Suppose that L is a finite Galois extension of the local field K, with Galois group G. If is a character of G, then the Artin conductor of is the number
where Gi is the i-th ramification group (in lower numbering), of order gi, and χ(Gi) is the average value of on Gi.[1] By a result of Artin, the local conductor is an integer.[2][3] Heuristically, the Artin conductor measures how far the action of the higher ramification groups is from being trivial. In particular, if χ is unramified, then its Artin conductor is zero. Thus if L is unramified over K, then the Artin conductors of all χ are zero.
The wild invariant[3] or Swan conductor[4] of the character is
in other words, the sum of the higher order terms with i > 0.
Global Artin conductorsedit
The global Artin conductor of a representation of the Galois group G of a finite extension L/K of global fields is an ideal of K, defined to be
where the product is over the primes p of K, and f(χ,p) is the local Artin conductor of the restriction of to the decomposition group of some prime of L lying over p.[2] Since the local Artin conductor is zero at unramified primes, the above product only need be taken over primes that ramify in L/K.
Suppose that L is a finite Galois extension of the local field K, with Galois group G. The Artin characteraG of G is the character
and the Artin representationAG is the complex linear representation of G with this character. Weil (1946) asked for a direct construction of the Artin representation. Serre (1960) showed that the Artin representation can be realized over the local field Ql, for any prime l not equal to the residue characteristic p. Fontaine (1971) showed that it can be realized over the corresponding ring of Witt vectors. It cannot in general be realized over the rationals or over the local field Qp, suggesting that there is no easy way to construct the Artin representation explicitly.[5]
Swan representationedit
The Swan characterswG is given by
where rg is the character of the regular representation and 1 is the character of the trivial representation.[6] The Swan character is the character of a representation of G. Swan (1963) showed that there is a unique projective representation of G over the l-adic integers with character the Swan character.
^ abManin, Yu. I.; Panchishkin, A. A. (2007). Introduction to Modern Number Theory. Encyclopaedia of Mathematical Sciences. Vol. 49 (Second ed.). p. 329. ISBN978-3-540-20364-3. ISSN 0938-0396.
Artin, Emil (1930), "Zur Theorie der L-Reihen mit allgemeinen Gruppencharakteren.", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg (in German), 8: 292–306, doi:10.1007/BF02941010, JFM 56.0173.02, S2CID 120987633
Artin, Emil (1931), "Die gruppentheoretische Struktur der Diskriminanten algebraischer Zahlkörper.", Journal für die Reine und Angewandte Mathematik (in German), 1931 (164): 1–11, doi:10.1515/crll.1931.164.1, ISSN 0075-4102, S2CID 117731518, Zbl 0001.00801
Fontaine, Jean-Marc (1971), "Sur les représentations d'Artin", Colloque de Théorie des Nombres (Univ. Bordeaux, Bordeaux, 1969), Mémoires de la Société Mathématique de France, vol. 25, Paris: Société Mathématique de France, pp. 71–81, MR 0374106
Serre, Jean-Pierre (1967), "VI. Local class field theory", in Cassels, J.W.S.; Fröhlich, A. (eds.), Algebraic number theory. Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the International Mathematical Union, London: Academic Press, pp. 128–161, Zbl 0153.07403
Snaith, V. P. (1994), Explicit Brauer Induction: With Applications to Algebra and Number Theory, Cambridge Studies in Advanced Mathematics, vol. 40, Cambridge University Press, ISBN0-521-46015-8, Zbl 0991.20005
Swan, Richard G. (1963), "The Grothendieck ring of a finite group", Topology, 2 (1–2): 85–110, doi:10.1016/0040-9383(63)90025-9, ISSN 0040-9383, MR 0153722
Weil, André (1946), "L'avenir des mathématiques", Bol. Soc. Mat. São Paulo, 1: 55–68, MR 0020961
April 10, 2024
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In mathematics the Artin conductor is a number or ideal associated to a character of a Galois group of a local or global field introduced by Emil Artin 1930 1931 as an expression appearing in the functional equation of an Artin L function Contents 1 Local Artin conductors 2 Global Artin conductors 3 Artin representation and Artin character 4 Swan representation 5 Applications 6 Notes 7 ReferencesLocal Artin conductors editSuppose that L is a finite Galois extension of the local field K with Galois group G If x displaystyle chi nbsp is a character of G then the Artin conductor of x displaystyle chi nbsp is the number f x i 0gig0 x 1 x Gi displaystyle f chi sum i geq 0 frac g i g 0 chi 1 chi G i nbsp where Gi is the i th ramification group in lower numbering of order gi and x Gi is the average value of x