ( K ) = 123g23/(g23 - 27g3’)
(24)
remains unchanged by such substitutions. Proof. Let y‘2
= 4x13
- g2’x’
- g3’
be some other equation generating K . Then (x‘) = X ‘ O ’ - ~ with a first degree prime divisor 0‘. The reflection automorphism o0,+ maps x’ into an element with denominator 02, that is, into an element a + fix with a, B E k , , fi # 0. Thus there is no loss of generality in assuming that x’ = a + Bx. Equation (25)then shows that y’ is of the form y’ = y + 6 x + EY with y, 6, E E k,, E # 0. These quantities substituted into (25) must yield (21), which is only possible if a = y = 6 = 0 and p 3 = E‘. Hence fl = t 2 , E =t3 with some nonzero t E k,. This is the substitution (22) which leads to (23). v Let the characteristic be 2 2 , 3. For every c E ko there exists an elliptic function field K with the invariant j ( K ) = c. Proof. For c equations
=
123 and c = 0 the fields in question have the defining yz = 4(x3 - x),
y2
= 4(x3 -
I).
The older literature calls the first of these lemniscatic, as the field is found in the computation of the arc of the lemniscate, while the second field is called equianharmonic. For any c other than 123 or 0, the field defined by y z = 4x3 - 3c(c - 1 2 3 1 ~ c(c - 123y
has that c as its invariant. 7 Let the characteristic be 2 2 , 3. If twofields, K and K , defined by Eqs. (21) and (25), have the same invariants, j ( K ) =j ( K ) ,then there exists an extension k, of the constant field ko such that the extensions K, = Kk, and K,’ = K k l are isomorphic. This k, can even be chosen such that [k,:k,] S 2,4, or 6, depending upon whether 123 # j # 0,j = 123, or j = 0. Proof. A substitution (22) with tin a suitable extension k J k , can be found yielding g2’ = g 2 . The equality of the invariant then assures that g3’= & g 3 .
202
Iv. ALGEBRAIC FUNCTIONS
OVER THE COMPLEX NUMBER FIELD
Should the minus sign hold, a further substitution (22) with t = (is still necessary, but then (25) has been transformed to (21). This shows the equations g3' = t - 6 g 3 g2' = t - 4 g 2 , to be solvable in a suitable k,. In the three respective cases, a second, fourth, or sixth root must be adjoined to k , . 7 These same considerations also show that the automorphisms of K which leave o fixed must be of the form (22) and even satisfy the equations g 2 = t - 4 g 2 ,g 3 = t - 6 g 3 . Let the characteristic #2,3. Then for 123 # j ( K ) # 0, j ( K ) = n3, j ( K ) = 0, respectively, the group of automorphisms leaving a jirst degree divisor of K invariant is the cyclic group of order 2,4, or 6 elements. This proves an assertion made in §2,3. But the cases with characteristic 2 and 3 are also needed there. Such an automorphism, though, satisfies xu = a
+ /Ix,
y" = y
+ 6x + ~ y , BE # 0.
By substitution into (20) it is seen that only a finite number of values are possible for LT,... . The exact procedure is left to the reader.? 7. ELEMENTARY ELLIPTIC FUNCTIONS We conclude with several formulas, referring to other texts for the proofs.$ Consider the classical case, and let K be defined by the Weierstrass normal form (21). Then, the Weierstrass p-function
relates x to the integral u of the first kind. The prime over the summation symbol indicates that the pair p = v = 0 is to be left out. Up to transforms under p(u) + o(p(u) + /I with constant tl # 0, /I, this p(u) is the only analytic function of the residue classes of the complex numbers mod w with only one pole, of the second order at u = 0 mod w , corresponding to the denominator divisor o2 of x. The coefficients of (21) are then
f It can be found explicitly in the second paper by Deuring mentioned on page 200. K. KNOPP,Funkrionenfheorie, Vol. 11. Goschen, Leipzig, und Berlin, 1926. English translation, Dover Press, New York, 1945. A. HURWITZand R. COURANT, Funkfionenfheorie,3rd ed., Berlin, 1929. Revised edition, Springer-Verlag, Berlin, 1964.
$2. FIELDS OF ELLIPTIC FUNCTIONS
203
The series (26) and (27) converge absolutely and uniformly in u, o,,c u b , if u remains in a compact region of the u-plane not containing any lattice points po, + v q , and the quotient z = w,/ob remains in a compact region of the z-plane not containing any real points. The verification that du = dx/y is the differential of the first kind is an easy application of (21). But this means that y = dx/du = ~ ’ ( u ) ,so that p(u) satisfies the differential equation
P’(u)2 = 4 P W 3 - g2&3(u) - 9 3 The series (27) are now seen to be the coefficients g 2 and g 3 by substitution of (26) in this equation. 7 This implies the following. For any pair o,, wb with nonreal quotient, there exists an elliptic function field whose integral of the first kind has that pair as its periods. The integral x dx 1‘ du (4x3 - g2x - g p
=jop
I
is of the second kind. Its only pole is at the point 0 , and it is of the first order with residue 1. Any integral of the second kind is of the form cir + Pu + z, with a, E ko , z E K. The proof is contained in the final theorem of 111,§5,4. Exercise. Express the reflection and translation automorphisms explicitly in terms of u and the functions x = ~ ( u )y, = p’(u) [use Eqs. (7) and (8)].
NOTES In $2,2 and 3 we essentially followed a paper [2] by Hasse. The finiteness of the group of automorphisms leaving a point fixed is proved differently there, though. The paper cited is only one of a series, whose full contents will come up later. We shall then also discuss the extensive literature related to it, as well as that of other topics concerning elliptic functions (V, $6). In this book we omit discussion of two problems in the theory of elliptic functions. The first problem is the connection between elliptic functions whose periods are related by equations 0,’
= aw,
+ pob
9
ob’
= yo,
+ 6wb
9
a6 - Py # 0
with rational a, P, y, 6. An introduction to this question can be found in Tricomi’s text [7], while Igusa [S] treats it from a modern point of view. The second problem is the description of all prime divisors of the first degree for the case that ko is the rational, or, more generally, an algebraic number field. A theorem of Mordell [6] shows that they form a group, in the
204
Iv. ALGEBRAIC FUNCTIONS OVER
THE COMPLEX NUMBER FIELD
sense of $2,2, with a finite system of generators (see also Weil [S]). A new proof was given by Hasse [3], while A. Weil generalized the theorem to fields of higher genus. Newer results as well as extensive reference to other literature are found in Cassels [I]. Hofmann [4] surveys Euler’s work in this field.
REFERENCES J. W. S. CASSELS, Proc. Internat. Congr. Mathematicians 1962, Stockholm, pp. 234-246. Zur Theorie der abstrakten elliptischen Funktionenkorper 11, J. Reine Angew. H. HASSE, Math. 175, 69-88 (1936). H. HASSE,Der n-Teilungskorper eines abstrakten elliptischen Funktionenkorpers als Klassenkorper, nebst Anwendung auf den Mordell- Weilschen Endlichkeitssatz, Math. Z . 48,48-66 (1942). J. E. HOFMANN, Uber zahlentheoretische Methoden Fermats und Eulers, ihre Zusammenhiinge undihre Bedeutung, Arch. History Exact Sci. 1, 122-159 (1961). J . 4 . IGUSA,On the transformation theory of elliptic functions, Amer. J. Math. 81, 436452 (1959). On the rational solutions of the indeterminate equations of the third and L. J. MORDELL, fourth degrees, Proc. Cambridge Philos. Scc. 21 (1922). F. TRICOMI, Funzioni ellittiche, Bologna, 1937. [German revised edition, edited and translated by M. KRAFFT,Ellbtische Funktionen. Leipzig, 19481. A. WEIL,Sur un thdorPme de Mordell, Bull. Sci. Math. 54, 1-10 (1930).
$3. The Group of Divisor Classes of Degree 0 In fields of genus g = 0 this group consists only of the unit element, according to 11,$5,1 and 111,$2. For fields with g = 1 we at least determined this group for the classical case in $2,4 and 5. The method used there is not applicable to higher genus, though, for the translation automorphism ( $2,3) has no analog. Such can only be found again in certain associated function fields of g variables.? We attain our goal by a different method, for which we must first make several preparations. 1 . THERIEMANN PERIOD MATRIX
Our considerations concern an algebraic function field K with constant field C, the trivial case in which the genus g = 0 being excluded. Canonical incisions are made in the Riemann surface W, as in $1,3. The differentials are considered divided into the three kinds, as in 111,$5,1. Let du, (i = 1, ..., g) be a basis of differentials of the first kind, and du, be differentials of the second kind satisfying Eqs. 111,$5,( 12).
t For the extensive theory of abelian function fields or abelian varieties see S. Lang Abelian Varieties. New York, 1958. A complete bibliography is found there. Classical function-theoreticalmethods are not used there, as opposed to the book by F. Conforto, Abelsche Funktionen und algebraische Geometrie. Berlin-G6ttingen-Heidelberg,1956.
$3. THE GROUP OF DIVISOR CLASSES OF DEGREE 0
205
For every differential of the first kind du, form the integral of thefirst kind du,
u =JP:
with the lower limit po held fixed while the upper limit p is allowed to vary over %. Integrals of the second and third kinds are introduced correspondingly. The path of integration must be given before these integrals are uniquely defined, a continuous deformation of that path being permitted though, so long as all residues are 0, that is, for differentials of the first and second kind. Given only the limits of (I), two such integrals may differ byanintegral over a closed path w. Now, by $1 any closed path w is made up of the paths a,, 6 , . By calling the integral over w the period of u corresponding to w, we can say that all the periods are linear combinations of the base periods
with rational integers as coefficients. The matrices R
=
(It), ,):(
ll = (RH),
H=
(3)
the first two having 2g rows and g columns and the last with 2g rows and columns, compile these periods. R is called the Riemann period matrix. The periods q;, ~,.t; do not change when differentials of functions of K are added to the dvj, as these have periods 0. And, the duj are determined only up to these by the du, . The scalar products of differentials of the first or second kind du, dv, introduced by 111, &(lo), can, with the help of the Cauchy integral theorem, now be written as (du, du) =
2 res,(u
du) = (2ni)-
P
I
’
u du,
(4)
taken over the canonical system of incisions a,, b, . The relationship of the function u on a, and a,’, and on b, and bit, is expressed by u(ai’) = u(ai)
+
du = u(ai)
+ o:,
s.. i
u(b,’)
= u(bJ
+
du = u(bi) -
Ia,
du = u(bi) - 0;.
Hence the integral on the right in (4), taken over the paths a i , b , , a,’, b,’, is
s
u do = -w:
do f.i
+ W:
do = 0:11ib Ibi
- w:v;.
206
IV.ALGEBRAIC FUNCTIONS OVER THE COMPLEX NUMBER FIELD
All that is left to compute (4) is summation over the i. Hence we can express the relations 111, §5,(12) in the form
The single matrix equation
n‘Jn = 2niJ,
(5)
where J is as defined in the appendix to Chapter I, §l,(l), compiles these results. This means that (2ni)’/211is a complex symplectic matrix. Equation (5) expresses the Riemann period relations, also called the Legendre relations in the elliptic case. Replacing the canonical incision system used by another such system leads to periods that are linear combinations of the old ones, with rational integers as coefficients. Indeed, all closed paths can be combined from the a i , b i . This implies that the new period matrix is 111 = Sn,with an integral 2g by 2g matrix S. S must also be symplectic, as (2ni)1”I11 is symplectic. Under a change of canonical incisions the period matrix transforms by a matrix S in the symplectic modular group P.t 2. A HERMITIAN METRICFOR DIFFERENTIALS OF THE FIRST KIND
Let z be some variable of K. Then % can be seen as a covering space of the complex z-plane, and the decomposition into real and imaginary parts, z = x + iy, has meaning. Now, for two differentials du and dv of K consider the surface differential
it represents a surface differential on %, which is independent of the variable z used. For, if z’ is some other variable, then
du
I , 1’1
du dv d z --d x ’ d y ’ = du -dz‘ dz‘ dz dz’
a(x’, y‘)
1
d x dy7
the Cauchy-Riemann differential equations giving the functional determinant
t Conversely, to every symplectic modular matrix S there exists a canonical system of incisions. Cf. C. L. Siegel, Ausgewiihlte Fragen der Funktionentheorie, Lecture Notes, Vol. 11, p. 108 ff. Gottingen, 1954.
$3. THE GROUP OF DIVISOR
CLASSES OF DEGREE 0
207
The differential du dij is finite everywhere on ‘3 for two differentials of the first kind, and its integral taken over the entire Riemann surface, {du, dv} =
ss
(7)
du dij,
91
exists. This dejines a hermitian metric in the space of diyerentials of the first kind, that is, it has the following properties: {a1 du,
+ a2 du2 , duJ
{ d u , do} = {do, d u } ,
= al{dul, d u 3 )
{du,du} > O
+ a 2 { d u 2 ,&I, if du # O .
The surface differential can also be expressed in terms of divergence. The indefinite integrals Jdu = u and jdv = v are introduced, and then du
+
du
div(uEx, uVY) = uxEx uyOy= ux(ijx+ icy) = 2u,Vx = 2 - d z dz * By the Gauss integral theorem the surface integral (7) is equal t o the integral
‘I 2
~ ( d 6y -~ ijy d x ) =
taken along the path of the canonical system of incisions. This line integral can be expressed in terms of the periods exactly as the integral $u do of the last section. This yields
The submatrix (at)of the Riemann period matrix is nonsingular, for otherwise there would exist a first kind differential du # 0 with periods 0 on the incisions a , . This would imply {du, du} = 0 in (8), contradicting the property of the metric. By applying the linear transformation with the matrix (w,~)-’ to the dui, the matrices (myj) and (ofj) pass to E and T =(W!~)(W;~)-~.
(9)
The matrix T is symmetric and its imaginary component is the matrix of a positive definite quadratic form (or, for short, its imaginary component is positive definite). Proof. By the above it is no loss of generality to assume that (myj) = E and (ofj)= T. The symmetry of T follows from the fact that (2ni)’’’II is a
208
IV.
