Tata Lectures on Theta I

247 Pages·2007·2.085 MB·English

by  David Mumford (auth.)

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Tata Lectures
on Theta I
David Mumford
With the collaboration of
C. Musili, M. Nori, E. Previato,
and M. Stillman
Reprint of the 1983 Edition
Birkhauser
Boston • Basel • Berlin
David Mumford
Brown University
Division of Applied Mathematics
Providence, RI02912
U.S.A.
Originally published as Volume 28 in the series Progress in Mathematics
Cover design by Alex Gerasev.
Mathematics Subject Classification (2000): 01-02,01A60,11-02,14-02,14K25,30-02,32-02,33-02,
46-02 (primary); 11E45,11G10,14C30, 58F07 (secondary)
Library of Congress Control Number: 2006936982
ISBN-10: 0-8176-4572-1 e-ISBN-10: 0-8176-4577-2
ISBN-13: 978-0-8176-4572-4 e-ISBN-13: 978-0-8176-4577-9
Printed on acid-free paper.
©2007 Birkhauser Boston BirkMuser 11$)
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David Mumford
With the collaboration of
C. Musili, M. Nori, E. Previato,
and M. Stillman
Tata Lectures
on Theta I
Birkhauser
Boston • Basel • Berlin
David Mumford
Department of Mathematics
Harvard University
Cambridge, MA 02138
Library of Congress Cataloging-in-Publication Data
Mumford, David.
Tata lectures on theta I.
(Progress in mathematics ; v. 28)
Includes bibliographical references.
Contents: 1. Introduction and motivation : theta
functions in one variable ; Basic results on theta
functions in several variables.
1. Functions, Theta. I. Title. II. Series: Progress
in mathematics (Cambridge, Mass.); 28.
QA345.M85 1982 515.9'84 82-22619
ISBN 0-8176-3109-7 (Boston)
ISBN 3-7643-3109-7 (Basel)
CIP- Kurztitelaufnahme der Deutchen Bibliothek
Mumford, David:
Tata lectures on theta / David Mumford. With the assistance of C. Musili
... - Boston; Basel; Berlin : Birkhauser.
1. Containing introduction and motivation: theta functions in one
variable, basic results on theta functions in several variables. -1982.
(Progress in mathematics ; Vol. 28)
ISBN 3-7643-3109-7
NE:GT
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Graph of Re -S^z, y^r) ,
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TABLE OF CONTENTS
Introduction ix
Chapter I. Introduction and motivation: theta functions in 1
one variable
§1. Definition of /&\Z,T) and its periodicity in z 1
§2. ^x,it) as the fundamental periodic solution to the 4
Heat equation
§3. The Heisenberg group and theta functions with 5
characteristics
§4. Projective embedding of (E/& + TLi by means of theta 11
functions
§5. Riemann's theta relations 14
§6. Doubly periodic meromorphic functions via ^KZ,T) 24
§7. The functional equation of ^1(Z,T) 28
§8. The Heat equation again 33
§9. The concept of modular forms 34
§10. The geometry of modular forms 44
§11. /iP-as an automorphic form in 2 variables 53
§12. Interpretation of H/I\ as a moduli space 60
§13. Jacobi's derivative formula 64
§14. Product expansion of t£ and applications 66
§15. Representation of an integer as sum of squares 74
§16. Theta and Zeta 83
§17. Hurwitz maps 92
Appendix: Structure of the inverse limit J-£ 95
§18. Hecke operators 103
References and Questions 116
Chapter II. Basic results on theta functions in several 118
variables
§1. Definition of and its periodicity in z 118
§2. The Jacobian variety of a compact Riemann surface 13 5
§3. xP'and the function theory on a compact Riemann surface 146
Appendix: The meaning of A 162
§4. Siegel's symplectic geometry 171
§5. tP'as a modular form 18 9
Appendix: Generators of Sp(2g,Z) 202
§6. Riemann's theta formula and theta functions associated 211
to a quadratic form
§7. Theta functions with harmonic coefficients 227
ix
Introduction
This volume contains the first two out of four chapters
which are intended to survey a large part of the theory of theta
functions. These notes grew out of a series of lectures given
at the Tata Institute of Fundamental Research in the period
October, 1978, to March, 1979, on which notes were taken and
excellently written up by C. Musili and M. Nori. I subsequently
lectured at greater length on the contents of Chapter III at
Harvard in the fall of 1979 and at a Summer School in Montreal
in August, 1980, and again notes were very capably put together
