In mathematics, an Artin L-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group G. These functions were introduced in 1923 by Emil Artin, in connection with his research into class field theory.
Their fundamental properties, in particular the Artin conjecture described below, have turned out to be resistant to easy proof.
One of the aims of proposed non-abelian class field theory is to incorporate the complex-analytic nature of Artin L-functions into a larger framework, such as is provided by automorphic forms and the Langlands program.
So far, only a small part of such a theory has been put on a firm basis.
's ring of integers, there is an Euler factor, which is easiest to define in the case where
is a slight modification of the characteristic polynomial, equally well-defined, as rational function in t, evaluated at
a complex variable in the usual Riemann zeta function notation.
is ramified, and I is the inertia group which is a subgroup of G, a similar construction is applied, but to the subspace of V fixed (pointwise) by I.
One application is to give factorisations of Dedekind zeta-functions, for example in the case of a number field that is Galois over the rational numbers.
In accordance with the decomposition of the regular representation into irreducible representations, such a zeta-function splits into a product of Artin L-functions, for each irreducible representation of G. For example, the simplest case is when G is the symmetric group on three letters.
Since characters are an orthonormal basis of the class functions, after showing some analytic properties of the
Artin L-functions satisfy a functional equation.
, which is L multiplied by certain gamma factors, and then there is an equation of meromorphic functions with a certain complex number W(ρ) of absolute value 1.
It has been studied deeply with respect to two types of properties.
Firstly Robert Langlands and Pierre Deligne established a factorisation into Langlands–Deligne local constants; this is significant in relation to conjectural relationships to automorphic representations.
The question of which sign occurs is linked to Galois module theory.
of a non-trivial irreducible representation ρ is analytic in the whole complex plane.
If the Galois group is supersolvable or more generally monomial, 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 Erich Hecke's work.
Langlands used the base change lifting to prove the tetrahedral case, and Jerrold Tunnell extended his work to cover the octahedral case;[3] Andrew Wiles used these cases in his proof of the Modularity conjecture.
Richard Taylor and others have made some progress on the (non-solvable) icosahedral case; this is an active area of research.
The Artin conjecture for odd, irreducible, two-dimensional representations follows from the proof of Serre's modularity conjecture, regardless of projective image subgroup.
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
The Artin conjecture then follows immediately from the known fact that the L-functions of cuspidal automorphic representations are holomorphic.
The Aramata-Brauer theorem states that the conjecture holds if M/K is Galois.
is equal to the Artin L-functions associated to the natural representation associated to the action of G on the K-invariants complex embedding of M. Thus the Artin conjecture implies the Dedekind conjecture.
The conjecture was proven when G is a solvable group, independently by Koji Uchida and R. W. van der Waall in 1975.