(Progress in Mathematics 20) Glenn Stevens (auth.)-Arithmetic on Modular Curves-Birkh盲user Boston (1988).pdf

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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
~ibrary of Conqress Cataloqing in Publication Data
Stevens, Glenn, 1953-
Arithmetic 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.
512' .72 82-4306
AACR2
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 ; )
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 -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
. A Calculus of Special Values 48
. The Cocycle TT f and Periods of Modular Forms 51
. Eisenstein Series 55
. Periods of Eisenstein Series 66
Chapter 3. The Special Values Associated to Cuspidal Group