Artin conjecture

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Template:Major 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

Statement

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

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.

External links