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/Q be an elliptic curve. Now, we work with an analytic function Lp(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 Lp(E,s) at s=1 is equal to the rank of the Mordell-Weil group E(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 Lp(E,s) is at most the rank of E(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/ .