Arithmetic on Modular Curves

233 Pages·1988·5.24 MB·English

by  Glenn Stevens (auth.)

D
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.
QA243.S77 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 ; 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 TTf 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: X1(13),XO(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 values of associated
L-functions. A very precise conjecture has been formulated for elliptic
curves by Birc~ and Swinnerton-Dyer and generalized to abelian varieties
by Tate. The numerical evidence is quite encouraging. A weakened form of
the conjectures has been verified for CM elliptic curves by Coates and
Wiles, and recently strengthened by K. Rubin. But a general proof of the
conjectures seems still to be a long way off.
A few years ago, B. Mazur [26] proved a weak analog of these con-
jectures. Let N be prime, and be a weight two newform for r 0 (N) .
For a primitive Dirichlet character X of conductor prime to N, let
i\ f (X) denote the algebraic part of L (f , X, 1) (see below). Mazur showed in
[ 26] that the residue class of Af (X) modulo the "Eisenstein" ideal gives
information about the arithmetic of Xo (N). There are two aspects to his
work: congruence formulae for the values Af(X) , and a descent argument.
Mazur's congruence formulae were extended to r 1 (N), N prime,
by S. Kamienny and the author [17], and in a paper which will appear shortly,
Kamienny has generalized the descent argument to this case.
The purpose of this monograph is to describe generalizations of the
x
congruence formulae to other congruence groups. The present work differs in
two ways from the earlier approach. First of all, the modular units, which
Mazur uses to produce his parabolic cohomology classes, are replaced by
their logarithmic derivatives which are weight two Eisenstein series. This
makes the congruence formulae more natural, and reduces lengthy calculations
with Dedekind sums to well known results concerning special values of
Dirichlet L-functions. Since Eisenstein series appear in many settings, we
may also anticipate that these methods will generalize, e. g. to the higher
weight case or to Hilbert modular forms. The second difference is that the
decomposition of the cuspidal group in the factors of the modular Jacobian is
used to determine which congruences are satisfied by special values of
L -functions of which cusp forms.
Let r be a congruence group of type (N l' N2) and of level
N lcm(N1,N2) (see §1.1). Let X be the associated modular curve,
o
r,
f be a normalized weight two newform for and Af <;:; Pic (X) be the
abelian subvariety associated to f ([ 42] , Theorem 7. 14). Let (!)(f) <;:; a:
be the ring generated by the eigenvalues of the Hecke operators acting on f.
Let 11" <;:; End (Pic0 (X)j~) be the ring of endomorphisms generated by the
Hecke operators. Then ~(f) is identified with the ring of endomorphisms
of Af obtained by restriction of 11" to Af • Let r c;; (!)(f) be a prime
ideal and k = (!)(f)jr. Let Af [p] denote the r -torsion subgroup of
Af • Then Af[ r] is a k-vector space. Assume r satisfies
xi
(*)
If Af is an elliptic curve this condition is trivially satisfied. in general (*)
holds for all but finitely many primes P in ~(f).
a: a:.
Let 1r act on through the natural homomorphism 1r'" ~(f) <;;;
Then integration with respect to the differential form w (f) on X whose
pullback to 3( is f (z) dz gives rise to a 1r -homomorphism
a:
---;> L
'Y ----;;> w (f)
1-1
+ -
There are periods Of ' Of € a:* and 1r -homomorphisms
~(f) such that the images are not contained in P and
0: P
Of course are determined only up to multiplication by a -unit in
~(f) •
TI-IEOREM (Shimura [43]): Let X be a nontrivial primitive Dirichlet
character of conductor prime to N. Then
A ( ) dfn T (x) L (f I X I 1) € ~(f)[ X]
f X 2"iOsgn(X)
f
m. - 1
XL 2" ia/m
where T (X) x(a) e X o
a = 1