Arithmetic on Modular Curves PDF

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.