The study of L-functions (such as the Riemann zeta-function) and their arithmetic properties has long been a central focus of number 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 significant arithmetic information. The most celebrated example of this is the Birch-Swinnerton-Dyer Conjecture (BSD) which 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. The 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, essentially a systematic study for a prime p of
the variation of the p-parts of the special values and of the related arithmetic data for a. 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.