### Friday February 2, 2007, 1:45 PM , Goldsmith 226

**Speaker**: Joel Bellaiche (Columbia)

**Title**: Extensions of Galois representations

**Abstract**:
A great number of results and open questions
Êin algebraic number theory can be reformulated, and thus unified,
in terms of existence, or
non-existence, of extensions with prescribed
properties between given irreducible representations of an absolute
Galois group.

In the first part of this talk, I want to explain how two old and
classical diophantine problems
(solving the Pell-Fermat equation, and counting rational points on
elliptic curves) can be reformulated in term of the existence of suitable
extensions of Galois representations, and to formulate the general
conjecture, due to Bloch and Kato, that predicts the dimension of the
space of extensions between two general Galois representations.

In the second part, I want to show how such extensions can be constructed
using families of automorphic forms.