# Artin conjecture

*This article or section of article is sourced from*:Wikipedia

*There are other Artin conjectures, but they are not within the scope of this wiki*

## Contents

## Statement

The Artin conjecture on Artin L-functions states that the Artin L-function is analytic (viz, holomorphic) in the whole complex plane, for any nontrivial irreducible Galois representation .

## Facts

### True for one-dimensional representations

The Artin conjecture is true for one-dimensional representations of the Galois group. In this case, the Artin L-function reduces to the Hecke L-function.

### True for two-dimensional representations

The Artin conjecture has been settled for all two-dimensional representations:

- The cyclic and dihedral case follow from Hecke's work
- Langlands did the tetrahedral case
- Tunnell did the octahedral case, extending work of Langlands

### Meromorphicity

The Brauer induction theorem states that every character of a finite group occurs as a rational linear combination of characters induced from cyclic groups. This along with Galois theory, shows that all the Artin L-functions are meromorphic.

### Proof in the function fields case

The Artin conjecture has been settled in the case of function fields.