Table Of Contentlun-ichi 19usa
Theta Functions
Springer-Verlag New York Heidelberg Berlin 1972
Jun-ichi 19usa
The Johns Hopkins University, Baltimore. Md. 21218, USA
GeschiiftsfUbrende Herausgeber:
B.Eckmann
Eidgeniissische Technische Hochschule ZUrich
B. L. van der Waerden
Mathematisches Institut der U niversitiit ZUrich
AMS Subject Classifications (1970)
Primary 14-XX, 13-XX, 32·XX
Secondary 10-XX, ~2-XX, 43-XX, 81-XX
ISBN -13 :978-3-642-65317-9 e-ISBN-13: 978-3-642-65315-5
DOl: 10.1007/978-3-642-65315-5
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
concerned. specifically those of translation, reprinting, re-use of iIIustrations. broadcasting, reproduction
by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copy
right Law where copies are made for other than private use, a fee is payable to the publisher, the amount
of the fee to be determined by agreement with the publisher. @by Spr!nger-VerlagBeriinHeidelberg 1972.
Softcover reprint of the hardcover 1st edition 1972
Library of Congress Catalog Card Number 74-183900.
Die Grundlehren der
mathematischen Wissenschaften
in Einzeldarstellungen
mit besonderer Beriicksichtigung
der Anwendungsgebiete
Band 194
Herausgegeben von
J. L. Doob . A. Grothendieck . E. Heinz· F. Hirzebruch
E. Hopf . W. Maak . S. MacLane . W. Magnus· J. K. Moser
D. Mumford· M. M. Postnikov· F. K. Schmidt· D. S. Scott
K. Stein
Geschiifts/iihrende Herausgeber
B. Eckmann und B. L. van derWaerden
lun-ichi Igusa
Theta Functions
Springer-Verlag Berlin Heidelberg New York 1972
Jun-ichl 19usa
The 10hns Hopkins University, Baltimore, Md. 21218, USA
Geschilftsfiihrende Herausgeber:
B.Eckmann
EidgenOssische Technische Hochschule Zlirich
B. L. van der Waerden
Mathematisches Institut der Universitiit Zlirich
AMS Subject Classifications (1970)
Primary 14-XX, 13-XX, 32-XX
Secondary 100XX, 22-XX, 43-XX, 81-XX
ISBN -13: 978-3-642-65317-9 e-ISBN -13: 978-3-642-65375-5
DOl: 10.1007/978-3-642-65375-5
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction
by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copy
right Law where copies are made for other than private use, a fee is payable to the publisher, the amount
of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1972.
Library of Congress Catalog Card Number 74-183900. Softcover reprint of the hardcover 1st edition 1972
Foreword
The theory of theta functions has a long history; for this, we refer
the reader to an encyclopedia article by A. Krazer and W. Wirtinger
("Sources" [9]). We shall restrict ourselves to postwar, i.e., after 1945,
periods. Around 1948/49, F. Conforto, c.L. Siegel, A. Well reconsidered
the main existence theorems of theta functions and found natural proofs
for them. These are contained in Conforto: Abelsche Funktionen und
algebraische Geometrie, Springer (1956); Siegel: Analytic functions of
several complex variables, Lect. Notes, I. A. S. (1948/49); Well: Theoremes
fondamentaux de la theorie des fonctions theta, Sem. Bourbaki, No. 16
(1949). The complete account of Weil's method appeared in his book of
1958 [20]. The next important achievement was the theory of compacti
fication of the quotient variety of Siegel's upper-half space by a modular
group. There are many ways to compactify the quotient variety; we are
talking about what might be called a standard compactification. Such a
compactification was obtained first as a Hausdorff space by I. Satake in
"On the compactification of the Siegel space", J. Ind. Math. Soc. 20,
259-281 (1956), and as a normal projective variety by W. L. Baily in 1958
[1]. In 1957/58, H. Cartan took up this theory in his seminar [3]; it was
shown that the graded ring of modular forms relative to the given modular
group is a normal integral domain which is finitely generated over C.
