Taylor-Wiles at 2 and icosahedral Artin representations

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.