ALGEBRAIC FUNCTIONS OVER THE COMPLEX NUMBER FIELD
symplectic matrix, by the appendix to Chapter I, §1,(3). Furthermore, (8) becomes
But this is a real symmetric matrix. The metric (7) being definite, our second assertion is also proved. 7 Let (i i) be an element of the symplectic modular group P.In §1,(3) of the appendix to Chapter I it was seen that then the same is true of (i :). But it is by such a matrix that the periods of the du, are transformed, according to the last section, under a change of incisions. Equation (9) then becomes T' =(AT B)(CT D)-'.
+
+
This lends concrete importance to the substitutions that were investigated in $1 of the appendix to Chapter I. 3. ABELIAN INTEGRALS OF THE THIRD KIND Our next task is to construct special differentials of the third kind, by which all others can be represented. Let p , q be arbitrary points on %, not coinciding. We need a differential dw'(p, q) whose divisor (dw'(p, 9)) = bp-'q-' with integral b prime to p and q. The class of b is Wpq which, by the Riemann-Roch theorem, has a dimension dim(Wpq) > dim(Wq). Thus, such a divisor b, prime at least to p , exists. The corresponding differential has a pole of order 1 at p , and can have, at the most, another such pole at q. The residue theorem of 111,§4,7 assures that this pole really does occur, and that the residues at p and q are equal but opposite. Multiplication by a constant sets the residues of dw'(p, q) at p and q to 1 and -1. This property of dw' ( p , q) is not lost when it is replaced by dw ( p , q)'= dw' ( p , q) - du with some differential du of the first kind. We try to choose the latter in such a manner that the periods of the indefinite integral W(P,
9) =
s
M P , 4)
along the incisions a , , b, of a canonical system for %, as in §1,3, become pure imaginary. This will turn out to be possible in a unique way, and then dw(p, q) is called the normal integral of the third kind for the points p , q. In the search for such a du we start with the normal integrals uj of the first kind, whose period matrix Q has the two parts (w;) = E, ( o z ) = T . Let the periods of w'(p, q) be qf, $. Express the desired differential, or
g3. THE GROUP OF DIVISOR CLASSES OF DEGREE 0
209
rather its integral, formally as u = cajuj, so that the aj can be determined from the equations Re(t$) = Re
c ajwt1,
Re(?!) = Re
( j
c ajwt).
(i
The first system of equations has the matrix (coyj) = E, and thus the solution Re(aj) = Re(q;). The imaginary components remain without influence, and are then computed by -Re(&
+ c Re(crj) Re(wfi) = c Im(crj) Im(wpj). i
j
The matrix on the right is (-i/2)(T - T) and, being the coefficient matrix of a definite quadratic form by the last section, it has a nonvanishing determinant. 7 Let some arbitrary differential dw have the poles pv with the residues e,, where = 0 because of the residue theorem. The differential dw, = dw x e , dw(p,, q) has poles of residue 0 only (where we take dw (4,q) = 0), and is thus of the second kind. By 111, §5,4 then, dw, is a linear combination of g differentials of both the first and second kinds as well as an exact differential (that is, the differential of a function of K ) .
ce,
4. ABEL’STHEOREM A divisor of degree 0, x=
n
vv/qv,
V
where pv and q, are an equal number of prime divisors, is a principal divisor if and only if there exist 2g rational integers e;, e:, such that for j =1, ...,g the e m t i o n
holds. The duj here are an arbitrary basis of differentials of thejirst kind and the wt, their periods, computed in 43,l. The particular pairing of the p, , qv is irrelevant, as already seen in §2,5. Proof. (a) First let x = ( x ) be a principal divisor and let all the pv and qv differ from another. Then log x = [dx/x is an integral of the third kind. The Riemann surface % is assumed to have a system of canonical incisions, as in §1,3, that do not cut the poles of dx/x. Further incisions, from some border point 1: to the points p,, q,, are made without intersections. Along the border of the cut-up surface take the integrals
s
(2ni)-’ log x dui,
(12)
210
Iv. ALGEBRAIC FUNCTIONS OVER THE COMPLEX NUMBER FIELD
which must vanish according to the Cauchy integral theorem. The integrals (12) consist of two parts. The first of these, over the incisions a , , b,, can be expressed as in §3,1 by the periods q,", q: corresponding to log x and by the periods of the uj as
Now, log x is determined up to integral multiples of 2ai. Hence qt = -2aie,", q: = -2nie,b with integral e,", e:. This gives the right side of (11). The other parts of the integrals (12) are (2ai)-11(
jrp"
log x duj
V
+
s,.
' log x duj +
Irq' +1:" log x duj
log x d u j ) *
where the paths rp, and pvr lie on the different edges of the same incisions. On the second of these, 2ni must be subtracted from log x . The integrals over rq, and q,r correspond, but here 2ai must be added. Thus these integrals contribute the sum
-2(
~ r p ' d u j - ~ r q " d u j=) - x [ : : d u j
V
V
to (12). This proves ( 1 1 ) for the case that the p, and q, all differ. If, say, pi = ... = p,,, and the other p differ from pl, then dx/x has the residue h at pl, so that the second part of (12) contributes -hjrp'duj. This is in concurrence with the asserted equation ( 1 l), where p1 occurs h times as upper limit. (b) Now let (1 1 ) hold with certain rational integers e,", e.: Using the normal integral w(p,, q), of the third kind, set
w
=
C W(P,
9
9,)'
V
This is an integral of the third kind with logarithmic singularities at the pv,q, but no others. It is diminished by 2aih, every time pv is encircled in the negative sense, where h, is the number of points pp equal to pv ; and it is augmented by 2nih,', every time q, is encircled in the negative sense, where h,' has the analog meaning. If the periods of w on the incisions a , , 6,are also multiples of Zai, then x = ew is a univalent function on % with the zeros pv and poles q, of correct multiplicity. A remark made in 01 then assures that x E K so that x is the principal divisor (x). The theorem would then be proved. Hence we need now only show that the periods of w at the incisions a,, b, are multiples of 27ri. The first part of the proof can be used literally, replacing
$3. THE GROUP OF DIVISOR CLASSES OF DEGREE 0
211
log x by w to yield the equation U
= (2719-l
C (q: 1=1
oFj - q: ofj).
Introducing the abbreviations
we may use (11) to write
c U
(cl"O;j
- C,"Ofj)
= 0.
1=1
Now clearly our theorem does not depend upon the use of a particular basis for integrals of the first kind. According to §3,2 we can choose a basis which has (w;) = E. Equation (13) becomes
We finally apply the fact that the qp, q: were constructed so as to be pure imaginary. Then, the cp, C: are real, and the equations conjugate to (14) also hold. They can be subtracted from (14), yielding -i
C cp(orj - zFj) = 0. 1
But the determinant of this system of equations cannot vanish, as its matrix represents a positive definite form, by §3,2. Thus, the c: =0, and by (13) the C: = 0 as well. The proof is finished. 7
5. THEJACOBIAN VARIETY With any divisor class A of degree 0, represented, say, by the divisor n p , qV-l, associate the vector c(4 =
(21:; 21:; du1,
du,)
a * - ,
(15)
V
V
in the g-dimensional vector space C ' , some path of integration being arbitrarily fixed for the present. Further, introduce the 2g period vectors 0:
= (Ofl,
...,
0:
= (Ofl, ...,0").
(16)
They are linearly independent over the real numbers, a consequence of our
Iv. ALGEBRAIC FUNCTIONS OVER THE COMPLEX NUMBER FIELD
212
demonstration that the system of Eqs. (13) can have no nontrivial real solution. These period vectors span a point lattice D in CB,which forms an additive group. The residue classes of the points of CBmodulo D form a finite variety 3,the so-called 2g-dimensional (real) torus. J is called the jacobian variety of K. As the factor group CQ/Dof the additive vector group CBwith respect to 0, J itself is an additive abelian group. The Abel theorem states that (15) associates every divisor class of degree 0 with a unique (i.e., independent of the representative np,q;' of A) point of CB/D= J. Moreover, we can prove the following. Equation (15) deJines a one-to-one mapping of the divisor classes of K of degree 0 onto the entire jacobian variety. Furthermore, this mapping satisfies c(AB) = c(A)
+ c(B).
(17)
This theorem gives a complete mastery of the divisor class group of degree 0, for the classical case. In formula (17) we also have the addition theorem for abelian integrals, which took the form §2,(19) for g = 1. For, in that case multiplication of classes and addition of points coincide by the convention of 52,2. Proof. Let n p v q ; ' , ns,t;' be representatives of the classes A and B. Then c(AB) =
(5;
dui f
zj::
du,) E c(A)
+ c(B) mod 0.
Taking the vectors as points in J, this is (17). Now let g arbitrary points pl, ..., p s be given on %, and let z, be local uniformizing variables at the p v . There then exist constants E, > 0 such that to every value of z, E C with lz,l< E, there corresponds only one point py' in a suitable neighborhood of py, and conversely. Construct g algebraically independent copies K,, ..., KB of K, with fixed isomorphisms K, + K. Compute the integrals
=Iqv ?V*
U,(ZS
.
du,
in K, They are analytic functions of the z, when lz,l< E,, and their derivatives with respect to z, are the functions du,/dz, E K, . The du, being linearly independent and the columns of the matrix (du,/dz,) lying in algebraically independent fields, the determinant of that matrix is a nonvanishing function in the composite of those fields. Setting this determinant to 0 thus defines an algebraic variety, which certainly can cover no complete neighborhood of the point z, = .'. = zo = 0.
$3. THE GROUP OF DIVISOR
CLASSES OF DEGREE 0
213
Thus in some sufficiently small neighborhood of that point a complex g-dimensional domain (an open connected point set) of the z-space exists on which lauj/dz,l # 0. It is mapped onto a domain 53, of J by the uj(z,). But each point of the domain also corresponds to a system of g points pl', ..., pe' on %, by the isomorphism K,+ K . This means that a point of R, corresponds to every divisor class. Abel's theorem assures that it can be only one, of course. Now let R be the set of all points of J to which divisor classes correspond. By (17), R is a subgroup of J. Furthermore, R contains a domain 53,. If fo E R,, then R, - to is a domain of R containing the neutral element, and for simplicity we can say A, is such. Were R # J there would exist a frontier point f of A. Were f E A, then so would f Ro c R, and f 53, would be a domain containing f, contrary to the assumption that f is a frontier point. But, were f 4 52, then also f + R, $ R, leading to the same contradiction. Thus R # J is refuted and the theorem proved. 7
+
+
NOTES
As we already mentioned in §2,8 for the case of elliptic functions, the divisor classes of degree 0 form a finitely generated group if ko is a finite algebraic number field. The newest proof of this fact is found in [2]. It is essential for the theorem that the group of divisor classes of degree 0 is abelian and also represents an algebraic variety defined over k , . Such a group is called an abelian variety. Abel's theorem gives an analytic model for this group for the case k, = C in the 2pdimensional period torus. Orientation into the theory of abelian varieties is given in [l]. The theory of abelian integrals has been generalized as follows. Let D(: I;) be a representation of a group 8 of fractional linear substitutions W,(l) to which a field of automorphic functions belongs. Now, consider one-column matrices of differentials that transform in the manner az+b
a b
ar+b
'( a 3=) D ( c d)u(T) dT unden elements (:
i) of 8.Indefinite integrals U(T) =
s:.
U(T)
dz
undergo substitutions of 6 as follows:
U(-)aT CT
+b +d
+
= D('
c
b)U(~) D (: d
f;),
214
Iv. ALGEBRAIC FUNCTIONS OVER THE COMPLEX NUMBER FlELD
where the D3(", :) are constant one-column matrices, depending upon the element involved, called the periods. The system of these periods represents a generalization of the Riemann period matrix (3). Information concerning the present state of this theory is found in a paper by Shimura [4] (cf. also [3]). Shimura remarks that under certain assumptions these periods can be associated with analytic models of abelian varieties. To date nothing is known about the meaning of these varieties for the function fields.
REFERENCES S. LANG,Abelian Varieties, Interscience tracts in pure and applied mathematics, No. 7. New York-London, 1959. Rational points of abelian varieties over function fields. Amer. [2] S. LANGand A. N~RON, J. Math. 81.95-118 (1959). [3] M. KUGA and G. SHIMURA, On vector differential forms attached to automorphic funcfions, J. Math. Soc. Japan 12,258-270 (1960). [4] G. SHIMURA, Sur les integrales attachkes aux formes automorphes, J. Math. Soc. Japan 11,291-311 (1959).
[I]
$4. Modular Functions
The field of modular functions is among the few examples for which the uniformization is completely known. The arithmetic definition of the modular group = r', and with it of the Riemann surface of modular functions, permits the deep application of this theory in arithmetic. 1. THEMODULAR SURFACE
In the following we build up on $01 and 2 of the appendix to Chapter I and on IV,$l. The modular group consists of the fractional linear mappings T-+$=--
+b cr+d-(:
a?
:)T,
a,b,c,dEZ,ad-bc=l,
(1)
of the upper half-plane Sj onto itself. For (1) we introduce the abbreviation LT,where L can be any two-row matrix. We also remark that the mapping (1) and the determinant ad - bc define the matrix (: :) up to the sign, thus exactly bivalently. The mappings (1) represent motions of the PoincarC model Sj of the hyperbolic plane. A fundamental domain 5 exists and was found in the appendix to Chapter I, $1. Identification of the frontier points of 5 equivalent under r yields a surface R. To it is added the point im. This makes 3 a closed surface of genus g = 0, the modular surface. fi is an infinite-sheeted covering space of R, where a
$4. MODULAR
21 5
FUNCTIONS
point T E !ijis said to lie over a point T~ E 3, that is, over T~ E 8, whenever T and T~ are equivalent under r. The neighborhood of a point T E $ lies ~ smoothly over a neighborhood T~ E 8 except for the points over u1 = ico,
u2 = ( = $( - 1 + iJ3),
u3 = i
for which the number of sheets of !ij cyclically connected are u1 = 00, v2 = 3, u3 = 2, respectively. We proceed to make % into a Riemann surface. For any inner point z0 of 8 call q,, = T - t o a local uniformizing variable. This can even be done for almost all frontier points. For 8 in the form we defined it, equivalent parts of the boundary are mapped onto each other by the substitutions (1) with the matrices
They leave the points u, ( p = 1,2,3), respectively, fixed. The fundamental domain can be changed in such a manner as to cut off part of a boundary and to attach an equivalent part to an equivalent boundary. This is a hyperbolic motion, and thus a conformal mapping, so that qro= T - to passes into a power series qi, = clqro+ ... with c1 # 0. Thus at all points other than the u, the local uniformizing variables may be taken the same. For these exceptional points, though, set
- eZnir ,
qim -
q, = T
qc = T
+ CzT + CzzT - 3(
= (T - O 3 / T ( T
+ I),
+ c3z- 2i = ( T - i)’/T.
