In
mathematics, an Artin ''L''-function is a type of
Dirichlet series
In mathematics, a Dirichlet series is any series of the form
\sum_^\infty \frac,
where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series.
Dirichlet series play a variety of important roles in analyti ...
associated to a
linear representation
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essenc ...
ρ of a
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
''G''. These functions were introduced in 1923 by
Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent.
Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing ...
, 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 Galoi ...
is to incorporate the complex-analytic nature of Artin ''L''-functions into a larger framework, such as is provided by
automorphic form
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...
s and the
Langlands program
In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic n ...
. 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
In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory — ...
of number fields, the Artin
-function:
is defined by an
Euler product In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhar ...
. For each
prime ideal in
's
ring of integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often d ...
, there is an Euler factor, which is easiest to define in the case where
is
unramified
In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) ...
in
(true for
almost all
In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the mathem ...
). In that case, the
Frobenius element
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphis ...
is defined as a
conjugacy class
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other w ...
in
. Therefore, the
characteristic polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ...
of
is well-defined. The Euler factor for
is a slight modification of the characteristic polynomial, equally well-defined,
:
as
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
in ''t'', evaluated at
, with
a complex variable in the usual
Riemann zeta function notation. (Here ''N'' is the
field norm In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield.
Formal definition
Let ''K'' be a field and ''L'' a finite extension (and hence an algebraic extension) of ' ...
of an ideal.)
When
is ramified, and ''I'' is the
inertia group
In number theory, more specifically in local class field theory, the ramification groups are a filtration of the Galois group of a local field extension, which gives detailed information on the ramification phenomena of the extension.
Ramificat ...
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 ]coinvariant
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphi ...
s, the largest quotient space
Quotient space may refer to a quotient set when the sets under consideration are considered as spaces. In particular:
*Quotient space (topology), in case of topological spaces
* Quotient space (linear algebra), in case of vector spaces
*Quotient ...
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
Artin may refer to:
* Artin (name), a surname and given name, including a list of people with the name
** Artin, a variant of Harutyun
Harutyun ( hy, Հարություն and in Western Armenian Յարութիւն) also spelled Haroutioun, Harut ...
shows, when ''G'' is an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...
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 (e.g. ). The set of all ra ...
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-function
In mathematics, the Dedekind zeta function of an algebraic number field ''K'', generally denoted ζ''K''(''s''), is a generalization of the Riemann zeta function (which is obtained in the case where ''K'' is the field of rational numbers Q). It ca ...
s, 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
In mathematics, and in particular the theory of group representations, the regular representation of a group ''G'' is the linear representation afforded by the group action of ''G'' on itself by translation.
One distinguishes the left regular r ...
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 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 In the mathematical field of representation theory, a trivial representation is a representation of a group ''G'' on which all elements of ''G'' act as the identity mapping of ''V''. A trivial representation of an associative or Lie algebra is a ...
) 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
:
:
:
: