Gergely Zábrádi


Non-commutative Iwasawa theory and the Birch--Swinnerton-Dyer conjecture

Take an elliptic curve with rational coefficients. According to the Mordell--Weil theorem the points on the curve over a number field form a finitely generated abelian group. The rank of this group is a very important arithmetic invariant---if we regard the curve as a diophantine equation then this quantity measures the `number of solutions'. The conjecture of Birch and Swinnerton-Dyer provides us with an extremely useful formula for the rank of the curve---the order of vanishing of the complex L-function at the symmetry point of its functional equation. In my talk I will be trying to shed some light on this mysterious relationship.
The main idea of Iwasawa theory is that instead of sticking to a fixed number field one investigates the above conjecture in a tower of Galois extensions whose Galois group is a p-adic Lie group. The connection between the arithmetic of elliptic curves and the special values of L-functions is provided by the so-called p-adic L-function whose---mostly conjectural---properties I would like to talk about.