As T traverses the part of a neighborhood of T~ = ioo, [,i lying in the interior of 8, q = qim,qc,q1 traverse a full neighborhood of q = 0 in the q-plane. As these functions are holomorphic for T # ioo, [, i and have nonvanishing derivatives, our convention gives % a conformal structure. The definition of 3 can also be formulated otherwise. The points of % are the totalities r T o of equivalent points LT, for all L E r, in addition to ria. This formulation has the advantage of being free from an arbitrary choice of a fundamental domain of r. On the other hand, a fundamental domain is a welcome aid to the intuition. In effect there is no difference between the definitions. 2. COVERING SPACES OF THE MODULAR SURFACE
Now let H be a subgroup of
r with finite index and n
r =j u= 1HL,,
II =
[T:H],
216
IV.
ALGEBRAICFUNCTIONSOVER THE COMPLEX NUMBER FIELD
a decomposition into right cosets. If, as before, 5 is taken as a fundamental domain of r, then n
is a fundamental domain of H. Here L,g stands for the set of points Ljto, t o traversing all points of 5.To verify this, simply multiply both sides of (3) by H noting (2). A surface RH is now defined by the “points” Ht,, once the HLjico are added as above ( j = I, ..., n). More than one point must generally be added, namely, all Ljioo not equivalent by H. The points of R consist of the totalities rt,, and these decompose by (2), each into n totalities HLjr,, that is, into n points of RH. We speak of each point of RH lying over some point of R. This makes RH into an n-sheeted covering space of R and, as such, SHis also a Riemann surface. We must investigate the nature of this covering space more closely. It will turn out, in particular, that RHhas only a finite number of ramification points over R. Furthermore, 91His connected, as two points Ht, and Htl can be connected by a curve in RH, by so connecting t oand t1 in 8. RHis covered by 8 just as R is. As the only ramifications of 43 lie over the points To,, all ramifications of 91Hwith respect to S must lie over these same points. In order to discuss them more closely, assume further that H is a normal subgroup of r. Let Nl denote the smallest natural number such that
Then the cosets HClpA4 and HCl’M are disjoint for 0 5 v # p < N, and any M E r, and if H is a normal subgroup, the same is true for the cosets HMClp and HMC,’. This implies the existence of a decomposition into cosets nl
NI-1
r =j =Ul v U HMjC1”, =O
nl = nN;’.
This means, incidentally, that n is divisible by N1. As C, leaves the point o1 = ioo fixed, only the nN;’ points HM,ioo of ‘?ItHlie over the point rim of R, and these are ramification points of order Nl. This means that we may = exp(2niN;’t) as a local uniformizing variable here. choose qH,fa, Further, let Nu(a= 2, 3) be the smallest natural numbers such that C? E H. The same argument shows that nN; points of SH lie over the points rt; and ri of 9l, and that they are ramification points of orders Nu.Local ~ and uniformizing variables are qH,( = t - t; and qH,[ = t - i or ( I ~=,qc qH,i = qi ,depending upon whether Nu= v, or 1.
$4. MODULAR FUNCTIONS
217
9l is homeomorphic to a sphere and thus of genusg = 0. !KH is an n = [r:HI sheet covering of 93, with ramification points over, and only over, D, (0 = 1, 2, 3). They number exactly n/Nu and are of order N u . The genus gH is given by §1,(2) where now the e,(((@))= Nu (4 = I , ..., n/N,). This gives gH= f ( n N ; ' ( N , =I+-
:(
- 1) + I I N ; ' ( N , - 1) + i1N;'(N3 - 1)) - n + 1,
I - - -, - I- N,
1 N2
(4)
N,
We further show that, in general, N2 = 3, N, = 2. Clearly the Nu are divisors of u, = 3 or 2. Now, were N2 = N 3 = 1, we should have gH c 0. Were N 2 = 3, N , = I , we should have gH= 1 - (n/2N1) - ( 4 6 ) 2 0,
so that n < 6. But n is divisible by N 2 = 3. This leaves n = 3. Furthermore, as Nl also divides n, the possibilities Nl = 1 or 3 remain. The first would make gH negative. Finally, N2= I , N3 = 2 would imply gH= 1 - ( n / 2 N 1 )- ( 4 4 ) 2 0
so that n < 4. The same argument then shows n = 2, and Nl = 1 or 2, the first case again making gH negative. Summing up we can state the following. The genus of 91H,for normal H c r, is generally given by
(t
where N = Nl is the smallest natural number such that T;')EH,the exceptions being [T:H] = Nl = N2 = 3, N, = 1, gH = 0, and [T:H] = Nl = N3 = 2, N2 = 1, gH = 0. The points HMjico of !KH are called the cusps of SH. The points HMjico of H are its cusps. The ( H E H) belonging to a fundamental domain number of cusps is, in the case of a normal subgroup, given by OH
=
[r:H]/N.
(6)
3. CONGRUENCE SUBGROUPS A particular normal subgroup is the principal congruence subgroup H = T ( N ) of level N, defined for every natural N by the congruence
H=
("c d' )= &(o1
01) mod N;
(7)
that is, H consists of all matrices H E r satisfying this congruence. In general, any subgroup is called a congruence subgroup if it contains some
218
Iv. ALGEBRAlC FUNCTIONS OVER THE COMPLEX NUMBER FIELD
T ( N ) . To investigate these T ( N ) as well as for later purposes we need the following lemma.
Lemma. For every solution of the congruence a6 - By
(a
there exists an integral matrix
= 1 mod N
(8)
1;) of determinant 1 such that
Proof. Because of (8) the g.c.d. of a, p, N is 1. Thus there exists a p’ such that a and j? + /I” have 8.c.d. 1. Hence, we may assume that (a, p) = 1. There then exist y’, 6‘ such that ad’ - fly’ = 1. Now set a =a,
b = b,
c =y
+ y’(1
- aS + fly),
d =6
+ S’(1
- a6 + By)
which gives ad - bc = 1 and the congruence (9). As a consequence of the lemma we have the following. Thefactor group
~ J Z (= N r/r(N) )
( 10)
is isomorphic to the group of two-row matrices (; $)of determinant 1, whose entries are residue classes mod N , where matrices are to be identified if they are the same up to multiplication by - 1. For N = 2 this last reservation is meaningless, as - 1 = 1 mod 2. The order of this factor group is equal to the index [r:T ( N ) ] .In particular
[r:r(~)] = 4~~ n (1 - q - 2 )
for N g 3
q/N
=6
(1 1)
for N = 2,
where q traverses all prime divisors of N. We find the number on the right, without the factor 3, in three steps. (a) Let N = q be prime. Then a6 - by = 0, mod q has exactly q3 q2 - q solutions, so that a6 - f l y f 0 mod q has q4 - q3 - q2 q solutions. The number of solutions of (8) is l/(q - 1) as many, i.e., q3 - q. (b) Let N = $ be a power of a prime. A solutionah, ..,,6his found by setting ah = ah-1 $-‘a, ... in a solution a,,-lr ..., 6h-l of (8) with q”-’ in place of q” and finding a, ... which solve
+
+
+
a6h-1+6uh-l-pyh-,
-7ph-1 = ( I -ah-18h-1 +ph-l’yh-l)ql-hmodq.
The number of solutions of this linear congruence is q3. The number of solutions of this case can now be found by induction on h. (c)If N is the product of powers q” of primes, then the number of solutions of (8) is equal to the product of the numbers of solutions for the factors $. Finally, due to the ambiguity of the sign in (7), we must divide by 2 unless N = 2.
$4. MODULAR
219
FUNCTIONS
As the exceptions to formula (5) cannot occur here, since neither C, nor C3 belong to any T ( N ) , N > 1, we may compute the genus by that formula. By a similar computation we find that the number of cusps of !Xr(N) as given by (6), namely, O N = [r:r(N)]/N, (12) is equal to the number of pairs (a, c ) of residue classes mod N having g.c.d. 1 with N , where (a, c) is identified with (-u, - c ) for N 2 (by calculating the latter number along with [r:T ( N ) ] ) . Thus, there exists a one-to-one correspondence between these pairs and the cusps T(N)(: 1;)ioo = r ( N ) a / co f % r ( N ) . t From the equation
=-
it also follows that the factor group %JI(N)= T/T(N) permutes these cusps T(N)a/c and the pairs (a, c ) in the same manner. In particular, we clearly have: %JI(N)is faithfully represented by the permutations sufered by the pairs (a, 4.
A local uniformizing variable at the cusp r ( N ) a / cis q = q r ( N ) , r ( N ) a c - l = exp[2ni~-'(dz - b)(-cT
+ a ) - ' ] = exp[2niN-'~-'(T)]
where b and d are to be determined for given a, c by ad - bc = 1. The ambiguity of this computation shows up in a constant factor of q, and is thus without significance. Indeed, the substitution (: f;) of T which maps T(N)ioo onto r(N)u/c also transforms q into q r ( N ) , r ( N ) i m . 4. MODULAR FORMS
By a modular form of the dimension -k for a subgroup H E r we mean a function M(T),defined in the complex upper half-plane $3 and analytic there except for singular points, satisfying the functional equations
(14)
where x(: 1;) is a root of unity depending upon (; f;). It is called the character of M(T). Of particular interest are modular forms with character 1, but they
t The notation r(N)a/cmust not mislead us into believing that this cusp is uniquely determined by the ratio (I : c mod N .
220
Iv. ALGEBRAIC FUNCTIONS OVER THE COMPLEX NUMBER FIELD
(-A -0
into (14) cannot occur if k is odd. Indeed, substitution of (: 5;) = givesX(-A -1. on the space of all functions in r In (14) we defined an operator for every substitution (1). Its easily verifiable properties are
We will choose either the group r itself or one of its congruence subgroups T ( N ) as H from now on. A modular form is said to be meromorphic at the cusp r(N)ac-l or r(N)ioo if it permits an expansion m
M(7) = (-c7
+ a ) - k C cnqn n=no
m
or
=
1 cnqn
n=no
(16)
in q = qrtN),r(N)lr,c or q = qr(N), r(N)im convergent for sufficiently small q. Modular forms are called meromorphic if they are meromorphic in the interior of the upper half-plane as well as at all cusps. It suffices to check this at all the cusps of a fundamental domain of T ( N ) , as the behavior (16) at the other cusps then follows from the functional equation (14). A modular form is called holomorphic or integral if it is holomorphic in the interior of 8 and only terms with n 2 0 occur in the expansions (16). In what follows we will consider only integral modular forms, so that we may omit the specification " integral." The modular forms of a fixed dimension - k for the same subgroup H E r form a vector space whose dimension will be calculated for certain cases in §4,6. As special modular forms of dimension -k for T ( N ) we define for all natural k > 2 the Eisenstein series
the prime over the summation indicating that the pair m , = mz = 0 is omitted even if it satisfies the conditions mi = a, mod N. This series converges absolutely and uniformly in every compact set of 5 for k > 2. Its expansions (16) will be given explicitly in g,7, as their calculation here would require much computation and interrupt our line of thought. The series vanish identically for 2al = 2uz F 0 mod N and k odd, for then each term is cancelled by its opposite.
$4. MODULAR FUNCTIONS
22 1
It is easy to verify that
= Gk(T; a,a
+ a,c, a,b + aid, N )
(18)
for every (: :) E r. Equation (18) carries with it the functional equation (14) for every (: f ; ) ~T(N), with :) = (+ depending upon whether d=+ImodN. 7 We say that Gk(?;a,, a , , N) is primitive if the g.c.d. (a,, a , , N) is 1. Primitive Eisenstein series can vanish identically only if N 5 2 and k is odd. Call two Eisenstein series Gk(T;a,, a,, N ) and G k ( t ;a,', a,', N ) formally equal if a, = a,'mod N and az = a,' mod N or a, = -a,' mod N and a, = -a,' mod N, otherwise formally diferent. Formally different Eisenstein series then correspond one to one with the pairs (a,, a,) of residue classes mod N such that (al, a,, N) = 1, where two pairs (a,, az) and ( - a , , - a z ) are counted as one. In &t,7 we shall undertake a computation to show the formally different Eisenstein series for k > 2 to be linearly independent. Anticipating this result, a remark in the last section permits us to observe the following. The number of linearly independent primitive Eisenstein series of an even dimension k is the same as the number of cusps. If N > 2, the same holds true for odd k. ; ] - I r associated with an element Equation (18) shows that the operator (: :)E r permutes the Eisenstein series in the same manner as (g i) itself permutes the pairs (a,, a,) or (g :) the cusps. By $4,3 then, the operators :]-Ir are a group of permutations of the G k ( t ;a,, a , , N ) faithfully representing the group 9Jl(N).
~(z
5. THEFIELDOF MODULARFUNCTIONS Meromorphic modular forms of dimension 0 are modular functions. One speaks OF such with respect to a subgroup H c r, and in particular of modular functions of level N if H c T(N). For N = I we even omit mention of the level. The modular functions of a group H are clearly a field KH . We will now investigate the fields K , and Kr"). As early as $2,(27) we considered the special Eisenstein series
222
Iv. ALGEBRAIC FUNCTIONS OVER THE COMPLEX NUMBER FIELD
There we wrote a,,wb instead of wl, w, . These expressions are homogeneous of degrees -4 and -6 in both variables wl, w , . The function $2,(24), in particular,
is a modular function. The function (20) maps the modular surface % onto the complex number sphere, conformally and one to one. Every modular function is a rational function ofjiz). Proof. By §2,6 there exists an elliptic field of functions L over the constant field C with any given y E C as its invariantj(L) = y. Any two such fields are related by a transformation 92,(22). The two base periods w, = wl, = w2 are multiplied, under it, by t - ’ , because du = dx/y. But their ratio remains unchanged. If wl’ = awl bw,, w, = cwl d o , is another basis of the period module, then z’ = wl’/wz’ = :)z. This means that exactly one point Tz,withj(z) = y, corresponds to every value of y. The mapping of % onto the number sphere given byj(z) is conformal everywhere. Indeed, it is certainly conformal almost everywhere, that is, for those z = ro wherej(7) is regular, a‘j(t)/dt # 0 and 7 - T~ is a local uniformization. If the mapping were not conformal at some point, then in a neighborhood of that point the equationj(t) = y would have to have several solutions in the fundamental domain 8. This is not the case. The denominator of (20), which is the discriminant of the polynomial in 42,(21), does not vanish in 8, according to $2,6, so that the point 7 = ioo of % is mapped onto the infinitely distant point of the number sphere. An arbitrary (meromorphic) modular function f(7) assumes the value 0 only finitely often on %. For, as % is a compact point set, an infinite number of zeros would have a cluster point, which is not permissible for a meromorphic function. The same argument shows that a meromorphic function has only finitely many poles. Let the number of zeros and poles of the modular function f(7) be no and n , . We maintain these numbers to be equal. The proof is found by taking the integral
+
(2ni)-’
(z
Sf’(.)
