In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 In mathematics, non-abelian class field theory is a catchphrase, meaning the extension of the results of class field theory, the relatively complete and classical set of results on abelian extensions of any number field ''K'', to the general Galois ...
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
In mathematics, the Langlands program is a set of conjectures about connections between number theory, the theory of automorphic forms, and geometry. It was proposed by . It seeks to relate the structure of Galois groups in algebraic number t ...
. So far, only a small part of such a theory has been put on a firm basis.
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
's
ring of integers, there is an Euler factor, which is easiest to define in the case where
is
unramified in
(true for
almost all
In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if X is a set (mathematics), set, "almost all elements of X" means "all elements of X but those in a negligible set, negligible subset of X". The meaning o ...
). 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''.
[It is arguably more correct to think instead about the coinvariants, the largest quotient space fixed by ''I'', rather than the invariants, but the result here will be the same. Cf. Hasse–Weil L-function for a similar situation.]
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
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example,
The set of all ...
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
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
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.
More precisely for
a
Galois extension of degree ''n'', the factorization
:
follows from
:
:
:
: