Kevin Buzzard, University of Cambridge
Artin's conjecture on representations of Galois groups
As part of his study of finite-dimensional complex
representations of certain finite Galois groups, Emil Artin
in 1923 showed how to associate to each such representation
a differentiable complex function, defined on the half-plane
Re(z)>1. He conjectured that in many cases these
functions should have an analytic continuation to the whole
complex plane. Many mathematicians have worked on this
conjecture but its resolution still seems very far off.
In my talk I shall explain the conjecture and how little we
know about it. No specialist knowledge of number theory or
complex analysis will be required.