Lethbridge Number Theory and Combinatorics Seminar

Let K/k be a Galois extension of number fields with Galois group G, and let rho be a non-trivial irreducible representation of G of dimension n. The Artin holomorphy conjecture asserts that the Artin L-function attached to rho extends to an entire function.

It is well-known that when n=1, this conjecture follows from Artin reciprocity. Also, by the works of Langlands and many others, we know that this conjecture is valid for n=2 under certain conditions. However, in general, the Artin holomorphy conjecture is wildly open.

In this talk, we will discuss how elementary group theory plays a role in studying the Artin holomorphy conjecture and introduce the notion of "nearly supersolvable groups". If time allows, we will explain how such groups lead to a proof of the Artin holomorphy conjecture for Galois extensions of degree less than 60.