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
:
:
:
:
Functional equation
Artin L-functions satisfy a
functional equation. The function
L(\rho,s) is related in its values to
L(\rho^*, 1 - s), where
\rho^* denotes the
complex conjugate representation. More precisely ''L'' is replaced by
\Lambda(\rho, s), which is ''L'' multiplied by certain
gamma factors, 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 and
Pierre Deligne 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 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. The Artin root number is, then, either +1 or −1. The question of which sign occurs is linked to
Galois module theory.
The Artin conjecture
The Artin conjecture on Artin L-functions (also known as Artin's holomorphy conjecture) 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 characters — and in particular for
Dirichlet L-functions. More generally Artin showed that the Artin conjecture is true for all representations induced from 1-dimensional representations. If the Galois group is
supersolvable or more generally
monomial, then all representations are of this form so the Artin conjecture holds.
André Weil 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'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 used these cases in his proof of the
Modularity 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 implies that all Artin L-functions are products of positive and negative integral powers of Hecke L-functions, and are therefore
meromorphic 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) for all
n \geq 1 . More precisely, the Langlands conjectures associate an automorphic representation of the
adelic group GL
n(''A''
Q) to every ''n''-dimensional irreducible representation of the Galois group, which is a
cuspidal representation if the Galois representation is irreducible, such that the Artin L-function of the Galois representation is the same as the
automorphic L-function
In mathematics, an automorphic ''L''-function is a function ''L''(''s'',π,''r'') of a complex variable ''s'', associated to an automorphic representation π of a reductive group ''G'' over a global field and a finite-dimensional complex represent ...
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 fields, then the quotient
s\mapsto\zeta_M(s)/\zeta_K(s) of their
Dedekind zeta functions is entire.
The Aramata-Brauer theorem states that the conjecture holds if ''M''/''K'' is Galois.
More generally, let ''N'' be 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, independently by Koji Uchida and R. W. van der Waall in 1975.
See also
*
Equivariant L-function
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