Arithmetic On Modular Curves (progress In Mathematics)

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. Front Matter....Pages i-xvii Background....Pages 1-42 Periods of Modular Forms....Pages 43-75 The Special Values Associated to Cuspidal Groups....Pages 76-106 Congruences....Pages 107-125 P-adic L-functions and Congruences....Pages 126-165 Tables of Special Values....Pages 166-210 Back Matter....Pages 211-217 Background -- Periods Of Modular Functions -- The Special Values Associated To Cuspidal Groups. Glenn Stevens. Bibliography: P. 211-214.