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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 pol ...
''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 lar ...
, in connection with his research into
class field theory In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credit ...
. 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 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 num ...
. So far, only a small part of such a theory has been put on a firm basis.


Definition

Given \rho , a representation of G on a finite-dimensional complex vector space V, where G 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 &mdash ...
L/K of number fields, the Artin L-function: L(\rho,s) 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 Leonhard Eul ...
. For each
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
\mathfrak p in K'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 deno ...
, there is an Euler factor, which is easiest to define in the case where \mathfrak p 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) as ...
in L (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 math ...
\mathfrak p ). 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 endomorphism m ...
\mathbf (\mathfrak p) 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 wor ...
in G. 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 chara ...
of \rho( \mathbf (\mathfrak)) is well-defined. The Euler factor for \mathfrak is a slight modification of the characteristic polynomial, equally well-defined, :\operatorname(\rho(\mathbf(\mathfrak)))^= \operatorname \left I - t \rho( \mathbf( \mathfrak)) \right , 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 rat ...
in ''t'', evaluated at t = N (\mathfrak)^ , with s a complex variable in the usual
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
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 ''K ...
of an ideal.) When \mathfrak 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 automorphism ...
s, 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 L(\rho,s) is then the infinite product over all prime ideals \mathfrak 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 commut ...
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 ration ...
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 rep ...
into
irreducible representation In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W,W ...
s, 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 \m ...
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 L/K a Galois extension of degree ''n'', the factorization :\zeta_L(s) =L(s,\rho_)= \prod_ L(\rho,s)^ follows from :L(\rho,s) = \prod_ \frac :-\log \det \left -N (\mathfrak)^ \rho \left ( \mathbf_\mathfrak \right ) \right = \sum_^\infty \frac N(\mathfrak)^ :\sum_\deg(\rho) \text(\rho(\sigma)) = \begin n & \sigma = 1 \\ 0 & \sigma \neq 1 \end :-\sum_\deg(\rho) \log \det \left -N \left (\mathfrak^ \right ) \rho \left ( \mathbf_\mathfrak \right ) \right = n \sum_^\infty \frac = - \log \left \left (1-N(\mathfrak)^ \right )^ \right /math> where \deg(\rho) is the multiplicity of the irreducible representation in the regular representation, ''f'' is the order of \mathbf_\mathfrak and ''n'' is replaced by ''n/e'' at the ramified primes. Since characters are an orthonormal basis of the class functions, after showing some analytic properties of the L(\rho,s) we obtain the
Chebotarev density theorem Chebotarev's density theorem in algebraic number theory describes statistically the splitting of primes in a given Galois extension ''K'' of the field \mathbb of rational numbers. Generally speaking, a prime integer will factor into several ideal ...
as a generalization of
Dirichlet's theorem on arithmetic progressions In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers ''a'' and ''d'', there are infinitely many primes of the form ''a'' + ''nd'', where ''n'' is als ...
.


Functional equation

Artin L-functions satisfy a
functional equation In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
. The function L(\rho,s) is related in its values to L(\rho^*, 1 - s), where \rho^* denotes the
complex conjugate representation In mathematics, if is a group and is a representation of it over the complex vector space , then the complex conjugate representation is defined over the complex conjugate vector space as follows: : is the conjugate of for all in . is ...
. More precisely ''L'' is replaced by \Lambda(\rho, s), which is ''L'' multiplied by certain
gamma factor The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativit ...
s, and then there is an equation of meromorphic functions :\Lambda(\rho,s)= W(\rho)\Lambda(\rho^*, 1 - s), with a certain complex number ''W''(ρ) of absolute value 1. It is the Artin root number. It has been studied deeply with respect to two types of properties. Firstly
Robert Langlands Robert Phelan Langlands, (; born October 6, 1936) is a Canadian mathematician. He is best known as the founder of the Langlands program, a vast web of conjectures and results connecting representation theory and automorphic forms to the study o ...
and
Pierre Deligne Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Pr ...
established a factorisation into
Langlands–Deligne local constant In mathematics, the Langlands–Deligne local constant, also known as the local epsilon factor or local Artin root number (up to an elementary real function of ''s''), is an elementary function associated with a representation of the Weil group of ...
s; this is significant in relation to conjectural relationships to
automorphic representation 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. Also the case of ρ and ρ* being equivalent representations is exactly the one in which the functional equation has the same L-function on each side. It is, algebraically speaking, the case when ρ is a real representation or
quaternionic representation In mathematical field of representation theory, a quaternionic representation is a representation on a complex vector space ''V'' with an invariant quaternionic structure, i.e., an antilinear equivariant map :j\colon V\to V which satisfies :j^ ...
. The Artin root number is, then, either +1 or −1. The question of which sign occurs is linked to
Galois module In mathematics, a Galois module is a ''G''-module, with ''G'' being the Galois group of some extension of fields. The term Galois representation is frequently used when the ''G''-module is a vector space over a field or a free module over a ring ...
theory.


