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.

1. Background. - 1. 1. Modular Curves. - 1. 2. Hecke Operators. - 1. 3. The Cusps. - 1. 4. $$
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$$-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.