2/4/2005

Christopher Skinner
University of Michigan

Main Conjectures and Modular Forms

The study of L-functions (such as the Riemann zeta-function) has long been a central focus of numbey theory, especially analytic number theory. But over the last half-century it has become increasingly clear that values of these function at special points (special values) reflect arithmetic information. The most celebrated examples of this are the Class Number Formula for zeta functions for number fields and the Birch-Swinnerton-Dyer Conjecture (BSD) for elliptic curves. The latter predicts that the order of vanishing at s=1 of the L-function L(E,s) of an elliptic curve E over a number field K is equal to the rank r(E) of the group of K-rational points on E. A refined version of this conjecture also expresses the leading coefficient of the Taylor series of L(E,s) around s=1 in terms of arithmetic data coming from E.
This talk will be about work related to proving parts of the refined BSD, at least when L(E,1) is non-zero. These results for L(E,1) are obtained through Iwasawa theory. More precisely, I will report on work relating the p-adic L-function of an elliptic curve (or even a holomorphic eigenform for GL(2)) that is ordinary at a prime p to the characteristic ideal of the associated p-adic Selmer group. The Main Conjecture for elliptic curves (or modular forms) asserts that the latter is generated by the former. K. Kato has shown that the characteristic ideal contains the p-adic L-function in many cases. I will discuss joint work with E. Urban that uses the arithmetic of automorphic forms on unitary groups, esp. U(2,2), to show the opposite inclusion.