+
dr/j(z)=0
over the boundary of a small circle f in % not containing zeros or poles of f ( r ) . It is, of course, simultaneously the boundary of the complementary region 93 - f. Hence the value of the integral is known, from the familiar theorems of function theory, to be &(no - nm), where poles and zeros are counted with their respective multiplicities. Hence, no = n , .
$4. MODULAR FUNCTIONS
223
Now let vi and ni be the zeros and poles of f ( 7 ) . The function
is then a modular function without poles or zeros, if j ( 7 ) - j ( i a ) is given meaning by setting it toj(t)-'. This function must be a constant, for otherwise we could find a y E C such that g(7) - y has zeros, but does not vanish identically. But this function would have no poles, and thus contradict what was shown above. This proves the theorem. 7 Thus the modular functions form a rational function field K r = C(j(7)).
(21)
Now assume N > 1. Adjoin to K r the quotients of the primitive Eisenstein series of level N and some fixed dimension k, excluding all k = 1 mod 2 if N = 2. The field extension Kr(N) thus found coincides with the field of all modularfunctions of level N . It is a Galois extension of K , with the group W ( N ) .
Proof. It is clear from the last section that these quotients are invariant under the substitutions (1) for (: : ) E T ( N ) , but that they permute under arbitrary substitutions (1) of r. It was seen at the end of the last section that these permutations even represent m(N)faithfully. This means that Kr(N)/Kr is normal with that group. Letf(7) be an arbitrary modular function of level Nand let L , = E, L 2 , ..., L, with n = [!W(N):I ] = [Kr(N): Kr] represent the cosets of T ( N )in r. The elementary symmetric functions of the f(Li(7))remain invariant under r and thus produce an equation of degree n for f ( 7 ) over K r . Were f(7) $ & ( N ) , there would exist a function generating the field Kr(N)(f (7))/Kr.No generality is lost if we assume f ( 7 ) to be that function. Then f ( 7 ) satisfies an irreducible equation of degree higher than [Kr(N):Kr] over K , , a contradiction. 6. MODULARFORMSAND DIFFERENTIALS M(7) dth, where h is a natural number and M(7) is a modular form with character x = 1 and dimension -2h for some group H G r,is invariant under H. For, the functional equation (14) can be written a7+b d a 7 + b h A4 ( T d ) = M(7)'
(dta)
If conversely 4 7 ) is a modular function and 47)
d j ( ~=) u~ ( ~ ) ( d j ( ~ ) /dzh d t )=~ M(7) dzh
Iv. ALGEBRAlC FUNCTlONS OVER THE COMPLEX NUMBER FlELD
224
a differential of degree h, then M ( t ) is such a modular form. Thus there is a one-to-one correspondence between modular forms with character 1 and differentials. We want to determine the denominator of the divisor that corresponds, by 111,&1, to such a modular form. First let N > 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.
THE CORRESPONDENCES
239
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
v. CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
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.
CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
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.
THE CORRESPONDENCES
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
v. CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
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.
CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
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
v. CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
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
v. CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
(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(<’) with coefficients in K[u, v ] . As 0 permutes the cosets E M , transitively A(<’) is irreducible in K(u, u), and the residue ring of KK‘[u, u] mod A(<’) is generated over K[u, u ] by adjoining a root of A(<’), for instance ux(R7) + uy(Rt). Now the coefficients Ai(7, 7’)in (27) generate a K x K’ideal AK K p having the property that the residue ring of K x K‘ mod A K K # is generated over K by adjoining x ( R r ) and ~ ( R T This ) . means that the residue ring is the field K,as was to be proved. 7 We can complement our algebraic description of this correspondence A , by a geometric description. In doing so we assume the points of the Riemann surfaces R,%, R’ of K , R , K’ to correspond one to one with the totalities @ ( T ) , 6 ( 7 ) , @’(T) of points (from among a certain range of T) equivalent under 6, 8,0’,respectively, as we saw to be the case in IV,§4,2 for the group 6 = r and its subgroups of finite index. By (26), over the point B ( T ) of R there lie the points EM,(?) of % which in turn lie over the points Q’M,(r) of 9’.This means that
<
e’(W +
u 0’Mv(7)
v= 1
is the point mapping of R onto R‘associated with AR . It is generally desirable to have correspondences of R with itself, which can be easily multiplied. This is achieved by following through with the mapping of R’ onto R given by 0 ’ ( ~ ) + R 0 ’ ( z=) 0 R ( r ) associated with the isomorphism f ( ~-f(Rr). ) The correspondence of points is now written
Conversely, map the point B ( T )of R onto the point R-’O(r) = Q’R-’(z) of R’. Over it there lie the points BM,’R-’(r) of % which lie over the points OMv’R -’(~of ) R.Thus
252
v. CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS e(R)
6(z)
--*
6(r)AR* =
U~M,'R-~(T)
v=l
(29)
represents the Rosati adjoint correspondence of $1,4. It is interesting to note the formal analogy between formulas (27) and (28). The formation of a system of generators for the ideal associated with AR can be read directly from (28). The same notation can be applied to represent composed correspondences A = m,A, whose prime components A , arise from substitutions R, as above. The cosets occurring in the right side of (28) must now be taken by their proper multiplicities m,: @(?)A = U m, x BR,M,,(?),
with @(?)A, = U R,M,,(?). V
P,V
$2. Representations of Correspondences in the Space of Differentials
1. DEFINITIONS As in $1, K, K', etc., will always denote algebraic function fields with exact constant field k o . In $2 the latter will always be assumed perfect. Then by 111, $4 the divisors of differentials of K, etc., are always divisors of the fields and belong to the canonical class. To a correspondence A between K and K' we now define a linear mapping of the differentials of degree h (h = 0, & I , f2,...) of K onto differentials of K'. As in $1,3 we again use several steps, starting with the simplest case in which g ( A ) = 1. The residue class mapping K x K ' / A K x Kthen , leads to an isomorphism o f K onto a subfield KO' of K'. By dw" = dw"A we denote, in this case, the image of d d in KO/, but take it as a differential in the entire field K'. This is achieved as follows: let uo' du: be that image in KO' and and u' a separating element in K ' ; then uo' duo'h = uo' (duo'/du')' du''. If K' is inseparable over KOf,duo'/du' = 0. Therefore we must assume throughout $2 : if A contains inseparable prime correspondences, h must be 20. The rest of the steps follow those of $1,3 exactly, and in the general case they yield a linear mapping dwh + dwfh= d d A with the properties dw' ( A + B) = d W h A
+ d W h B,
(dWh
A)B = d W h (AB).
(1)
The ring R of correspondences of K with itself is thus represented by linear mappings of the space of diFerentials dw" of K. If the characteristic is prime and h > 0 the representation is of the residue class ring of R modulo the ideal RP v P*R generated by the correspondence P of $1,7. The second of Eqs. ( I ) s h v s that we need only prove that dw"P = d d P * = 0. For the first term, dw" must first be mapped isomorphically into a subfield KO' c K' and taken as a differential of K'. But K'/Ko' is purely inseparable so that the differentials in A ' of all functions of KO'vanish. Thus dw" P = 0.
$2.
CORRESPONDENCES IN THE SPACE OF DIFFERENTIALS
253
To compute dwh P* we start from PK K , = K x K'(xP - x', y p - y'). Adjoining the pth roots of x' and y' we get an extension K,'/K', and P* becomes P* = p Q with Q K x K , = K x K'(x y -(/?). Now the first equation of (1) gives dw" P* = p dwhQ = 0. 7 As we wish a representation of finite degree, we limit ourselves to the effect of correspondences on differentials of the first kind. It will turn out that these differentials keep that property if only certain limitations are made in the case of h # 1. To this end we distinguish certain prime divisors p,, ..., pm of K arbitrarily, and, given a prime correspondence A between K and a field K' isomorphic to K for which K = K x K ' / A K x Kise separable over K and K', we say that correspondence is of bounded ramification if it satisfies the following three conditions. (a) The images p,,A in K' are composed of the p,,' which are associated with the p,, under the isomorphism K' K . (b) Any ramifications of R / K and R/K' that exist lie over the places p,, and p,,', respectively. (c) The differents are of the form
-g?,
where p=
n $, i
p' =
ni
p;'
are the decompositions of the prime divisors p of K and p' of K' in K. If A = BP' or A = P*'B with maximal I and a prime correspondence B, then the field extensions K / K and R/K' are separable for B instead of A , as was shown in §1,7. We then say A is of bounded ramification if this is so of B. Finally an arbitrary correspondence A = m,A, is said to be of bounded ramification if all prime correspondences A , occurring satisfy the criterion. The Rosati adjoint A* of a correspondence A of bounded ramification is also of bounded ramification. The special form (2) of the different is often said to be regular, or R / K and K / K ' are said to ramijy regularly at all places. According to 111, $2 this is the case whenever k , is of characteristic 0 and algebraically closed. But the latter condition is not even necessary, for the ramification indices ei and the different remain invariant under the algebraic closure of k , carried out as an extension of constants of K . The concept of regular ramification was introduced in 11,$6,2 for Galois field extensions in a different manner. We now hold the power product
(3) of the prime divisors distinguished above fixed, in order to operate with the a = (pl
pJh-l
254
V.
CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
notion of a differential dw" of degree h and of the first kind with respect to a, as introduced in 111,§5,1. We prove the following theorem. The correspondences of K with itself map all differentials of the$rst kind of K among themselves. The correspondences of bounded ramijication map all differentials dw" of thejrst kind with respect to a among themselves. Remember that, in the case of inseparable correspondences, h > 0 is required. It naturally suffices to prove this for a prime correspondence A, where the separability ofR/Kand K / K may even be assumed, for otherwise the assertion is a trivial consequence of the vanishing of dwh A. Now, in this special case the considerations of §1,6 must be applied in the sense of differentials to determine dwh A. That is, dw" is taken as a differential of R and the sum of the differentials conjugate to dw" with respect to K' is formed. Letting denote a separating nonconstant element of K' this sum is XI
dWth= s K / K , (dWh/dXth) dXfh,
(4)
is the trace. Thus we must study the behavior of dw" taken in the where field R and then its trace. According to 111, $4,( 16) the differential dw'h is associated with the divisor (dWrh),.= (dw'/dx')hb:'/k,,(x,)jm t - Z h
(5)
in K', where jw'is the denominator divisor of holds also for the divisor (dWh)Kin R, namely, (dWh)K= (dw/dx'lhbhK'/k,-,(x')jm
XI.
1-2h
An analogous formula h
bR/R' 9
(6)
where the transitivity formula 11,§4,(9) has been applied. In the same sense we have the formula (dwh)K = (dWh),bk/K. (7) The fact that dufi = (dw"), is a differential of the first kind with respect to a means ~,,((dwh),) 2 1 - h for all prime divisors p,, of a and v,((dw"),) 2 0 for all other prime divisors. Now we consider dw" = ( ~ i w h as ) ~ a differential of R and use (7) as well as the assumption of bounded ramification. The above conditions on the local orders of (dw"), are then translated into the following conditions on the local orders of (dw")K:
2 efli(1- h) VPr,((dWh)K)
+ h(ePl- 1) = elti- h 2 1 - h
for all prime divisors pPi of the F, in R and ve ((dw"),) 2 0 for all other prime divisors of R. To extract our assertion from this statement, we choose a prime element x' for one of the p,'. (It is a separating element of K' since we assume the constant field to be perfect; therefore there are no inseparable prime divisors.) Let 1, be the exact power of p,,' dividing the different bK'/ko(xO), and eli the
$2.
