Arithmetic on Modular Curves

by G. Stevens

Overview

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 c- 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.

Product Details

ISBN-13:	9780817630881
Publisher:	Birkhäuser Boston
Publication date:	01/01/1982
Series:	Progress in Mathematics , #20
Edition description:	1982
Pages:	217
Product dimensions:	5.98(w) x 9.02(h) x (d)

Table of Contents

1. Background.- 1.1. Modular Curves.- 1.2. Hecke Operators.- 1.3. The Cusps.- 1.4. $$ % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf % gDOjdaryqr1ngBPrginfgDObcv39gaiuqacqWFtcpvaaa!41F4! \mathbb{T} $$-modules and Periods of Cusp Forms.- 1.5. Congruences.- 1.6. The Universal Special Values.- 1.7. Points of finite order in Pic0(X(?)).- 1.8. Eisenstein Series and the Cuspidal Group.- 2. Periods of Modular Forms.- 2.1. L-functions.- 2.2. A Calculus of Special Values.- 2.3. The Cocycle—f and Periods of Modular Forms.- 2.4. Eisenstein Series.- 2.5. Periods of Eisenstein Series.- 3. The Special Values Associated to Cuspidal Groups.- 3.1. Special Values Associated to the Cuspidal Group.- 3.2. Hecke Operators and Galois Modules.- 3.3. An Aside on Dirichlet L-functions.- 3.4. Eigenfunctions in the Space of Eisenstein Series.- 3.5. Nonvanishing Theorems.- 3.6. The Group of Periods.- 4. Congruences.- 4.1. Eisenstein Ideals.- 4.2. Congruences Satisfied by Values of L-functions.- 4.3. Two Examples: X1(13), X0(7,7).- 5. P-adic L-functions and Congruences.- 5.1. Distributions, Measures and p-adic L-functions.- 5.2. Construction of Distributions.- 5.3. Universal measures and measures associated to cusp forms.- 5.4. Measures associated to Eisenstein Series.- 5.5. The Modular Symbol associated to E.- 5.6. Congruences Between p-adic L-functions.- 6. Tables of Special Values.- 6.1. X0(N), N prime— 43.- 6.2. Genus One Curves, X0(N).- 6.3. X1(13), Odd quadratic characters.