James Cogdell:
The local Langlands correspondence for GL(n) and the
symmetric and
exterior square epsilon factors
Abstract. Artin introduced his non-abelian L-functions for representations
of the Galois group in a series of papers in 1923--1931. He was able to
define the local Euler factors for all primes and define the Artin
conductor that appears in the functional equation, but the Artin root
number remained mysterious. It was factored by Deligne in 1971 as part of
his proof of the existence of the local epsilon-factors that appear
in the functional equation of the Artin L-functions. One way to try to
understand these L-functions and epsilon-factors is to find a
corresponding analytic object, an automorphic form, whose L-function and
espilon-factors match the arithemtic ones. This is the content of
the local Langlands correspondence. This correspondence should be robust
and preserve various parallel operations on the arithmetic and analytic,
such as taking exterior or symmetric square. In collaboration with F.
Shahidi and T-L. Tsai, we have recently shown that indeed the local
epsilon-factor that appear in the functional equation are preserved
under these operations. The proof is an application of local/global
techniques and the stability of these factors under highly ramified
twists. In this talk I will attempt to explain a bit about Artin
L-functions, automorphic forms, the local langlands correspondence, and
the techniques we use in our proof.