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| In [[mathematics]], 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 the 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]] is to incorporate the complex-analytic nature of Artin ''L''-functions into a larger framework, such as is provided by [[automorphic form]]s and [[Langlands' philosophy]]. So far, only a small part of such a theory has been put on a firm basis.
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| ==Definition==
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| Given <math> \rho </math>, a representation of <math>G</math> on a finite-dimensional complex vector space <math>V</math>, where <math>G</math> is the Galois group of the [[finite extension]] <math>L/K</math> of number fields, the Artin <math>L</math>-function: <math>L(\rho,s) </math> is defined by an [[Euler product]]. For each [[prime ideal]] <math> \mathfrak P </math> in <math>L</math>'s [[ring of integers]], there is an Euler factor, which is easiest to define in the case where <math> \mathfrak P </math> is [[unramified]] in <math> L </math> (true for [[almost all]] <math> \mathfrak P </math>). In that case, the [[Frobenius element]] <math> \mathbf{Frob} (\mathfrak P) </math> is defined as a [[conjugacy class]] in <math>G</math>. Therefore the [[characteristic polynomial]] of <math> \rho( \mathbf{Frob} (\mathfrak{P})) </math> is well-defined. The Euler factor for <math> \mathfrak{P} </math> is a slight modification of the characteristic polynomial, equally well-defined,
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| :<math> \operatorname{charpoly}(\rho(\mathbf{Frob}(\mathfrak{P})))^{-1}
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| = \operatorname{det} \left [ I - t \rho( \mathbf{Frob}( \mathfrak{P})) \right ]^{-1}, </math>
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| as [[rational function]] in ''t'', evaluated at <math> t = N (\mathfrak{P})^{-s} </math>, with <math>s</math> a complex variable in the usual [[Riemann zeta function]] notation. (Here ''N'' is the [[field norm]] of an ideal.)
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| When <math> \mathfrak{P} </math> 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''.<ref group="note">It is arguable more correct to think instead about the [[coinvariant]]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.</ref>
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| The Artin L-function <math>L(\rho,s) </math> is then the infinite product over all prime ideals <math> \mathfrak{P} </math> of these factors. As [[Artin reciprocity]] shows, when ''G'' is an [[abelian group]] these ''L''-functions have a second description (as [[Dirichlet L-function|Dirichlet ''L''-function]]s when ''K'' is the [[rational number]] field, and as [[Hecke character|Hecke ''L''-function]]s in general). Novelty comes in with [[abelian group|non-abelian]] ''G'' and their representations.
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| One application is to give factorisations of [[Dedekind zeta function|Dedekind zeta-function]]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]] into [[irreducible representation]]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]] 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.
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| ==Functional equation==
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| Artin L-functions satisfy a [[functional equation (L-function)|functional equation]]. The function ''L''(ρ,''s'') is related in its values to ''L''(ρ*, 1 − ''s''), where ρ* denotes the [[complex conjugate representation]]. More precisely ''L'' is replaced by Λ(ρ, ''s''), which is ''L'' multiplied by certain [[gamma factor]]s, and then there is an equation of meromorphic functions
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| :Λ(ρ, ''s'') = ''W''(ρ)Λ(ρ*, 1 − ''s'')
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| 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 Langlands and Deligne established a factorisation into [[Langlands–Deligne local constant]]s; this is significant in relation to conjectural relationships to [[automorphic representation]]s. Also the case of ρ and ρ* being [[equivalent representation]]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]] 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 {{harv|Perlis|2001}}.
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| ==The Artin conjecture==
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| The '''Artin conjecture''' on Artin L-functions states that the Artin L-function L(ρ,''s'') of a non-trivial irreducible representation ρ is analytic in the whole complex plane.<ref name=Mar18>Martinet (1977) p.18</ref>
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| This is known for one-dimensional representations, the L-functions being then associated to [[Hecke character]]s — and in particular for [[Dirichlet L-function]]s.<ref name=Mar18/> More generally Artin showed that the Artin conjecture is true for all representations induced from 1-dimensional representations. If the Galois group is [[supersolvable]] then all representations are of this form so the Artin conjecture holds.
