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.