|Speaker :||Peng-Jie Wong (PIMS, University of Lethbridge)|
|Title :||Group theory and the Artin holomorphy conjecture|
|Time :||2017-12-06 (Wed) 11:00 - 12:30|
|Place :||Seminar Room 638, Institute of Mathematics (NTU Campus)|
Let $K/k$ be a Galois extension of number fields with group $G$, and let $\rho$ be a non-trivial irreducible representation of $G$ of dimension $n$. Nearly a century ago, Artin conjectured 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 (=the class field theory). Also, by the works of Langlands and many others, several significant progress has been made for $n=2$. However, in general, the Artin holomorphy conjecture is still open.
In a slightly different vein, via his induction on characters, Brauer obtained the meromorphy for all Artin $L$-functions. As Brauer's proof is pure group-theoretic, it is expected that group theory should play a role in the game.
In this talk, we shall emphasise how to use elementary group theory to study the Artin holomorphy conjecture. In particular, we shall introduce the notion of ``nearly supersolvable group'', which can be seen as a generalisation of supersolvable groups (and hence abelian groups). If time permits, we will explain how such groups lead to a proof of the Artin holomorphy conjecture for Galois extensions of degree less than 60.
(For the most of this talk, only the knowledge of undergraduate level algebra will be assumed.)