CORRESPONDENCES IN THE SPACE OF DIFFERENTIALS
255
ramification index of a prime divisor pMiof p,‘ in R. With (6) and (2) we now have
2 -
V ~ , ~ ( ( ~ W / ~ X ’1) ~ )h
- eh,l,h - (eli - l ) h = 1 - elih(lp+ 1 )
which can also be written h .rh- 1 +I,h
>
) = 1 - eli. v,,i((dw/dx ) This means that (dw/dx’)hx’l-h+Iflh lies in the p,’-component of the inverse of the different b K I K ,But . by 11,@4,2,this component bi/lK,,p,pis the complement (3b,,)&Rp of the integral domain of all p,’-integral elements of R. Thus the traces with respect to K’ of all these elements are p,’-integral: v,‘,(sR/K‘(((dW/dX’)h)X’h-
+Iph
)) 2 0,
which is equivalent to V,,.(SR/K,((dW/dX’)h)) 2 1 - h - 1,h. Combining this with (5) we have already proved that Vp,,(dWfh) = Vp,,(SR/K,((dW/dX’)h)dX’h) 2 1 - h for all p. We must carry out the same considerations for the nonramified prime divisors p’ of K‘, starting out with v,,((dwh)K)2 0. Let x’ again be a prime element for p’, and p‘ divide the different bR,lko(x,) to the power I ; due to (6), the assumption can be written VPi((dW/dX’)h X“h) 2 0
leading to traces satisfying v , , ( ~ ~ , K * ( ( d w / d x2‘ )-Ih, ~))
which is equivalent to v , , ( d ~ ’2~ 0, ) because of (5). The proof is complete. 7 For a later application we need the following corollary to the theorem just proved: the correspondences of bounded ramification of K with itself map all diferentials dw’-h of the Jirst kind with respect to a-‘ among themselves. Although replacing h by h - 1 is not generally permissible here, this is possible in (4) through (7). The proof now proceeds through the inequalities ~ , ~ ( ( d w ’ - ~2) Kh )- 1 ,
2
- l)ePi+ (ePi- 1)(1 - h) = h - 1 ,
=( h
- l)ehi(l, + l),
V ~ , , ( ( ~ W ’ - ~ ) K )(h
256
V. CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
Because the traces of integral elements are integral,
vP;(sg/K, (dw' -h/ddl -')) 2 ( h - 1)(1, + l), V,p,(SK/K,(dWl-h/dX'l-h) d$l-h) 2 ( h - 1)(1, + 1) + (1 - h)l, = h - 1. The considerations concerning the other prime divisors can be repeated with 1 - h instead of h. 7 The correspondences of bounded ramification generate a subring R, of the full ring R of correspondences, this subring clearly depending upon the distinguished divisor a. The correspondences of bounded ramification include, as special cases, the unramified correspondences, that is, those composed of prime correspondences with unramified extensions K/K and K/K'.Examples will be found in the next two sections. Exercise. If the requirement for bounded ramification is reduced to the divisibility of the differents by at most # e i - ') or pi(er'-'I, respectively, with an arbitrarily given natural integer r 2 1, then setting a = (P~...)'~-' retains the validity of the theorems. 2. THECLASSICAL CASE Throughout this section we take k, = C and h = 1. Let du, be a basis of differentials of the first kind. For an arbitrary correspondence A of K with itself the du, A are differentials of the first kind and satisfy a system of equations
c #
du, A =
j= 1
aij duj
(8)
with coefficients aij E C. We have implicitly identified the fields K and K', which is quite natural here. Equations (1) assure us that this gives a representation of the ring of correspondences by g row matrices in C. This representation is related to another, based on a topological property of correspondences. First let A be a prime correspondence. In §1,6 we saw that such an A furnishes a conformal mapping of the Riemann surface 93 of K onto itself, ramification points being allowed. As this mapping is continuous it maps closed paths onto closed paths. Also, the relation between a region and its boundary is retained. Both of these statements need only be verified locally, the global statement following. (The arguments in detail are left to the reader.) This shows that A maps homology classes of closed curves onto such classes. Now by IV,§1,3the canonical system of incisions a,, bi is a basis of these homology classes. We can apply A to it, so that a linear system of equations
$2. CORRESPONDENCES IN
257
THE SPACE OF DIFFERENTIALS
with rational integers m;,, etc., results. It is not difficult to see how these equations behave for arbitrary correspondences A = C m,A, taken as sums of prime correspondences. Equations (9) thus yield a representation of the ring of correspondences in 29 row matrices with rational integers as coefficients. To understand the relation of (8) to (9) it is better to think of (8) in its exact defined sense, writing
the dul’ being the basis associated with dul of a field K‘ isomorphic to, and corresponding with, K. Once again let A be a prime correspondence. Let the variables of K’ traverse one of the period paths a, or b, and form the integrals IV, §3,(2) thus traced for the right side:
For the left side, remember the definition of duj A. Let z’ be a local uniformizing variable in a neighborhood 3’of a place of the Riemann surface ‘3’. By the last section, then, d u j A = C (duj/dz’)(’) dz‘
(10)
V
is a certain sum taken over g ( A ) functions (duj/dz’)(’)algebraically conjugate with respect to K‘. The neighborhood 3’is the image of g ( A ) inverse images 3,in %, and theseare theimages of 3’by the Rosati-adjointcorrespondenceA*. Now, form the integral of the differential (10) over a path c’ lying in 3’. Each summand on the right could just as well be taken in ‘3 by replacing c’ with a path c, lying in one of the neighborhoods 3,. Suitable numbering of the differentials (duj/dz’)(’)in 3, makes them equal to duj . Thus the sum on the right is the integral jduj extended over the image of c‘ under the Rosati conjugate correspondence A*. Letting c‘ exhaust one of the cycles a,, b , , this yields
The homology classes aiA*, biA*can here be expressed in the a i ,b, as in (9). We simplify the notation by writing the matrices which occur as m;, etc. The equations
1w$ajl = I
1
m;ppj + C II$CI$~, 1
C w!lajl = 1
1
m!pfj + C n!lwFj 1
result. The complex conjugate equations must also hold, the rational integers of the matrices mi being self-conjugate. All the equations thus found can be
258
v. CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
compiled to a single matrix equation
or, in shorter notation,
(transposition being indicated by the superscript t). We want to use this The equation with A* in place of A. The matrix
is nonsingular. For we know w; has an inverse by IV, §3,2, and that
where T was introduced in IV,§3,(9). As the imaginary component of the matrix T is positive definite, T - T is nonsingular. We can now compile results. Let D l ( A ) be the representation of the ring of correspondences defined by ( 8 ) and H ( A ) be that defined by (9). A nonsingular matrix P, (composed of the Riemann period matrix and its complex conjugates) exists such that H(A)P1 = P,
D1(A*)'
0
holds; that is to say, the representation H ( A ) is equiualent to the sum of the representation D1(A*)' and its complex conjugate.
3. CONTINUATION ; REPRESENTATIONS OF ROSATI-ADJOINT CORRESPONDENCES We must now prove the important formula {du A, du} = {du, du A*},
(11)
where is the scalar product of differentials of the first kind defined in IV, §3,(7). We may limit ourselves to prime correspondences, and use the notation of §1,6. The scalar product on the left of (1 1) is computed as an integral over the Riemann surface '3' by carrying du to K' by the isomorphism. The differential du A is computed by (10). The index v occurring in the following indicates {..a}
82.
CORRESPONDENCES IN THE SPACE OF DIFFERENTIALS
259
the various sheets of % lying over %'. In all these sheets do has the same value, and we can take
as the sum of the differentials du dfi in 3,over all the points of % lying over a single point of 8'. This leads to {du A , do} =
fs
du dij.
91
The same transformation applies to the right side of (1 I). There % and %' are interchanged so that A must be replaced by A*. 7 If a basis dui such that {dui , duj} = dij (= 1 or 0 for i = j , i # j , respectively) and the abbreviation dui A* = a:. duj
c j
are used, then (8) and (1 I) yield {dui A , d u j } = a i j = { h iduj , A*}
= a;i.
Thus in a suitable basis the Rosati-adjoint correspondence can be represented by ihe Hermite-adjoint matrix. 4. THETRACE
Let the divisor a be fixed as in (3). A basis of the differentials of the first kind with respect to a is given by dw:. In $1,2 we associated with any correspondence A of bounded ramification a system of equations dw? A = 1 a i j dw; i
with aii E k, . The rest of this paragraph will be concerned with computing the trace of the representation of the ring R, of correspondences with bounded ramification. To this end we shall apply the results of III,§5 and therefore make the assumptions on a formulated there which can now be expressed as: (i) either the number m of distinct prime factors of a satisfies the inequality (m + 2(g - I))(h - 1) 2 0 and a does not lie in ihe (1 - h)th power of the canonical class, or (ii) a lies in the (1 - h)th power of the canonical class. It suffices to compute the trace of prime correspondences because of ( l ) , nonvanishing values being found under this hypothesis only if the corresponding field extensions KIK and RIK' are separable. We may also eliminate the unit correspondence A = D which maps every differential onto itself;
260
v. CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
its trace is therefore equal to the number gh,. of linearly independent dw; which was calculated in 111,§5,1. The representation( 12)suffers no change under an extension of the constant field ko . We may therefore assume throughout that ko is algebraically closed. We use the basis d’-hltli of principal part systems complementary to the dw; to write the trace as
which follows from HI,@,(13). The key to our computation is the application of 111, @,6, where the principal part systems
ddll - h BOh=
d ’ l - h ~ idw;, ’
dd”-hBlh = G d ’ ~ ‘ ’ dXh -~
i
were introduced. The second of these is exact and compensates the poles of the first, as well as having a pole at the diagonal. Otherwise it has no further pole under assumption (i) or exactly one further pole of first order with a given residue under assumption (ii). Apply the correspondence A with respect to (the variables of) K to both of these principal part systems. On the one hand we have, from (13),
On the other, dd”-hBlhAcan be formed, confusion being best avoidi.3 by interpreting A for the time being as a correspondence between K and an algebraically independent K“. Our assumption that A # D assures that the resulting exact principal part system in K’K“ has no pole at the diagonal. Thus we may now identify K‘ and K”, so that we have a differential du’ of degree 1 in K : du’ = ( d d ” - h Q 1 h A ( K + K , , J K , , = K , . (14) By construction du’ has all the poles of (dd”-hB,h and then others. Because the sum of all residues of du’ is 0, the trace is the sum of the residues, taken negatively, of those other poles. We have assumed implicitly that none of these other poles coincide with those of d”-hltli’.But this we may, in fact, assume, for by III,§5 this principal part system can be chosen with poles at arbitrarily given places. These further poles of du‘ under discussion will turn out (with a single exception) to be thefixedpoints of the correspondence A. They are precisely those prime divisors p of K whose images p’ by the isomorphism K z K’ occur in the prime decomposition of p A . The differential (14) is computed by formula (4), which requires that dJ1 - h Elh= G d’x’’-’ dxh be taken as a differential of degree h in the common extension field R of K and K‘ determined by A, and that then its trace
82.
CORRESPONDENCES IN THE SPACE OF DIFFERENTIALS
261
be formed with respect to K‘. It seems useful to write dx‘ in place of d’x’. This trace s ~ / ~ , (dxldx’)”) (G d ~ is ’a differential ~ of degree h in K‘,and du’
= s R / K * ( G ( d x l d ~ ’ ) dx‘. ~)
(15) To investigate the behavior of nu’ at any place p‘ not a poleof a d’l-hmi’, set
G = ( u ’ / ~ ) ( x-
XI)-’
+H.
The a‘ E K‘ here is such that v,.(a‘) = vpr(a’),and x‘ is a prime element for p’, while a and x are the corresponding elements in K by the isomorphism. It can be seen that the first summand on the right yields singularities of du’ at the fixed points of A. We assert that the second summand yields no contribution to the singularities of du’ outside of the singular places of the d l - h m i . Certainly H is integral for all divisors of K K I K not already divisors of K‘, by 111, §5,6. Thus H E K x K’. This means that in the residue class formation mod A R which is equivalent to the embedding of K and K‘ in a common extension R, only the prime divisors p‘ and p of K K I K and KK’IK‘ that are already prime divisors of K’ and K must be considered. As for these, we see from 111, §5,6 that h 0, = v,(aH dxh) 2 0 v p G’ - 1 ~ d ~ t l)-> (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.
CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
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
v. CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
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 $,(A*) + s,(A*). By §2,2 this is the trace of the topologically defined representation H(A). Thus, its trace is s(H(4) =
t M. EJCHLER, Eine (1957).
-Cf* + g ( A ) + g ’ ( 4
(26)
Verallgemeinerung der Abelschen Integrale, Math. 2. 67, 267-289
266
v. CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
This formula was first proven by Hurwitz.? It is actually a special case of a formula of far greater generality, valid for finitely multivalued mappings of arbitrary manifolds onto themselves. Any textbook of topology gives further information concerning this formula, the Lefschetz j x e d point theorem. Also noteworthy is a theorem of Deuring’s:$ 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
v. CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
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
v. CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
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.
CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
(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 u<(T)using an integral transformation. But this would have no advantage here, where we are concerned only with formal properties of the Fourier coefficients CQ,.
$3.
MODULAR FUNCTIONS
277
m
which, indeed, we can by using (13). Now, because of (1 1) the product of the series (30) is
C
Z(S) =
(t,,(n))nh-'-S,
(n,N)= 1
so that (29) is a consequence of (26). T Conversely, knowledge of the Fourier coeficients ci,,, brings with it that of the coeficients tij(n). The proof is accomplished by first applying to Eq. (26) the relations for the (t,(n)) corresponding to (1 l), finding
where n and n' are held prime to N and m is composed of prime divisors of N. The linear independence of the ui(T) assures that the rank of the matrices (ci,J,I = 1,2,. .., formed by their coefficients coincides with the dimension d of the module of the u i ( T ) . Thus d values of I can be found such that the sub~) with them is nonsingular. These I are then decomposed matrix ( c ~ ,formed to I = mn,where m is composed of the prime divisors of N and n of the others. Now (31) can be solved for the nfh-'tij(n'),which are thus linear forms in the c~.,,,,,,,~- with coefficients independent of n'. 6. RAMANUJAN'S CONJECTURE
From ( 5 ) and (8) we have
T,* = A,,T,,. In the space of normed modular forms for a real character x(n) this implies T,,* = x(n)- 'T,, = f T,,. The theorem of §3,4 shows that the module of these cusp forms can be transformed so that the (tij(n))are all diagonal matrices. The eigenvalues r i ( n ) of these (tlj(n))are real or imaginary numbers depending upon whether x(n) = 1 or - 1. This means that the matrix (28) can be
278
v. CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
transformed to the diagonal matrix with axis elements “(s)
=
fl(1
- ti(p)ph-l-s
+ X(p)pZh-l-2s)-l*
(33)
PXn
We are particularly interested in the decomposition 1 - T,(p)ph-’-’ -k X(p)p2h-1-Zs = (1
- wi(P)ph-
-“I(1 - [X(P)p/oi(P)lPh-
- ’)
(34)
of the individual factors of (33). For the convergence abscissa of the infinite products (33) and (28) depends upon the rate of growth of the roots o i ( p ) . The easiest example is where N = 1, h = 6 (so that k = 12), x(n) = 1. The rank of the module of associated modular forms is one, and the modular function is
called the discriminant, and coincides with the discriminant gZ3- 27g,’ of the Weierstrass normal form of the elliptic functions of base period, o,= 1, wb = T [(cf. IV, 52,(21)]. On the basis of extensive numerical computation Ramanujan conjectured that the associated Dirichlet series m
~ ( s )=
2 r(n)n-’
n=1
has the product development (28), that is, that
which was first proved by Mordell. Ramanujan also conjectured that the root o ( p ) of the associated equation (34) are complex numbers of absolute value
Iw(p)I =(p)”’. But, despite much effort, this conjecture has not been verified to date. For the history and rich literature regarding this problem, see the references below, especially Van der Blij [l]. Ramanujan’s conjecture has been extended to the general case: loi(p)l = ( p ) ’ / ’ . The first result along this path is due to Petersson [S], while the deepest penetrations were made by Igusa (111, §6,7,[3]) and Shimura [9]. These papers all consider the case h = 1 (k = 2) only. A very special result of Hecke’s falls into this connection, though, and holds for arbitrary h, [6]. A discussion of it would lead us too far from our subject. The form of (34) makes it clear, incidentally, that the generalized Ramanujan conjecture represents a best possible approximation of the convergence abscissa of the product (28). Along with an approximation of the convergence abscissa of (28) comes, because of (29), an approximation of the rate of growth of the Fourier
$3.