The relation between this theory and the theory of theta functions
comes from the following fact: Mter a suitable normalization, every theta
function becomes a linear combination of
L
0m(r, z)= e(t(p+ m') rt(p+m')+(p+m'nz+m"»),
peZB
in which e(x)=exp(2nix), m=(m' m") with m', m" in Rg, r is a point of the
Siegel upper-half space 6 i. e., a complex symmetric matrix of degree g
g,
with a positive-definite imaginary part, and z a variable point of O. If we
evaluate this theta function at z=O, we get a "constant" Om(r,O); we
restrict m to Q2g and call the corresponding function r -+ 0m(r, 0) on 6g
a theta constant. It was observed in 1964 that, up to a normalization, the
standard compactification is nothing else than the usual closure (in the
complex projective space) of the image of 6 under a mapping given by
g
a carefully chosen set of theta constants; this is our fundamental lemma
[5]. Around the same time, Siegel published his work on the moduli of
VI Foreword
abelian varieties (3rd in [16]); Siegel considered theta constants of the
following form:
(Ok! + ... +kg Bm•O ("C, z)/(o Z1)k! ••. (0 z/g)z=o
and proved, e. g., the important fact that a finite set of theta constants
"separates points" of the quotient variety.
On the other hand, the structure of the compactification is an attrac
tive subject to algebraic geometers; and the fundamental lemma has been
used as such for this purpose. In 1966 D. Mumford succeeded in devel
oping a purely algebraic theory oftheta functions which includes the case
of characteristic p 9= 2 [12]; his theory contained new results even in the
case of characteristic 0, e. g., the above theorem of separating points
without taking derivatives. This was an important achievement in the
direction of the purely algebraic theory of abelian varieties created by
Weil during the middle 40's.
The main objective of this book is to develop a theory of theta func
tions in characteristic 0 which includes all that we have said. Following
our practice, we shall give a complete statement of the main result: Let
A(x, y) denote a non-degenerate alternating form on R2g x R2g of the
furm
g
A(x,y)= Lei(X;Yg+i-Y;X +i)'
g
;=1
in which e1, .,. , eg are positive integers each e; dividing e;+ 1 for 1 ~ i < g;
let e denote the diagonal matrix with e; as its i-th diagonal coefficient.
Let G denote the subgroup of GL (R) which keeps A (x, y) invariant and
2g
Gz(e, 2e) the subset of G consisting of those elements
such that 0( = 19 + e a, f3 = e b, y = e c, ~ = 19 + e d for integer matrices a, b,
c, d in which b, c have even diagonal coefficients; G acts on 6 as
g
"C --+ (0( "C + f3 e)(y"C + ~ e)-1 e;
if e=O mod 2, Gz(e, 2e) forms a subgroup of G. If e=O mod 4, the quo
tient variety Gz(e, 2e) \ 6 is a complex manifold; if m' runs over a
g
complete set of representatives of zg e-1 /zg, the correspondence
gives rise to an injective, locally biholomorphic mapping of this manifold
into ~(C) for N + 1 = det(e); if X denotes the image, the closure X of X
in ~(C) with respect to the usual Hausdorff topology is a projective
variety of dimension !g(g+ 1) such that the boundary X -X of X is
Foreword VII
contained in a Zariski closed set of dimension at most h(g-I). This is a
theorem of considerable "depth" at least in the sense that the proof is
quite involved.
Some reader must have been puzzled by the highly specialized nature
ofthe functions that we have talked about. If we accept, however, abelian
varieties as natural objects, the above theta functions can be introduced
in a quite natural manner in connection with such varieties. This was
known classically. In the last decade, a certain group-theoretic meaning
of theta functions came to be known. It has turned out that a theorem
discovered in connection with physics, the so-called Stone-von Neumann
Mackey theorem, plays a central role; the same theorem also provides
the starting point to Mumford's theory. We owe this theory to V. Barg
mann [2], P. Cartier [4], G. W. Mackey [11], I.E. Segal [15], and Weil
[21]. This book contains a major aspect of this theory.
Chapter I deals with the theory just mentioned; we shall give an
intrinsic characterization of functions of the form e(Q(x») in which Q(x)
is a quadratic polynomial in x such that the imaginary part of the homo
geneous part of degree 2 is positive-definite. Chapters II-III deal with the
theory of Conforto, Siegel, Weil; we shall prove the theorem on the
projective embedding of a polarized abelian variety by theta functions.