displaystyle chi nbsp on Gi 1 By a result of Artin the local conductor is an integer 2 3 Heuristically the Artin conductor measures how far the action of the higher ramification groups is from being trivial In particular if x is unramified then its Artin conductor is zero Thus if L is unramified over K then the Artin conductors of all x are zero The wild invariant 3 or Swan conductor 4 of the character is f x x 1 x G0 displaystyle f chi chi 1 chi G 0 nbsp in other words the sum of the higher order terms with i gt 0 Global Artin conductors editThe global Artin conductor of a representation x displaystyle chi nbsp of the Galois group G of a finite extension L K of global fields is an ideal of K defined to be f x ppf x p displaystyle mathfrak f chi prod p p f chi p nbsp where the product is over the primes p of K and f x p is the local Artin conductor of the restriction of x displaystyle chi nbsp to the decomposition group of some prime of L lying over p 2 Since the local Artin conductor is zero at unramified primes the above product only need be taken over primes that ramify in L K Artin representation and Artin character editFurther information Galois module Artin representations Suppose that L is a finite Galois extension of the local field K with Galois group G The Artin character aG of G is the character aG xf x x displaystyle a G sum chi f chi chi nbsp and the Artin representation AG is the complex linear representation of G with this character Weil 1946 asked for a direct construction of the Artin representation Serre 1960 showed that the Artin representation can be realized over the local field Ql for any prime l not equal to the residue characteristic p Fontaine 1971 showed that it can be realized over the corresponding ring of Witt vectors It cannot in general be realized over the rationals or over the local field Qp suggesting that there is no easy way to construct the Artin representation explicitly 5 Swan representation editThe Swan character swG is given by swG aG rG 1 displaystyle sw G a G r G 1 nbsp where rg is the character of the regular representation and 1 is the character of the trivial representation 6 The Swan character is the character of a representation of G Swan 1963 showed that there is a unique projective representation of G over the l adic integers with character the Swan character Applications editThe Artin conductor appears in the conductor discriminant formula for the discriminant of a global field 5 The optimal level in the Serre modularity conjecture is expressed in terms of the Artin conductor The Artin conductor appears in the functional equation of the Artin L function The Artin and Swan representations are used to defined the conductor of an elliptic curve or abelian variety Notes edit Serre 1967 p 158 a b Serre 1967 p 159 a b Manin Yu I Panchishkin A A 2007 Introduction to Modern Number Theory Encyclopaedia of Mathematical Sciences Vol 49 Second ed p 329 ISBN 978 3 540 20364 3 ISSN 0938 0396 Snaith 1994 p 249 a b Serre 1967 p 160 Snaith 1994 p 248References editArtin Emil 1930 Zur Theorie der L Reihen mit allgemeinen Gruppencharakteren Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg in German 8 292 306 doi 10 1007 BF02941010 JFM 56 0173 02 S2CID 120987633 Artin Emil 1931 Die gruppentheoretische Struktur der Diskriminanten algebraischer Zahlkorper Journal fur die Reine und Angewandte Mathematik in German 1931 164 1 11 doi 10 1515 crll 1931 164 1 ISSN 0075 4102 S2CID 117731518 Zbl 0001 00801 Fontaine Jean Marc 1971 Sur les representations d Artin Colloque de Theorie des Nombres Univ Bordeaux Bordeaux 1969 Memoires de la Societe Mathematique de France vol 25 Paris Societe Mathematique de France pp 71 81 MR 0374106 Serre Jean Pierre 1960 Sur la rationalite des representations d Artin Annals of Mathematics Second Series 72 2 405 420 doi 10 2307 1970142 ISSN 0003 486X JSTOR 1970142 MR 0171775 Serre Jean Pierre 1967 VI Local class field theory in Cassels J W S Frohlich A eds Algebraic number theory Proceedings of an instructional conference organized by the London Mathematical Society a NATO Advanced Study Institute with the support of the International Mathematical Union London Academic Press pp 128 161 Zbl 0153 07403 Snaith V P 1994 Explicit Brauer Induction With Applications to Algebra and Number Theory Cambridge Studies in Advanced Mathematics vol 40 Cambridge University Press ISBN 0 521 46015 8 Zbl 0991 20005 Swan Richard G 1963 The Grothendieck ring of a finite group Topology 2 1 2 85 110 doi 10 1016 0040 9383 63 90025 9 ISSN 0040 9383 MR 0153722 Weil Andre 1946 L avenir des mathematiques Bol Soc Mat Sao Paulo 1 55 68 MR 0020961 Retrieved from https en wikipedia org w index php title Artin conductor amp oldid 1170996705 Swan representation, wikipedia, wiki, book, books, library,