Artin L-function - The Artin Conjecture

The Artin Conjecture

The Artin conjecture on Artin L-functions states that the Artin L-function L(ρ,s) of a non-trivial irreducible representation ρ is analytic in the whole complex plane.

This is known for one-dimensional representations — the L-functions being then associated to Hecke characters — and in particular for Dirichlet L-functions. More generally Artin showed that the Artin conjecture is true for all representations induced from 1-dimensional representations. If the Galois group is supersolvable then all representations are of this form so the Artin conjecture holds.

André Weil proved the Artin conjecture in the case of function fields.

Two dimensional representations are classified by the nature of the image subgroup: it may be cyclic, dihedral, tetrahedral, octahedral, or icosahedral. The Artin conjecture for the cyclic or dihedral case follows easily from Hecke's work. Langlands used the base change lifting to prove the tetrahedral case, and Tunnell extended his work to cover the octahedral case; Wiles used these cases in his proof of the Taniyama–Shimura conjecture. Richard Taylor and others have made some progress on the (non-solvable) icosahedral case; this is an active area of research.

Brauer's theorem on induced characters implies that all Artin L-functions are products of positive and negative integral powers of Hecke L-functions, and are therefore meromorphic in the whole complex plane.

Langlands (1970) pointed out that the Artin conjecture follows from strong enough results from the Langlands philosophy, relating to the L-functions associated to automorphic representations for GL(n) for all . More precisely, the Langlands conjectures associate an automorphic representation of the adelic group GL_n(A_Q) to every n-dimensional irreducible representation of the Galois group, which is a cuspidal representation if the Galois representation is irreducible, such that the Artin L-function of the Galois representation is the same as the automorphic L-function of the automorphic representation. The Artin conjecture then follows immediately from the known fact that the L-functions of cuspidal automorphic representations are holomorphic. This was one of the major motivations for Langlands' work.