Arithmetic on Modular Curves
One of the most intriguing problems of modern number theory is to relate the arithmetic of abelian varieties to the special values of associated L-functions. A very precise conjecture has been formulated for elliptic curves by Birc~ and Swinnerton-Dyer an
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Progress in Mathematics Vol. 20 Edited by J. Coates and S. Helgason
Birkhauser Boston· Basel· Stuttgart
Glenn Stevens
Arithmetic on Modular Curves
1982
Birkhiiuser Boston • Basel • Stuttgart
Author: Gl enn Ste'lens Department of ~athematics Rutgers University New Brunswick, New Jersey 08903
of Conqress Cataloqing in Publication Data Stevens, Glenn, 1953Arithmetic on modular curves. (Progress in mathematics; v. 20) includes bibliographical references. 1. Forms, Modular. 2. Curves, Modular. 3. L-functions. 4. Conqruences and residues. I. Title. II. Series: Progress in mathematics (Cal'lbridge, Mass.) ; 20. QA243.S77 512' .72 82-4306 AACR2
~ibrary
CIP-Kurztitelaufnahme der Deutschen Bibliothek Stevens, Glenn: Arithl'letic on I'lodular curves I Glenn Stevens. -Boston; Basel; Stuttqart : BirkhJuser, 1982. (progress in mathel'latics ; Vol.20) NE: GT
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, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the copyright owner. ©Bi rkh~user Boston, 1982 ISBN 978-0-8176-3088-1
ISBN 978-1-4684-9165-4 (eBook)
DOI 10.1007/978-1-4684-9165-4
Dedicated to Mrs. Helen Hammitt
Table of Contents
Introduction Chapter 1.
Background
1
1. 1.
Modular Curves
4
1. 2.
Hecke Operators
7
1. 3.
The Cusps
11
1. 4.
1[-modules and Periods of Cusp Forms
18
1. 5.
Congruences
24
1. 6.
The Universal Special Values
27
1. 7.
Points of finite order in Pic 0 (X (r»
32
1. 8.
Eisenstein Series and the Cuspidal Group
35
Chapter 2.
Periods of Modular Forms
43
2. 1.
L-functions
45
2.2.
A Calculus of Special Values
48
2.3.
The Cocycle TT f and Periods of Modular Forms
51
2.4.
Eisenstein Series
55
2.5.
Periods of Eisenstein Series
66
Chapter 3.
The Special Values Associated to Cuspidal Groups
76
3.1.
Special Values Associated to the Cuspidal Group
78
3.2.
Hecke Operators and Galois Modules
84
3.3.
An Aside on Dirichlet L-functions
90
3.4.
Eigenfunctions in the Space of Eisenstein Series
93
3.5.
NOTlvanishing Theorems
101
3.6.
The Group of Periods
103
viii
Chapter 4.
Congruences
107
4. 1.
Eisenstein Ideals
109
4.2.
Congruences Satisfied by Values of L-functions
115
4.3.
Two Examples: X 1 (13),X O(7,7)
122
Chapter 5.
P-adic L-functions and Congruences
126
5. 1.
Distributions, Measures and p-adic L-functions
128
5. 2.
Construction of Distributions
134
5.3.
Universal measures and measures associated to cusp forms
141
5.4.
Measures associated to Eisenstein Series
146
5.5.
The Modular Symbol associated to E
151
5.6.
Congruences Between p-adlc L-functions
157
Chapter 6.
Tables of Special Values
166
6. 1.
Xo (N), N prime :: 43
167
6.2.
Genus One Curves, Xo (N)
188
6.3.
Xl (13), Odd quadratic characters
205
Bibliography
211
Introduction
One of the most intriguing problems of modern number theory is to relate the arithmetic of abelian varieties to the special value
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