Weak almost monomial groups and Artin's conjecture

TL;DR

This paper introduces weak almost monomial groups, a new class of finite groups, and explores their role in Artin's conjecture, establishing conditions under which the conjecture holds at specific points.

Contribution

It defines weak almost monomial groups, proves their closure properties, and links these groups to criteria for the validity of Artin's conjecture at certain points.

Findings

Artin conjecture holds at s_0 if the monoid of Artin L-functions is factorial.

If s_0 is a simple zero for one Artin L-function and not for others, the conjecture is true at s_0.

Weak almost monomial groups generalize previous notions and are key to understanding Artin's conjecture.

Abstract

We introduce a new class of finite groups, called weak almost monomial, which generalize two different notions of "almost monomial" groups, and we prove it is closed under taking factor groups and direct products. Let $K/\mathbb Q$ be a finite Galois extension with a weak almost monomial Galois group $G$ and $s_0\in \mathbb C\setminus \{1\}$. We prove that Artin conjecture's is true at $s_0$ if and only if the monoid of holomorphic Artin $L$-functions at $s_0$ is factorial. Also, we show that if $s_0$ is a simple zero for some Artin $L$-function associated to an irreducible character of $G$ and it is not a zero for any other $L$-function associated to an irreducible character, then Artin conjecture's is true at $s_0$.