Title: A conjecture of Bloch-Kato and automorphic forms
<p>Speaker: Joel Bellaiche, Columbia University


<p>Abstract: To a continuous, geometric representation $\rho$ of the absolute Galois group of
a number field, over a $p$-adic field, is attached two objects of arithmetic
interest.
The first one is of algebraic nature : the so-called Selmer group of $\rho$,
a finite dimensional vector space that classify the arithmetically significant
extensions of the trivial representation by $\rho$. The second one is of
analytic nature :
the $L$-function of $\rho$ which is (conjecturally in most cases) a meromorphic
function on the complex plane.
The Bloch-Kato conjecture predicts a surprising and beautiful relation between
those two objects.

In my talk, I will explain and motivate the definition of the object introduced
above as well as the
statement of the conjecture. I will show using examples how many outstanding
theorem
and questions in number theory may be seen as special cases of its conjecture. I
will then
explain a method, used by several authors in recent years (including myself), to
deduce an important part
of this conjecture using congruences between automorphic forms, and discuss the
results obtained so far by this method.