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.
Read more about this topic: Artin L-function
Famous quotes containing the word definition:
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)