Mark Dickinson (University of Michigan)

In the 1920's Emil Artin conjectured that the L-function L(s,r)

associated to a continuous representation r: Gal(L/K)-->GL_n(C) of the

Galois group of an extension L/K of number fields is holomorphic

everywhere, except possibly at s=1. I will discuss a partial proof of

this conjecture in the rather special case when n=2, K=Q and r is an

icosahedral (equivalently, non-solvable) odd representation. This

complements results of Langlands, Tunnell et. al. (circa 1980) which gave

a complete proof of the conjecture for n=2 and solvable r (the case n=1

was proved by Artin himself). I will also explain how the Taylor-Wiles

construction at the prime 2 plays a crucial role in the proof.