Artin L-function - Definition

Definition

Given, a representation of on a finite-dimensional complex vector space, where is the Galois group of the finite extension of number fields, the Artin -function: is defined by an Euler product. For each prime ideal in, there is an Euler factor, which is easiest to define in the case where is unramified in (true for almost all ). In that case, the Frobenius element is defined as a conjugacy class in . Therefore the characteristic polynomial of is well-defined. The Euler factor for is a slight modification of the characteristic polynomial, equally well-defined,

as rational function in t, evaluated at, with a complex variable in the usual Riemann zeta function notation. (Here N is the field norm of an ideal.)

When 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.

The Artin L-function is then the infinite product over all prime ideals of these factors. As Artin reciprocity shows, when G is an abelian group these L-functions have a second description (as Dirichlet L-functions when K is the rational number field, and as Hecke L-functions in general). Novelty comes in with non-abelian G and their representations.

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 G has an irreducible representation of degree 2, an Artin L-function for such a representation occurs, squared, in the factorisation of the Dedekind zeta-function for such a number field, in a product with the Riemann zeta-function (for the trivial representation) and an L-function of Dirichlet's type for the signature representation.