Ebook: 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.

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This volume provides an in-depth study of the geometry of special values (at s = 1) of L-functions attached to modular curves. Its primary aim is to describe congruences satisfied by the special values modulo Eisenstein ideals. One chapter is devoted to constructing p-adic L-functions and proving congruences between them. The final chapter displays numerous tables of special values of L-functions. The results presented here have intriguing applica- tions to the conjecture of Birch and Swinnerton- Dyer, and shed light on the structure of the cuspidal divisor group. 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. '][-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 1Tf 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 Chapter 4. Congruences 107 4.1. Eisenstein Ideals 109 4.2. Congruences Satisfied by Values of L-functions 115 4.3. Two Examples: XI (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-adic 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. XI (13), Odd quadratic characters 205 Bibliography 211