|
Abstract: The Riemann zeta function has a meromorphic continuation to the whole complex plane with a simple pole at s=1 and no other poles. In other words, the zeta function has an analytic continuation to C apart from well-understood poles. Similar results are true for the L-functions attached to Dirichlet characters and, more generally, Hecke characters. Emil Artin reformulated these results as saying that the L-functions associated to 1-dimensional complex Galois representations had analytic continuation apart from well-understood poles and conjectured that the same should be true for n-dimensional complex Galois representations. This conjecture could now be regarded as a precursor to the Langlands programme. I will explain the embarassingly small number of positive results we know about Artin's conjecture. This talk will be for non-experts; I will spend about half the talk defining complex Galois representations and their L-functions, and the other half giving statements of results and sometimes indications of proofs.
|