MODULAR FUNCTIONS
279
coefficients ci,,,,of the cusp forms. In particular, assuming the conjecture to be correct, we have Ici,,,l = O(mh-*+e)
for every positive E , this also implying a corresponding improvement of the approximation of IV, &14,(29). 7. RESULTS FOR MODULAR FORMS OF ODD DIMENSION; NOTES
First let k be some odd natural number. The theory developed in §3,3-5 can be carried over to all modular forms satisfying the functional equations
depending upon whether d E f 1 mod N for all (: !) E T ( N ) . t Then for any residue class r prime to N and any G E T ( N ) we have u(z)[U,]-‘[GI
-’= u(r)[U,GU;
U,]
-’= x(G)u(r)[U,]-’
’.
= u (r)[G ] - ‘[U,] Thus if u(r) is such a function, then so is u(r)[Ur]-’, and we have a representation of the multiplicative group of residue classes modulo N in the space of modular forms. The commutativity of this group assures the reducibility of the representation to irreducible components of the first degree. Thus every modular form can be written as a sum of eigenfunctions of the [ U r ] - k , defined by u(r)&,l = U ( T ) [ v; ‘3 = x(r)u(r),
-’
where ~ ( ris) a character of the multiplicative residue class group modulo N. As suggested by (23), we let T,, operate on such modular forms by defining
the square root on the right being taken positive. The arguments of §3,1 again show this operation to preserve the property of being a modular form of character X(r), so that the considerations of §3,5 hold literally. This is the path followed by Hecke in his two papers, I and I1 [ S ] , even and odd k being treated uniformly. These investigations contain our principal theorem of $ 3 3 and go considerably farther. Hecke annexes the factor f N > 2 must be assumed. For N = 1 we also know from IV, 44,6 that no integral modular forms exist with character x # 1.
280
v. CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
- 1 on the right in (35) eliminating the root there and thus also arriving at a simplified final formula (28). But such a factor sacrifices the representation of the ring of correspondences generated by the T,, . The possibility of such a representation for odd k, however, is an open question anyway. Indeed, it would have to be shown that a relation rn,TnpTmo= 0 implies the corresponding relation for the representing matrices. Although (10) and (1 1) are still carried over correctly if ( ~ / d ) ' /is~ chosen positive, it could be that other relations hold outside of these. The possibility of defining similar linear operators in the space of integral modular forms of nonintegral dimension to an arbitrary multiplier system u(M) is considered by Wohlfahrdt [lo]. With regard to the Peterson scalar product, cf. IV, §4,9,[5]. From now on drop the assumption of k being odd. In hopes of applying the principal theorem of 43,5 to problems of number theory, especially in the realm of quadratic forms, a connection must be sought between correspondences and the other Fourier coefficients c,,,, composed of prime divisorsof the level. Suchaconnection is found in Hecke [S], Part 11. However, distinctions must be made in the treatment of the modular forms depending upon whether they are already invariant under certain extensions of the group T ( N ) or not, that is, whether they belong to certain fields between Kr and Kr(N).Hecke [5,7] could decompose the space of modular forms into subspaces of characteristic behavior under the Galois group of Kr(N)/Kr. The representations of the Tndecompose simultaneously (cf. [4a]). The applications of the theory to definite quadratic forms of a finite number of variables are compiled by Hecke in the monograph [7]. The general principle is to find all quadratic forms of a given number of variables 2, determinant IF1 and level N,and to form their theta series. The associated Eisenstein series and cusp forms of character ,,k/2
belong to the modular forms of character I(: i) = qkfor d = q = f 1 mod N. As soon as it is known that the theta series generate the same function module as the Eisenstein series and cusp forms, it is also known that all statements concerning the latter apply to quadratic forms. This premise may easily be checked in special cases. It was proved for the principal character x(: i) = 1 and sufficiently large prime levels in [3], and, allowing certain limitations, also for arbitrary levels not divisible by perfect squares, in [4]. The proof depends, in part, on a modification of the trace formula of 42 which is valid for modular functions of the subgroup r,,(N) of all (: i) E r with c = 0 mod N (cf. the author's paper cited on page 265). Despite the nonbounded ramification of modular correspondences of r , ( N ) the formula hardly differs from $2,(19). For the other part, an arithmetic analog of correspondences is
w. CASTELNUOVO’S INEQUALITY
28 1
constructed for quaternary quadratic forms, and its trace is computed as well. The assertion results from the fact that the traces coincide. For the theory of an arithmetic analog of correspondences for quadratic forms in an arbitrary number of variables, see [2]. It is perhaps strange that ) 1. Thus the analytic such could be found only under the hypothesis that ~ ( n = theory yields more. REFERENCES
F. VANDER BLU. The function 7(n) of S. Rarnanujan, Math. Student (New series) 18, Nos. 3, 4 (1951). M. EICHLER,Quadratische Formen und orthogonale Cruppen, Chap. IV. Berlin, GBttingen, Heidelberg, 1952. M. EICHLER, uber die Darstellbarkeit von Modulformen durch Thetareihen, J. Reine Angew. Math. 195, 156-171 (1956). M. EICHLER,Quadratische Formen und M o d u l f i t i o n e n , Acta Arith. 4, 217-239 (1958).
M. EICHLER, Einbe Anwendungen der Spurformel im Bereich der Modularkorrespondenzen, Math. Ann. (1966). In press. E. HECKE,uber Modulfunktionen und Dirichletsche Reihen mit Eulerscher Produktentwicklung, Part I, Math. Ann. 114, 1-28 (1937); Part 11, Math. Ann. 114, 316351 (1937). Math. Werke, Nos. 35, 36, GBttingen, 1959. E. HECKE,uber die Darstellung der Determinante einer positiuen quadratischen Form durch die Form, Vierteljschr. Naturforsch. Ges. Zurich 85, 6 4 7 0 (1940). Math. Werke No. 40. E. HECKE,Analytische Arithmetik der positiven quadratischen Formen, Kgl. Danske Videnskab. Selskab, Math.-Fys. Medd. 17, (1940). Math. Werke. No. 41. H. PEIERSSON,uber die Zerlegung der den ganzen Modulformen wn haherer Stufe entsprechenden Dirichletreihen in vollstiindige Eulersche Produkte, Acta Math. 80, 191-221 (1948).
G. SHIMURA, Correspondences modulaires et les fonctions 5 de courbes algJbriques, J. Math. Soc. Japan 10, 1-28 (1958). K. WOHLFAHRDT, uber Operatoren Heckescher Art bei Modulformen reeller Dimension, Math. Nachr. 16, 233-256 (1957).
$4. Castelnuovo’s Inequality 1. INTRODUCTION
In $4, K and K‘ will always be conservative and separably generated fields; this hypothesis is certainly satisfied whenever ko is perfect. In $2 we showed that a separable prime correspondence A, not the unit correspondence, together with its Rosati-conjugate correspondence A, satisfies the equation %(A + A*) = g ( A ) + g ‘ ( 4 -f(4, (1) where s1 is the trace of the representation of the correspondence in the module of differentials of degree 1 and of first kind, andf(A) is the number of
282
v. CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
fixed points as discussed in $2,5. By defining the number of fixed points of a composed correspondence by
f
=
c m,f(A,),
Eq. (1) also is valid for such sums. Even the unit correspondence D may be included. Its trace is known to be s,(D D*) = 29, so that setting
+
f ( D ) = 2 - 29 leads to the validity of (1). We saw in $2,2 that in the classical case
44 = g ( 4 + g ’ ( 4 -f(4
(2) is the trace of a particular representation H ( A ) of the ring of correspondences. H ( A ) is the sum of the two representations H , ( A ) , H,(A). Suitable choice of a basis of the representation module makes the matrices H,(A) of Rosatiadjoint correspondences hermitian adjoint :
Hi@)* = H,(A)‘,
i = 1,2,
Thus we have i = 1,2, Hi(AA*) = H,(A)H,(A)‘, and the traces of these matrices are 2 0. This means that s(AA*)
20
(3)
for any correspondence A , a fact called Castelnuovo’s inequality. Surprisingly this inequality (3) can be proved quite generally for arbitrary fields of functions. Note that, from their derivation, the right sides of (1) and (2) are rational integers. In a function field of prime characteristicp the left side of (1) is an element of k, so that then this equation requires that the residue class mod p be taken on the right. If the function s(A) is defined as the right side of (2), though, it becomes a rational integer independently of the characteristic; then (3) makes sense. Two proofs, valid generally, will be supplied in $4 for (3). In this general case we also call s(A) the trace of A , without necessarily requiring it to be the trace of a representation of the ring of correspondences. Nevertheless, we shall see that at least in certain special cases it is. 2. REDUCTION TO THE CLASSICAL CASE We will first use the already proved validity of inequality (3) in the classical case to prove its validity in the following more general case, The characteristic is taken to be 0, and the field K to be generated as K = k,(x, y ) with f ( x , y ) = 0. Further, let A , ( x , y ; x’, y’) (v = 1, 2, ...) be a system of generators of the K x K’-ideal defining the correspondence A , these A , being rational functions of their variables. But a finite number of constants occur in f ( x , y ) and the A(x, y ; x’, y’), and these, adjoined to the
& CASTELNUOVO'S I.
INEQUALITY
283
prime field Q, give a finitely generated field k, which can be embedded isomorphically into the field C of complex numbers. K , = k,(x, y ) with f ( x , y ) = 0 now defines a function field over kl and AY(x,y ; x', x ' ) a correspondence A , between K, and K,'. Computation of AIAl* is the same as that of AA*, and the numbers g(AIA1*), g'(AIA1*), and f ( A , A 1 * ) coincide with g(AA*), g'(AA*), andf(AA*). As k, c C we know (3) to hold with A , in place of A . Thus (3) must also hold for the situation originally at hand. Thus we have proved our inequality for all fields of characteristic 0. The inequality of Castelnuovo holds even if K and K' are not isomorphic. Then AA* is at least a correspondence between K and a field K" isomorphic to K, and the trace (2) is well defined. If K and K' are not isomorphic there do exist isomorphic finite extensions K,/K and K,'/K'. The lemma of $1,3 gives A"(A*)" = [K,' : K']AA*, so that K'may be replaced by K,' in the proof of (3). If K and K" are also replaced by K, and K,", then g(AA*)andg'(AA*) are multiplied by the factor [K, : K]. As will be shown [in Eq. (17)], the same is true for f ( A A * ) . These substitutions make K, K', and K" isomorphic, so that then (3) holds. These considerations, incidentally, do not depend upon the field characteristic. The entire proof can also be made useful for a field K of prime characteristic. It must be assumed, though, that K arises from a field KO of the same genus but of characteristic 0 by reduction with respect to a prime ideal p o of the constant field of KO,the ideal p o being regular in the sense of 111,$6,2, We will not discuss the question of the existence of such a KO for arbitrary K. The configuration occurs naturally in certain applications (e.g., $53-7). Our notation will now depart from that of 111, $6, in that fields of characteristic 0 and associated objects will carry subscript 0, which will be dropped after residue class formation modulo p o . In adapting (3) we shall use the fact, still to be demonstrated, that the number of fixed pointsf@) of a correspondence A is equal to the degree of a certain divisor f(A) of K associated with A , called the divisor ofjixedpoints. This proof could be based on $2, but it will be more convenient to relate it to later material of $43.Our point of departure is inequality (3) for characteristic 0, f ( A A * ) being taken as a divisor degree. The argument is as follows. Given is a correspondence A between K and K'. K'can also be considered a reduction mod p o of some field KO'of functions with characteristic 0, the regularity of p o in this second reduction not being required. 111, $6,6 assures the existence of a finite extension Khl/Ko' and of a correspondence A:' between KO and KA, which, under reduction mod p o , maps into that extension A" of A associated with the extension Kl'/K'. (If p o splits in the extension K6,/K0' the reduction is taken modulo one of its prime divisors.) By the lemma in $1,3 we have A"A*"= [K,': K']AA*. Here we form the trace (2) which is a linear combination of divisor degrees in
284
V. CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC
FUNCTIONS
KK“ and K. But these are invariant under reduction mod po. This means that s(A~’A:’‘)= [K1’:K’]s(AA*), which is 2 0, for the correspondence on the left involves fields of characteristic 0. Two points are left to be clarified in this argument: first, that the operations of multiplying correspondences and reduction mod po are commutable, and second, that the operations of formation of a fixed point divisor, to be described in 443,and of reduction mod po are commutable. The first case results from the definition of multiplication in 91,2 as a residue class formation of ideals with respect to ideals; the fixed point divisor will also be defined with such an operation. Thus both cases depend upon double residue formation, which is always commutative. That is, for a ring 0 and D-ideals a, b we have the isomorphism theorem Despite the simplicity of the idea behind this proof of Castelnuovo’s inequality, troublesome details were necessary, not to mention the assumption that a function field of prime characteristic could be obtained from a field of characteristic 0 by reduction mod po . Among the details was the proof of the trace formula of 92. Our second proof (in #4,6-8) will turn out to be quicker, but also more sophisticated. It will be based from the start on our new interpretation off(A) as the degree of the divisor of fixed points, and it will be precisely this interpretation which will then lead to applications in number theory (95).
3. EXTENSION OF THE NOTIONOF CORRESPONDENCE Let a prime correspondence P be given. All those A E KK’ whose denominators are not divisible by the ideal P K x Kform , a discrete local ring DP. In 11,$5 we associated a prime divisor of KK’ with such an Dp, and we will now identify P with that prime divisor. An arbitrary correspondence A = Zm,P, thus becomes a divisor of KK’, where divisor multiplication is now written additively. Now consider the complete group of divisors of KK’, but with a slight change of definition from 11, 95. There we proceeded from a rational subfield ko(x, x’) of KK’.A prime divisor 3,, of ko(x, x’) was associated with every prime polynomial p in x and x‘, and one more 3aowas added. In contrast we now introduce two “infinite” prime divisors 3m and 3,, whose local rings consist of those rational functions having nonpositive degrees in x and x‘, respectively. A prime polynomial p(x, x ‘ ) of degrees G and G‘ in x and x’, respectively, is now paired with the principal d‘visor -G
(P(X,x‘)) = 3p(x.x’) 3,
-G’
-
3d Divisors thus defined in ko(x, x’) decompose to prime divisors in any finite extension and, in particular, in KK‘.