The Artin conjecture

The Artin conjecture on Artin L-functions states that the Artin L-function L(\rho,s) of a non-trivial irreducible representation ρ is analytic in the whole complex plane. This is known for one-dimensional representations, the L-functions being then associated to
Hecke character In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of ''L''-functions larger than Dirichlet ''L''-functions, and a natural setting for the Dedekind zeta-functions and ce ...
s — and in particular for
Dirichlet L-function In mathematics, a Dirichlet ''L''-series is a function of the form :L(s,\chi) = \sum_^\infty \frac. where \chi is a Dirichlet character and ''s'' a complex variable with real part greater than 1. It is a special case of a Dirichlet series. By a ...
s. More generally Artin showed that the Artin conjecture is true for all representations induced from 1-dimensional representations. If the Galois group is
supersolvable In mathematics, a group (mathematics), group is supersolvable (or supersoluble) if it has an invariant normal series where all the factors are cyclic groups. Supersolvability is stronger than the notion of solvable group, solvability. Definition ...
or more generally
monomial In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called power product, is a product of powers of variables with nonnegative integer exponent ...
, then all representations are of this form so the Artin conjecture holds.
André Weil André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...
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 Erich Hecke (20 September 1887 – 13 February 1947) was a German mathematician known for his work in number theory and the theory of modular forms. Biography Hecke was born in Buk, Province of Posen, German Empire (now Poznań, Poland). He ...
's work. Langlands used the
base change lifting In mathematics, base change lifting is a method of constructing new automorphic forms from old ones, that corresponds in Langlands philosophy to the operation of restricting a representation of a Galois group to a subgroup. The Doi–Naganuma lift ...
to prove the tetrahedral case, and Jerrold Tunnell extended his work to cover the octahedral case;
Andrew Wiles Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specializing in number theory. He is best known for proving Fermat's Last Theorem, for which he was awar ...
used these cases in his proof of the Taniyama–Shimura 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 In mathematics, Serre's modularity conjecture, introduced by , states that an odd, irreducible, two-dimensional Galois representation over a finite field arises from a modular form. A stronger version of this conjecture specifies the weight and ...
, regardless of projective image subgroup.
Brauer's theorem on induced characters Brauer's theorem on induced characters, often known as Brauer's induction theorem, and named after Richard Brauer, is a basic result in the branch of mathematics known as character theory, within representation theory of a finite group. Backgrou ...
implies that all Artin L-functions are products of positive and negative integral powers of Hecke L-functions, and are therefore
meromorphic In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
in the whole complex plane. pointed out that the Artin conjecture follows from strong enough results from the Langlands philosophy, relating to the L-functions associated to
automorphic representation 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 for
GL(n) In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
for all n \geq 1 . More precisely, the Langlands conjectures associate an automorphic representation of the
adelic group In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group ''G'' over a number field ''K'', and the adele ring ''A'' = ''A''(''K'') of ''K''. It consists of the points of ''G'' having values in ''A''; the ...
GLn(''A''Q) to every ''n''-dimensional irreducible representation of the Galois group, which is a
cuspidal representation In number theory, cuspidal representations are certain representations of algebraic groups that occur discretely in L^2 spaces. The term ''cuspidal'' is derived, at a certain distance, from the cusp forms of classical modular form theory. In the con ...
if the Galois representation is irreducible, such that the Artin L-function of the Galois representation is the same as the automorphic L-function of the automorphic representation. The Artin conjecture then follows immediately from the known fact that the L-functions of cuspidal automorphic representations are holomorphic. This was one of the major motivations for Langlands' work.


The Dedekind conjecture

A weaker conjecture (sometimes known as Dedekind conjecture) states that if ''M''/''K'' is an extension of
number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...
s, then the quotient s\mapsto\zeta_M(s)/\zeta_K(s) of their
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 is entire. The Aramata-Brauer theorem states that the conjecture holds if ''M''/''K'' is Galois. More generally, let ''N'' the Galois closure of ''M'' over ''K'', and ''G'' the Galois group of ''N''/''K''. The quotient s\mapsto\zeta_M(s)/\zeta_K(s) 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 In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates ...
, independently by Koji Uchida and R. W. van der Waall in 1975.


See also

*
Equivariant L-function In algebraic number theory, an equivariant Artin L-function is a function associated to a finite Galois extension of global fields created by packaging together the various Artin L-functions associated with the extension. Each extension has many t ...


Notes


References


Bibliography

* Reprinted in his collected works, . English translation i
Artin L-Functions: A Historical Approach
by N. Snyder. * * * * * * * * {{DEFAULTSORT:Artin L-Function Zeta and L-functions Class field theory