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 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.

Inhalt

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.