$4, CASTELNUOVO’S
285
INEQUALITY
This defines a group R of divisors of KK’. It contains three distinguished subgroups: (a) the prime divisors of all 3p with p = p(x) dependent upon x alone, and with p = co, and their power products, it is the group f of divisors of K; (b) the analogous group f‘; (c) the group 6 generated by prime divisors of all 3p with p = p(x, x’) dependent upon both variables. Clearly R is the direct sum of f, f‘, and a. Any divisor ‘ill can be decomposed uniquely as %==++’+A, a € € , a’ef’, A E ~ . (4) We call a and a’ the constant and A the nonconstant components of %. The group 6 is isomorphic to the group of correspondences between K and K’;indeed, the two groups coincide if viewed under the convention made above. The divisors of more general form (4) are the correspondences in the extended sense. Two equivalence relationships are defined for them. The stricter requires the difference between the equivalent divisors, or correspondences to be a principal divisor. Two correspondences are said to be equivalent in the extended sense if their difference is of the form a + a‘ + (A) with a principal divisor (A). Two degrees g(%) and g’(%) are again defined, with
w = g(W + g(W,
g‘(’ill + 8 )= g’(W + g ’ ( b ) . (5) Indeed, the components a and A of (4) are divisors of the field KK’ of functions in one variable over the constant field K , and thus have degrees. We set
d’ill+
+
+
g’(’ill) = g‘(a’ A). g(%) = g(a A), This implies (5). These degrees are functions of the class of in the strict sense. We may also define multiplication of the correspondences in the extended sense. For algebraically closed k , this leads, as in §1,5, to a multiplication of correspondence classes in the extended sense, for
(a
+ a‘ + A)(b’ + b”+ B) = g‘(b‘ + B)a + Ab’ + g’(a’ + A)b”+ a‘B + AB.
(6) AB is defined as in §1,2, a‘B as in §1,5, and Ab’ as the obvious adaptation of the prqcess of §l,5 (residue formation for b’ mod A). The associative and distributive laws hold, and Eq. §l,(lO) as well as the lemma of §1,3 remain valid. These facts are all just incidental, and will not be applied later. Important later is a global representation of the divisors. The process used was already described in 11, §5,5, but will be repeated here, now eliminating the introduction of homogeneous coordinates at the cost of certain advantages and gaining others. We adjoin an indeterminate z and form the Kronecker divisors of KK’ with respect to the order lj = k,[x,x’] in the sense of 11, $1,6. These divisors are the cosets of the elements a(z) E KK’(z) with respect to the group of units @,(z)of the principal order fi(z) with respect to lj(z). Thegroup 9, of Kronecker divisors of KK’ with respect to lj is a homomorphic
286
V.
CORRESPONDENCES BETWEENFIELDS OF ALGEBRAIC FUNCTIONS
image of the divisor group R defined above; the divisors of R mapped onto the unit element are exactly those composed of prime divisors of the denominators of x and x'.
4. THEFIXED POINTS OF A
CORRESPONDENCE
From now on we suppose K and K to be isomorphic. Moreover, we determine a fixed isomorphic map of K onto K' leaving each element of ko invariant, once and for all, This isomorphism will be indicated by a prime: x-XI,
awa'.
To any correspondence N between K and K' we define a divisor f(N) of K to be the divisor of fixed points of N. The definition itself depends upon whether or not 21 contains the unit correspondence D in its prime decomposition. For D itself we set f(D) = -b, (7) with some arbitrarily determined divisor b in the class of differentials of K. If D does not occur in the prime decomposition of N we define the p-component f,(N) = pCpp(') for any place p of K in another manner. The representation of correspondences by Kronecker divisors is used, a special preparation being needed first. Choose some element x E K whose denominator is not divisible by p, and denote its isomorphic image x'. Form = ko[x,x'] and the principal order !jj(z) of KK'(z) with respect to lj(z), with the indeterminate z. It is easy to see that, when K and K' are identified, B(z) becomes the entire field K(z), and this is the reason why B(z) cannot serve our purposes. By &(z) we denote the set of elements of $(z) which are mapped onto elements of 3 ( z ) under K = K', where now 3 ( z ) is the principal order of K(z) with respect to i(z) = k,[x](z). That !ij,(z) is an integral domain is obvious. Moreover, we maintain it to be a principal ideal domain in the sense that any finite set of elements ai(z)E 5jl(z) not all divisible by the diagonal have a g.c.d. in fil(z)[a,(z),a2(z), ...I. This is proved by an easy adaptation of the argument of 11, §1,3. The elements ai(z) given, like any elements of B1(z),can be uniquely decomposed to
a i w = aro(z) + 6i(Z) with aio(z)E K(z) and 6,(z) divisible by D. The g.c.d. is a(z) = ao(z)
+ 6(z) = C aiO(z)zht+ C ai(z)zh', i
i
with suitable h i . It must still be shown that the quotients
are mapped onto elements of 3 ( z ) by K = K . But this is also obvious, as the ai(z)were assumed not all divisible by D, so that the quotients aio(z)ao(z)-' are defined and contained in 3 ( z ) .
§4.
CASTELNUOVO'S INEQUALITY
287
For simplicity's sake we again write @(z) in place of s l ( z ) . The divisor associated with a correspondence 2l not divisible by D is given by a finitely generated ideal and therefore by a Kronecker divisor with respect to !ij(z). As such it is represented by an element a(z), uniquely determined by % up to the factor of a unit E(Z)of @(z). Now identify the fields K and K' in the sense of the given isomorphism. The divisor D does not occur in the prime decomposition of 2l, so that a(z) is mapped onto a nonzero element E(z) of K(z). Moreover, elements of @(z) are mapped into 3(z), and the property of being a unit is retained by definition. This means that the coset a(z)E,(z) is a Kronecker divisor of K with respect to i. Its p-component is defined to be the p-component off(%). It is implicit that the f,(QI) really define a divisor f(%), that is, that they are almost all (1). It is not yet clear, though, that the definition is independent of the choice of x. But, say xl is some other element whose demoninator is not divisible by p. Then so is x, = xlx. The principal order 3, of K with respect to k0[x2] consists of all functions with denominators composed of the prime divisors of the denominator of x, . The analogous statement holds for the principal order @, of KK' with respect to Ij, = k o [ x , , x,']. Thus, .2f22 3, 9, 2 @, and for the groups of units E,,(z) 2 E,(z),E,,(z) 2 E,(z). The denominator of x, is not divisible by p, so that identification of K and K' makes the units of E,,(z) to Kronecker divisors with respect to i not divisible by p. Hence f,(%) comes out the same, if x is replaced by x, . The same argument can be applied for x1 and x, . The components of divisors not containing D satisfy
f,(%
+ 8)= f,(W + f,(dJ).
For if a(z) and p(z) are elements representing A and B, then a(z)B(z) represents the divisor 2l + d. The identification K = K' maps the product a(z)b(z) onto the product of the images, which is what must be shown. If D occurs in the prime decomposition of 2l exactly a times, then set f(%) = f(% - a D ) - ab.
(8)
This convention then yields f(%
+ 9) = fW) + f(W.
(9)
The following remark will be needed later: if QI is an integral divisor not containing D, then f(QI)is integral. This is an immediate rcsult of our construction. A correspondence (A) of the principal class in the strict sense satisfies
-
f((A)) 0. (10) Thus, 2l + f(2l) is a homomorphism of the classes of correspondences onto the classes of divisors.
288
v. CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
Proof. First consider only (A) free of factors of the diagonal D. The element A can then be taken as representative of the Kronecker divisor with respect to Ij for (A). This means
f(W)= (AK~=K).
(11)
Now, let D divide (A) exactly a times. By 111, $ 5 3 we see that the product A(x - x ~ ) is - ~free of D-components if x is a separating element of K. By assuming the validity of f((x - x') - 0 ) by (12) N
Eq. (8) implies (lo), as we already know that assertion holds for A(x - x')-'. It remains to prove (12). The ideal DKx g s defining D is generated by differences y - yf with y E K, a system of y = y o , y l , ... generating K over ko sufficing. Now, let (yi - y:) generate DK K , . Then
w = (Yo - Yo') +
(Y1
- Y O Z 't ...
is associated with a Kronecker divisor having only constant prime divisors aside from D, with exactly one factor D. As any y, may be replaced by y;', we may also assume all the y i to be integral at some place p. Finally, the differencep - pf for some prime element p of p could be chosen as one of the generators of DK K , allowing us to choose yo = p. Thus prepared, we use the corollary of III,95,5 to compute
The y being integral at the place p, so are the differential quotients dyldp. Thus, (13) gives us
(where we have inconsistently used the multiplicative notation to emphasize the relation between the quotient and differential quotient). By again using the corollary of 111, $5,5 (and once more in multiplicative notation) we have
Compiling these equations for all p , we have found that
fry)
=(ax),
and (12) is proved. 7
(14)
$4.
289
CASTELNUOVO'S INEQUALITY
We can now define the number offixedpoints of A as
f (W= s(f(W). (15) Because of (lo), f(W is a function of the correspondence class in the strict sense, and because of (9,
fW + 8)=f(W +f(W.
(16)
Another means of calculating the divisor f(2l) is easier to handle in applications, but would not have sufficed to prove (9) and (16), as we do not know whether the 3, x 3,'-ideals form a group. The p-component f,(%) for an integral divisor 9l can also be found asfollows: decompose % as in (4) and set
f,(W = f,(a
+ b' + 4= a, + 6, + f,(A),
where the 3,-ideal associated with f,(A) and with the same notations isfound as f,(A) = ideal of residue classes of AR R, n3px 3;. mod DR
Kt
in 3,.
(3, is the integral domain of elements of K integral at p.) Proof. Let A, be a system of generators of the ideal A K x K tAs . the A, could be multiplied with elements a, E K, b,' E K'so that the products liein3, x 3;, , and are divisible neither by p nor p', we may assume them already to be such. The polynomial a(z) = CAP' thus represents a Kronecker divisor 8 differing from A by a constant factor and, in particular, by a factor not divisible by p or p'. Thus f,(A) = f , ( 8 ) . But f,(8) is the p-component of the divisor of E(z) = Z(Ai)K,=K zi, while the (Ai)K,=Kare the residues of the A, in K with respect to the module D K x K ,The . assertion is thus proved. T We must finally get to the equation f(A'A*') = [K,: K ] f ( A A * ) , (17) which we used in §4,2 for the extension A' of the correspondence A between K and K' to a correspondence between Kl and K'. It is a consequence of f(B"') = CK,: KIf(B1, where the isomorphic extensions K l / K and Kl'/K' lead from B to B"'. The latter equation immediately follows from the fact that the degree of a divisor is multiplied by the degree of the extension when it is brought up to an extension field. 5. THECONNECTION WITH $2
We interrupt our line of thought to show that the number of fixed points defined by (1 5 ) coincides with the number
f( A ) =
c
fii
(18)
290
v. CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
given in $2, where the faare those of $2,(23).We again make the assumptions of $2, that k, is algebraically closed and that A is a prime correspondence which defines a common separable extension K = (K x K')/A, K t of the two fields K and K'. Formula (18) was, in particular, the foundation of §4,2, where it was used for the classical case. For the proof, we first select another field K" independent of, but isomorphic with, K and K'. Set &,(A) = ideal of residue classes of ( A K x , , x K") n (3,x 3;, x 3,"..) mod K x D,, in 3, x Ti,,.(19) x,,p
The algorithm at the end of the last section shows that f,(A) is gotten from this by identifying K = K", the distinction between the divisor !,(A) and the associated 3,-ideal being dropped. Now extend K to Kl until the extended correspondence is a sum A' = EA, of prime correspondences of degree g'(A,) = 1. We give a construction of such an extension. By §1,6, A defines a relation between K and K' such that R = (K x K')/AKX,,is a finite separable extension of both K and K'. Every x f E K' satisfies a polynomial equation f ( x ' ) = 0 over K, and A, is generated by all these polynomials. Each of these polynomials has a zero in R, so that a summand Al of degree g'(Al) = 1 can be split off A over K. Then, if Kl = K , K 2 , ... are all the conjugate fields of R over K and Kl is their composite field, the desired decomposition is found in K l . Now set this extension into (19), which is done by simply multiplying both sides by the principal order 3', of Kl with respect to 3,. For the extended ideal on the left we find x 3bt x Yip,) ~ , , ( A ' )= ideal of residue classes of ( A i l x K , x K")n mod K, x D R , x K , in 3,, x 3:". The separability of K/IR assures us that the prime correspondences A, are , mutually prime. Hence, distinct, and thus the ideals
The validity of this equation is not impaired by multiplication on both sides with K" and intersection with 31,x 3,' x Yi,,. Thus
sl,(A1) = nideals of residue classes of ((AP),,
x
K")n
x 3brx 3j.)
mod K, x
DK'xp
in 3', x Ti,,.
rKt
P
Because the A, are mutually prime we have, for ((A,),, x R , x K" u (Am)KIx K , x K") n so that, moreover,
@
#
6,
x 3bj x 3:,,)= 3',,x 3;, x 3:,,
$4.
Z,,(A')
29 1
CASTELNUOVO'S INEQUALITY
=nideals of residue classes of ((A,&,
xK,
x
K")n(3,, x 3;. x 3,",,)
P
mod K, x
DK'xp
in 3,, x 3;".
As g'(A,) = 1, the symmetry law for multiplication of $1,3 can be applied to the individual factors on the right, yielding
Z,,(A')
=nideals of residue classes of (K,
x DK.xKr,) n (3,, x
P
mod (A&,
xK,
36, x 3,",,)
x K" in 31,x 3;.,.
We saw that this gives f,(2l) if K and K" are identified; actually it gives the extended ideal of f,(q) in K,, but a distinction between them need not be made here. Thus &(A) =
n ideals of residue classes of K,
DK,
n(31p x 36.)
