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
:
:
:
:
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 al ...
.
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 mea ...
. 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 ...
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 constants; 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 ...
s. Also the case of ρ and ρ* being
equivalent 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 essen ...
s 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
In the mathematical field of representation theory a real representation is usually a representation on a real vector space ''U'', but it can also mean a representation on a complex vector space ''V'' with an invariant real structure, i.e., an a ...
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
...
. 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 ...
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 ...
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. ...
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 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 solvability.
Definition
Let ''G'' be a group. ''G'' is ...
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 expon ...
, 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. ...
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 o ...
's work. Langlands used the
base change lifting 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, 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.
Backgro ...
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 pole (complex analysis), pole ...
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 ...
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'' ...
GL
n(''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 co ...
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 ...
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 ...
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 (mathematics), group that can be constructed from abelian groups using Group extension, extensions. Equivalently, a solvable group is a ...
, 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 Approachby N. Snyder.
*
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{{DEFAULTSORT:Artin L-Function
Zeta and L-functions
Class field theory