University of Georgia
Mathematics Department Colloquium
Professor Mark Dickinson
University of Michigan
January 30, 2003
Modularity of icosahedral Artin representations
Abstract: In the 1920s, Emil Artin showed how to
attach an L-function to a finite-dimensional complex representation of
the absolute Galois group of a number field. This L-function is a
meromorphic function on the complex plane, which naturally encodes
arithmetic properties of the Galois representation. Artin further
conjectured that these L-functions should be holomorphic away from 1.
One important special case of Artin's conjecture can be reformulated
as describing a correspondence between weight one cuspidal modular
forms which are eigenforms for the Hecke operators, and
two-dimensional odd irreducible complex representations of the
absolute Galois group of the rationals. In this form the conjecture,
often known as the `Strong Artin conjecture', stands as one of two
notable classical cases of the Langlands conjectures for number
fields; the modularity of elliptic curves over the rationals is the
other.
I will describe recent progress on the Strong Artin conjecture, and
indicate some possible future directions towards a complete proof of
the conjecture.