Each polarized abelian variety determines a graded ring of theta functions,
which is unique up to a degree-preserving isomorphism; it is a normal
integral domain which is finitely generated over C. Roughly speaking, the
"structure constal}ts" of this C-algebra are the theta constants. Chap
ter IV is due largely to Mumford. We shall show that, under the condition
which corresponds to e=O mod 4, the image under the projective
embedding can be considered as the set of common zeros of explicitly
written quadratic polynomials. In Chapter V we shall prove the main
theorem that we have stated. We shall use the classical method by Siegel
to prove the algebraic dependence of modular forms.
In writing this book, we have tried to make the book readable by any
good second year graduate student in a relatively short time. This put a
yoke on our tongue. Except for some elementary properties of algebraic
varieties (to which we gave only brief indications of proofs), we have
assumed very little. We have avoided complex spaces, Chow's theorem
etc. by restricting ourselves to complex manifolds; we have avoided the
theory of local rings by restricting ourselves to standard theorems in
algebra. Also we have tried not to include those materials which the
reader can learn after finishing the book. For instance, we have excluded
the theory of" p-adic theta functions" by Tate and Morikawa; and the
theory of "Fourier-Jacobi series" by Pyatetski-Shapiro [14] appears
only implicitly. We have excluded the development in the last decade on
"Schottky's relations". We can say that Baily's paper, "On the theory of
VIII Foreword
O-functions, the moduli of abelian varieties, and the moduli of curves",
Ann. Math. 75, 342-381 (1962), belongs to this category. But more
definitely, there are important works by Andreotti-Mayer, Rauch and
Farkas etc., which we have excluded. We have done all this with the hope
that the reader can read through the book and thus gain a bridgehead
toward this interesting but difficult theory.
It is with pleasure that we mention that the book has been written, in
a more or less final form, while the auther was at the Institute for Advanced
Study in the academic year 1970/71. He acknowledges the Johns Hopkins
University for giving him the leave of absence and the National Science
Foundation for making the leave financially possible. He wishes to take
this opportunity to express his gratitude to those people who gave him
useful suggestions concerning the book. He owes specifically to K. Igusa
for helping him in fmishing the book, e. g., for providing the proof of
Chap. IV, Theorem 4. Also K.Igusa and C. Y. Lin read the manuscript
and corrected minor errors.
Towson, 1971 lun-ichi Igusa
Contents
Chapter I. Theta Functions from an Analytic Viewpoint . 1
§ 1. Preliminaries . . . . . . 1
§ 2. Plancherel Theorem for R" 5
§ 3. The Group A (X) . . . 8
§ 4. The Irreducibility of U . 11
§ 5. Induced Representations 14
§ 6. The Group Sp(X) . • 20
§ 7. The Group B(X) . . . 26
§ 8. Fock Representation. . 31
§ 9. The Set ~(X) . . . . . 36
§ 10. The Discrete Subgroup Ii . 42
Chapter II. Theta Functions from a Geometric Viewpoint 51
§ 1. Hodge Decomposition Theorem for a Torus. 51
§ 2. Theta Function of a Positive Divisor . 58
§ 3. The Automorphy Factor u~(z). . 64
§4. The Vector Space L(Q, I, l/I). . . . . 70
§ 5. A Change of the Canonical Base. . . 78
Chapter III. Graded Rings of Theta Functions . 86
§ 1. Graded Rings. . . . . . . . . . 86
§ 2. Algebraic and Integral Dependence . . 94
§ 3. Weierstrass Preparation Theorem . . . 99
§ 4. Geometric Lemmas . . . . . . . . . 107
§ 5. Automorphic Forms and Projective Embeddings . 112
§ 6. Polarized Abelian Varieties . . 118
§ 7. Projective Embeddings. . . . . . . . . 125
§ 8. The Field of Abelian Functions . . . . . 132
Chapter IV. Equations Defining Abelian Varieties 136
§ 1. Theta Relations (Classical Forms) . . . . 136
§ 2. A New Formalism. . . . . . . . . . . 142
§ 3. Theta Relations (Under the New Formalism) 146
§ 4. The Ideal of Relations . . . . . . ~ . . . 152
§ 5. Quadratic Equations Defming Abelian Varieties 166
