Christian Wuthrich (Nottingham): p-adic L-series and Iwasawa theory
Artin and Tate have shown a large part of the conjecture of Birch and Swinnerton-Dyer in the function field case in the 60s. Iwasawa theory for elliptic curves as initiated by Mazur tries to use similar to tools to approach the p-adic version of the Birch and Swinnerton-Dyer conjecture.
Let E/\mathbb{Q} be an elliptic curve. Now, we work with an analytic function L_p(E,s) taking values in the p-adic numbers. It is built on the values of the complex L-function and can be described explicitly using modular symbols. The conjecture says again that the order of vanishing of L_p(E,s) at s=1 is equal to the rank of the Mordell-Weil group E(\mathbb{Q}). The big advantage of the p-adic setting is that this p-adic L-function has a natural link to the arithmetic side via the so called "main conjecture" of Iwasawa theory about which we know quite a lot.
A big and difficult theorem by Kato shows that the order of vanishing of L_p(E,s) is at most the rank of E(\mathbb{Q}). It even says something about the size of the mysterious Tate-Shafarevich group. Furthermore one can generalize it to abelian extensions K/Q. It is suitable for explicit computations.
References
Mazur, B.; Tate, J.; Teitelbaum, J., On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer. Invent. Math. 84 (1986), no. 1, 1--48. math scinet
Stein and Wuthrich, Computations About Tate-Shafarevich Groups Using Iwasawa Theory, preprint http://wstein.org/papers/shark/ .