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| [[André Weil]] proved the Artin conjecture in the case of function fields.
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| 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 [[Hecke]]'s work. Langlands used the [[base change lifting]] to prove the tetrahedral case, and Tunnell extended his work to cover the octahedral case; Wiles used these cases in his proof of the [[Taniyama–Shimura conjecture]]. [[Richard Taylor (mathematician)|Richard Taylor]] and others have made some progress on the (non-solvable) icosahedral case; this is an active area of research.
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| [[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.
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| {{harvtxt|Langlands|1970}} pointed out that the Artin conjecture follows from strong enough results from the [[Langlands philosophy]], relating to the L-functions associated to [[automorphic representation]]s for [[GL(n)]] for all <math> n \geq 1 </math>. More precisely, the Langlands conjectures associate an automorphic representation of the [[adelic group]] GL<sub>n</sub>(''A''<sub>'''Q'''</sub>) 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 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.
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| ==See also==
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| *[[Equivariant L-function]]
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| ==Notes==
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| {{Reflist|group=note}}
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| ==References==
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| {{reflist}}
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| *{{Cite journal |first=E. |last=Artin |title={{lang|de|Über eine neue Art von L Reihen}} |journal=Hamb. Math. Abh. |volume=3 |year=1923 }} Reprinted in his collected works, ISBN 0-387-90686-X. English translation in [http://www.math.columbia.edu/~nsnyder Artin L-Functions: A Historical Approach] by N. Snyder.
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| *{{Citation | last1=Artin | first1=Emil | author1-link=Emil Artin | title=Zur Theorie der L-Reihen mit allgemeinen Gruppencharakteren. | language=German | doi=10.1007/BF02941010 | id={{JFM|56.0173.02}} | year=1930 | journal=Abhandlungen Hamburg | volume=8 | pages=292–306}}
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| *{{Cite journal |doi=10.1090/S0273-0979-1981-14936-3 |last=Tunnell |first=Jerrold |title=Artin's conjecture for representations of octahedral type |journal=Bull. Amer. Math. Soc. |series=N. S. |volume=5 |year=1981 |issue=2 |pages=173–175 }}
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| *{{Cite book |last=Gelbart |first=Stephen |chapter=Automorphic forms and Artin's conjecture |title=Modular functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn., Bonn, 1976) |pages=241–276 |series=Lecture Notes in Math. |volume=627 |publisher=Springer |location=Berlin |year=1977 }}
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| *{{citation|last=Langlands|first=Robert|title=Letter to Prof. Weil|year=1967|url=http://publications.ias.edu/rpl/section/21}}
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| *{{Citation | last1=Langlands | first1=R. P. | title=Lectures in modern analysis and applications, III | url=http://publications.ias.edu/rpl/section/21 | publisher=[[Springer-Verlag]] | location=Berlin, New York | series= Lecture Notes in Math | isbn=978-3-540-05284-5 | doi=10.1007/BFb0079065 | mr=0302614 | year=1970 | volume=170 | chapter=Problems in the theory of automorphic forms | pages=18–61}}
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| *{{citation | last=Martinet | first=J. | chapter=Character theory and Artin L-functions | pages=1-87 | title=Algebraic Number Fields, Proc. Symp. London Math. Soc., Univ. Durham 1975 | editor1-last=Fröhlich | editor1-first=A. | editor1-link=Albrecht Fröhlich | publisher=Academic Press | year=1977 | isbn=0-12-268960-7 | zbl=0359.12015 }}
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| ==External links==
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| *{{Springer|first=R.|last= Perlis|id=a/a120270|title=Artin root numbers}}
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| {{L-functions-footer}}
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| {{DEFAULTSORT:Artin L-Function}}
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| [[Category:Zeta and L-functions]]
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| [[Category:Class field theory]]
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She is known by the name of Myrtle Shryock. To collect cash is one of the issues I love most. Minnesota is where he's been living for many years. Bookkeeping is what I do.
Also visit my blog post; at home std testing