P
mod (AP)KlxKa in 31,. (20) According to 111, $S,S,if x is a prime element for p, then x - x' is a prime element for D. The residue classes of Kl DK, K. n 3,, x 3b, modulo the A, are the principal 3,,-ideals ( x - xP'),the xP' being the images of the x in the fields K , . Hence (20) can be written as fp(A)
=
n
(x - xp))p = ( I I K / d X - .'I), 9
P
the subscript p indicating that only the components at the place p, or the places lying over p, are to be taken. The subscript e can also be eliminated, then choosing x' E R = ( K x K')/AK This is exactly the configuration of $2,5, but with K and K' interchanged. The power series $2,(21) are set up for all fixed points of R lying over p, and it is seen that f,(%) is always divisible by the same power of p as nKIK(IIpf~). Here we have, of course, taken the fixed points in the sense of $2. Noting that, because ko is algebraically closed by hypothesis, all prime divisors of K are of the first degree and n~&) = p, we see that &,(A)) = Xf6, which verifies the assertion. Kt.
6. THETRACE For a correspondence in the extended sense, s(W = B(W + g'(W -f(W defines the trace. With (16) it satisfies s(% f
B) = s(%)
+ s(B).
For constant correspondences it is obvious that s(a) = s(b') = 0.
292
v. CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
Also for a correspondence (A) of the extended principal class the trace is .$(A)) = 0.
Hence, the trace is a linear function of the class of a correspondence in the extended sense. We also mention the immediate result, s(%) = s(%*). (22) The principal theorem, which we now want to prove, is stronger than inequality (3). In a conservative function jield K, s(%%*) >0 (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.
CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
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
The first sum can be computed by (25) and is i
i
V.
296
CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS.
Because of §1,(10) the second sum is 2
c S(PiPj*) c (g(P,) + g(Pj)) - 2 C f ( P , P i * ) =
i cj
= 2(rn
- l)g(A) -
* * a .
lcj
i#j
We shall demonstrate the inequality
C f(PiPj*) 5 (g’ + m - 2)g(A),
(31)
i<1
which then implies 2 C S(PiPj*) 2 2(1 - g’)g(A). i-=/
Taken with (30), this inequality is equivalent to (29). It is the proof of (31) in which the difficulty lies, and for which we needed the preparations of the last section. We start with rn elements wi’, constructed to suit the pi’ as in the lemma. The Pi define isomorphisms K’ -+ K j onto subfields of K of indices CK: KjI = dPj). (32) These isomorphisms map the wi’ onto elements wij E K j . The lemma showed that the wi’ could be chosen so that the w i j are integral at a given place p of K. We must now demonstrate three properties of the w i j : (a) their determinant ( w i j (# 0; (b) its denominator is divisible by the divisor of fixed points f(CicjPiPj*); and (c) the denominator of Iwij( is of degree s(g’ m - 2)g(A). This suffices, for the numerator cannot have a larger degree than the denominator, nor can f( Xi< Pipj*).The last statement is (31). (a) By definition
+
wi’ = wij + Hij,
nijE 0 mod P i ,
(33)
The w i j are integral at the place p, as are also the wi’ and the llij. Consider the wij mod p. Certainly there exist wij E ko congruent to the w i j mod p. Then nij = 0 mod p i ‘ , wij = wi’ + n i j , according to (33) and pPj = pj‘, so that the w i j are the residue classes of the wI’ mod pi’ in k o . We can apply (28) to the primed symbols to see that oij= 1 or 0, depending upon whether i = j or i # j. Hence, the determinants loljl and lwijl # 0. (b) We show the denominator of Iwijl to be divisible by f,( Pipj*) for every place q of K. By 3, we denote the integral domain of all elements of K integral at q, we set 3q,i= Ki n 3,,and write 3,’for the isomorphic image of 3q,iin K‘.The common denominator of the wi’ can still be adjusted so that the wi‘ lie in eachof thefinitelymany3,’forwhich f,( PIPj*) # 0. (The 0 here is the additive notation for the unit divisor.)
xixi
xi<
$4. CASTELNUOVO’S INEQUALITY
297
Now let x i traverse all elements of 3q,i and denote by xi the corresponding elements of K j under the isomorphisms x i c)x’ t)x i . Also, let x; be the associated elements in a field K” isomorphic to, but independent of, K. The differso that computation in the ences x i - xy generate the ideal (CPi’$j*)K.K,r, manner of §4,4 using ideal theory shows fq(YiVj*) to be generated by the differences x i - x i . Let, in particular, a prime element for the prime divisor qi of Ki , divisible by q, be chosen as x i . The degree of qi must be 1, as k, is algebraically closed, and by 111,41,2 there exist constants yiJp in ko such that
Remembering that the fields K j are isomorphic, we see that the y i j r do not depend upon j, so that
fq(ci<
A choice of r greater than the power of q contained in P i p j * )assures the differsnce between the determinants lwijl and lZ~y,,,xjp1to be divisible by q more often than fq( Pipj*). But it is well known that the latter j ( x i - xi), and thus determinant is a polynomial in x i and xi divisible by Pipj*). The same must be true of lwijl. also by fq( (c) For any fixed index j the w i j have a common denominator Ijj . Because of (27) its degree in K j must be
niC
xi<
d b j ) 6 g‘ + m - 2. The degree of bj in K must be g(Pj) times as much, because of (32). Thus the degree in K of the denominator divisor of the determinant lwijl must be IZ(g’ + m - 2)g(Pj) = (g’ + m - 2)g(A). The proof is complete. 7 -
9*. REMARKS CONCERNING THE RINGOF CORRESPONDENCE CLASSES By $1,4 the classes of correspondences form a ring. In §4,6 we saw that the trace is a function of these classes. If A # 0 is a class and m a natural number then mA = A + -..+ A # 0. For @A*) # 0 means that s((mA)(mA)*) = m2s(AA*) > 0. This makes it possible to write formal sums Zq,A, of correspondence classes A, with arbitrary rational q p , and to define addition and multiplication naturally. The result is the hypercomplex system 6 over the rational number field Q.
298
V.
CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
6 is semisimple. It must be proved that no nilpotent elements are contained in the center of 6.Say Nh-' # 0, N h = 0, and h 2 2. It is certainly no limitation of generality to assume N to be a correspondence class in the original ) 0, so that sense. The principal theorem says that s ( N ~ - ' N * ~ - '> N h - l p h - 1 # 0. Also, s(Nh-' N *h-l(Nh-'N*h-')*) > 0, so that N h - I p h - 1 ( Nh - 1 N * h - 1 )* # 0. Certainly (Nh-'N*h-')* = Nh-'N*"-'. By the center property of N we see that then N 2 h - 2 N * 2 h - 2# 0. But, as h 2 2 we have 2h - 2 2 h, which gives the contradiction desired. 7 The ring of correspondence classes can be represented by matrices of 2g rows if the characteristic is 0, as shown for the classical case in 84,l. (The representation decomposes to the sum of two representations of g rows.) $(A) is the trace of the representation. For any characteristic A. Weilt could prove the following. Let 1be any prime not the characteristic and consider the totality of divisor classes of K which give the principal class when multiplied by lh with some h > 0 (or, in multiplicative notation, whose lhth power is that class). It is a group, 6 , ,isomorphic to the additive group of vectors with 2g components, which are residue classes of 1-adic numbers modulo I-adic integers. The correspondence classes A define homomorphisms of 6, into itself, and are thus representable as matrices G ( A ) with 2g rows of I-adic integers. The trace of this representation G(A) coincides with s(A), which in turn implies that the representation is equivalent to one with rational coefficients, The analogous group 6, where 1 is the characteristic involves a more difficult configuration, which was investigated by J.-P. Serre.$
NOTES The first general proof of the principal theorem is due to Weil [4], but we followed the argument of Roquette [3]. Another proof is found by close examination of the representation G(A) mentioned in §4,9. The principal theorem then turns out to be a special case of a more general statement which is valid for all abelian varieties (cf. Lang's book, IV,§3,6 [l]). A comparatively short proof was given by Mattuck and Tate [2]. It is based on the Riemann-Roch theorem for two-dimensional algebraic varieties, which cannot enter the scope of this book. From this point of view the inequality s(AA*) > 0 is related to other facts of algebraic geometry (cf. Grothendieck 111).
t
VariPtPs abdiennes et courbes algibriques, Actualites Sci. Ind. No. 1064 (1948). $ Quelques propriitis des varidis abdiennes en charactCristique p . Amer. J. Math. 80, 715-739 (1958).
$5. APPLICATIONS IN NUMBER THEORY
299
A common feature of all these proofs is their resort to algebraic geometry of several dimensions and, in particular, to the intersection theory for subvarieties. Our proof limited the use of intersection theory. Actually, though, the number of fixed points of a correspondence A is the number $(A, D) of intersection points between the curve A = 0 on the “variety” K x K’ and the diagonal D which is defined by identification of K and K’:
More generally, the intersection number s(A, B ) of two correspondences A, B can be defined by the method of w,4, first choosing a prime correspondence B with g(B) = 1 but not occurring among the prime factors of A . It defines a mapping of K onto a subfield KO’c K’, and one can set
As in $1,3 (multiplicative symmetry), it can then be shown that
s(A, B ) = s(B, A). Another easy result is s(A, B) = s(AB, D).
This implies that f ( A B ) = f ( B A ) and thus the important law of symmetry for the trace, s(AB) = s(BA).
REFERENCES [l] A. GROTHENDIECK, Sur une note de Muttuck-Tute. J. Reine Angew. Math. 200, 208215 (1958). [2] A. MATTUCKund J. TATE,On the inequality of Custelnuovo-Severi, Abh. Hamburger Math. Sem. 22,295-299 (1958). [3] P. ROQUET~E, Arithmetischer Beweis der Reimunnschen Vermutung in Kongruenzzetufunkrionenkorpern beliebigen Geschlechts, J. Reine Angew. Math. 191.199-252 (1953). [4] A. WEIL,Sur les courbes ulgibriques et les vuriitis qui s’en diduisent, ActualitCs Sci. Ind. No. 1041 (1948).
$5. Applications in Number Theory 1.
THEZETA FUNCTION OF A FIELD OF FUNCTIONS
Let K be a field of functions over a finite exact constant field ko with q = p r elements; p is the characteristic. These fields have much in common with finite algebraic number fields. This relationship is particularly emphasized by the zeta function, which we now define and investigate. As for the zeta function of an algebraic number field, we must refer to the texts on algebraic number theory, but no such information will be presupposed here.
300
v. CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
In extreme contrast to the arithmetic case we can prove an analog of the Riemann conjecture for fields of functions, which turns out to be a result of Castelnuovo's inequality. In this regard the theory of functions proves to be easier than number theory, in the end. To a divisor a is associated its absolute norm
N(a) = q#(").
(1)
For a prime divisor a = p, the residue class field of the integral domain 3, of functions integral at p, modulo p, is an extension of ko of degree g(p). All residue classes modulo p are of the form Qioi, el E ko , where mi is a basis of the field. Thus, there are exactly N ( p ) residue classes. The relationship mentioned between algebraic number fields and the function fields being considered is primarily rooted in the finiteness of this residue class number; indeed, the residue classes of a number field modulo a prime ideal are finite in number. Obviously N(ab) = N(a)N(b). (2) The zeta function is defined by the sum
taken over all integral divisors a of K. We drop the problem of convergence for the moment, as it will solve itself in the quite explicit form in which we can write [(s) below. The uniqueness of prime decomposition of divisors together with (2) yields the Euler product decomposition
taken over all prime divisors p. The prime divisors arise from the numerator divisors of the prime polynomials p(x) in some variable x and the denominator divisor of x by means of the extension K = k,(x). Now, the number of prime polynomials of some fixed degree, indeed, the number of arbitrary polynomials of any degree, is finite. This nieans that there can be but a finite number of prime divisors in K of any degree. Hence the product (4) and with it the sum (3) is meaningful, the problem of convergence being postponed. It becomes opportune, at times, to use the variables u =q-5
in place of s, and to write Z(U)
The function U
=
w.
dlogZ(u) = N,un du n= 1
(5)
(6) (7)
$5. APPLICATIONS IN
NUMBER THEORY
30 1
can be interpreted as follows: for every n the coeficient N,, in the expansion gives the number of first degree prime divisors of the extended field Kk,,,, where ko,, is the familiar unique extension of ko of degree n. In fact the definition of Z(u) placed into the Euler products yields d log Z(u) U
du
=
c 2 g(p)u"~'", p v=l
so that
Nn
=
2 dp).
e(p)/n
All prime divisors p of K whose residue class fields are isomorphically represented in k,,,/k, decompose to prime divisors of the first degree in that field. But these are exactly the prime divisors whose degrees over k, divide n. And then, these p decompose into g(p) first degree divisors.
2. THEFUNCTIONAL EQUATION
The functional equation which we now want to derive for [(s) will turn out to be a result of the Riemann-Roch theorem. The number of divisor classes of a given degree n is the same for all n. For, if A , , A z ,... are differing classes of degree n, then AIA,-' = ( I ) , A 2 A 1 - ' , ... are differing classes of degree 0, and if B,, B2,... are differing classes of degree 0, then B,A,, BzA,, ... are differing classes of degree n. This number h is usually called the class number of K , and we will show it to be finite. Evidently all natural integers occurring as degrees of integral divisors are multiples of an integer r. In 111,§3,5 we saw that r = 1 or 2 in function fields of genus 0. Under the condition that the constant field k, is finite r = 1, since Eq. (14) in that paragraph is always solvable in such a k, . We shall prove here that, with a finite k , , r = 1 regardless of the genus. The result will be derived from properties of the zeta function, so we have to investigate the zeta function without this information. Yet one point is clear since the canonical class coniains integral divisors and has degree 2g - 2 ; r must be a divisor of 29 - 2. If the genus g of K is 0, then the class number h = 1 and there are n + 1 linearly independent integral divisors a], ..., a,,,, in the class of degree n. All integral divisors of this class are of the form a = CQiai with ei E k , , not all 0. As a depends only upon the ratios of the ei,there are exactly (q"" - l)(q - l)-' such a. Thus,
302
V . CORRRESPONDENCFS
BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
Now assume g > 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
v. CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
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,
APPLICATIONS IN NUMBER THEORY
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.
CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
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.
APPLICATIONS IN NUMBER THEORY
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.
CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
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
v. CORRESPONDENCES BETWEEN FIELDS OF ALGEBRAIC FUNCTIONS
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).
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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
