The Theory of Jacobi Forms
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Progress in Mathematics 55 Series Editors: Hyman Bass · Joseph Oesterlé · Yuri Tschinkel · Alan Weinstein
Martin Eichler · Don Zagier
The Theory of Jacobi Forms
ll
Progress in Mathematics. Vol. 55 Edited by J. Coates and S. Helgason
Springer Science+Business Media, LLC
Martin Eiehier Don Zagier The Theory of
Jacobi Forms
1985 Springer Science+Business Media, LLC
Authors Martin Eiehier im Lee 27 CH-4144 Arlesheim (Switzerland)
Don Zagier Department of Mathematies University of Maryland College Park, MD 20742 (USA)
Ubrary oj Congress Cataloging in Publication Data Eiehler, M. (Martin) The theory of J aeobi forms. (Progress in mathematies ; v. 55) Bibliography: p. 1. Jaeobi forms. 2. Forms, Modular. 1. Zagier, Don, 1951II. Title. III. Series: Progress in mathematies (Boston, Mass.} ; v. 55. QA243.E36 1985 512.9'44 84-28250
CIP-Kurztitelau/nahme der Deutschen Bibliothek Eichler, Martin: The theory of J aeobi forms I Martin Eiehier ; Don Zagier. Boston ; Basel ; Stuttgart : Birkhäuser, 1985. (Progress in mathematies ; Vol. 55)
NE: Zagier, Don B.:; GT ISBN 978-1-4684-9164-7 ISBN 978-1-4684-9162-3 (eBook) DOI 10.1007/978-1-4684-9162-3 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, e!ectronic, mechanical, photocopying, recording or otherwise, without prior permission of the Copyright owner.
© 1985 Springer Science+Business Media New York Originally published by Birkhäuser Boston, Inc in 1985.
9 8 7 6 5 4 3 2 1
V
T A B L E
I.
0 F
C 0 N T E NTS
Introduction
1
Notations . .
7
Basic Properties
8
Jacobi forms and the Jacobi group
2.
Eisenstein series and cusp forms .
17
3.
Taylor expansions of Jacobi forms
28
Application: 4. II.
III.
8
1.
37
Jacobi forms of index one
41
Hecke operators
Relations with Other Types of Modular Forms
.
57
5.
Jacobi forms and modular forms of half. . . . . integral weight . . . .
57
6.
Fourier-Jacobi expansions of Siegel modular forms and the Saito-Kurokawa conjecture
72
7.
Jacobi theta series and a theorem of Waldspurger
81
89
The Ring of Jacobi Forms
89
8.
Basic structure theorems
9.
Explicit description of the space of Jacobi forms. 100 Examples of Jacobi forms of index greater than 1
10.
Discussion of the formula for
11.
Zeros of Jacobi forms
..
dim Jk,m
113 121 133
Tables
141
Bibliography
146
INTRODUCTION The functions studied in this monogra9h are a cross between elliptic functions and modular forms in one variable.
Specifically,
we define a Jacobi form on SL 2 (~) to be a holomorphic function (JC = upper half-plane) satisfying the t\-10 transformatio n eouations
(1)
) z ( a-r +b cp CT +d ' CT +d
(2)
rjl(T, z+h+]l)
(cT+d)
k
e
2Tiimcz· CT +d
cp(T,z)
and having a Four·ier expansion of the form
2:: 2: n=O 00
cp(T,z)
(3)
c(n,r)
e2Tii(nT +rz)
rE~
r Here
k
2
~ 4nm
m are natural numbers, called the weight and index of
and
Note that th e function
respectively.
rp,
cp (T, 0) is an ordinary modular z-+rjl(
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