by E. Previato and M. Stillman, respectively. Both the Tata
Institute and the University of Montreal publish lecture note
series in which I had promised to place write-ups of my lectures
there. However, as the project grew, it became clear that it
was better to tie all these results together, rearranging and
consolidating the material, and to make them available from one
place. I am very grateful to the Tata Institute and the University
of Montreal for permission to do this, and to Birkhauser-Boston for
publishing the final result.
The first 2 chapters study theta functions strictly from the
viewpoint of classical analysis. In particular, in Chapter I,
my goal was to explain in the simplest cases why the theta
functions attracted attention. -I look at Riemann's theta function
T) for z € (E, xGH = upper half plane, also known as 1?" ,
and its 3 variants ^i'^f ' ll* We show how tnese can be used
0
to embed the torus (C/Z+Z«T in complex projective 3-space, and
X
how the equations for the image curve can be found. We then prove
the functional equation for with respect to SL(2,Z) and show
how from this the moduli space of 1-dimensional tori itself can
be realized as an algebraic curve. After this, we prove a beautiful
identity of Jacobi on the z-derivative of i?*. The rest of the
chapter is devoted to 3 arithmetic applications of theta series:
first to some famous combinatorial identities that follow from
the product expansion of •'$'; second to Jacobi's formula for the
number of representations of a positive integer as the sum of
4 squares; and lastly to the link between 4?" and £ and a quick
introduction to part of Hecke's theory relating modular forms and
Dirichlet series.
The second chapter takes up the generalization of the geometric
results of Ch. I (but not the arithmetic ones) to theta functions
in several variables, i.e., to ifHz,Q) where z G (Cg and
ft € .%* = Siegel's gxg upper half-space. Again we show how <&'
can be used to embed the g-dimensional tori X„ in projective
space. We show how, when ft is the period matrix of a compact
Riemann surface C, /L^is related to the function theory of C.
We prove the functional equation for and Riemann's theta
formula, and sketch how the latter leads to explicit equations
for X~ as an algebraic variety and to equations for certain
modular schemes. Finally we show how from ^z,fl) a large class
of modular forms can be constructed via pluri-harmonic
polynomials P and quadratic forms Q.
The third chapter will study theta functions when ft is a
period matrix, i.e., Jacobian theta functions, and, in particular,
xi
hyperelliptic theta functions. We will prove an important identity
of Fay from which most of the known special identities for Jacobian
theta functions follow, e.g., the fact that they satisfy the non
linear differential equation known as the K-P equation. We will
study at length the special properties of hyperelliptic theta
functions, using an elementary model of hyperelliptic Jacobians
that goes back, in its essence, to work of Jacobi himself. This
leads us to a characterization of hyperelliptic period matrices ft
by the vanishing of some of the functions /iP- f1(0, Q) . One of the
goals is to understand hyperelliptic theta functions in their own
right well enough so as to be able to deduce directly that
functions, derived from them satisfy the Korteweg-de Vries equation
and other "integrable" non-linear differential equations.
The fourth chapter is concerned with the explanation of the
group-representation theoretic meaning of theta functions and the
algebro-geometric meaning of theta functions. In particular, we
show how i?(z,ft) is, up to an elementary factor, a matrix coefficient
of the so-called Heisenberg-Weil representation. And we show how
the introduction of finite Heisenberg groups allows one to define
theta functions for abelian varieties over arbitrary fields.
The third and fourth chapters will use some algebraic
geometry, but the chapters in this volume assume only a knowledge
of elementary classical analysis. There are several other
excellent books on theta functions available and one might well
ask — why another? I wished to bring out several aspects of the
theory that I felt were nowhere totally clear: one is the theme