Artin conductor
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.
Local Artin conductors
Suppose that L is a finite Galois extension of the local field K, with Galois group G. If [math]\displaystyle{ \chi }[/math] is a character of G, then the Artin conductor of [math]\displaystyle{ \chi }[/math] is the number
- [math]\displaystyle{ f(\chi)=\sum_{i\ge 0}\frac{g_i}{g_0}(\chi(1)-\chi(G_i)) }[/math]
where Gi is the i-th ramification group (in lower numbering), of order gi, and χ(Gi) is the average value of [math]\displaystyle{ \chi }[/math] 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
- [math]\displaystyle{ f(\chi) - (\chi(1)-\chi(G_0)), }[/math]
in other words, the sum of the higher order terms with i > 0.
Global Artin conductors
The global Artin conductor of a representation [math]\displaystyle{ \chi }[/math] of the Galois group G of a finite extension L/K of global fields is an ideal of K, defined to be
- [math]\displaystyle{ \mathfrak{f}(\chi) = \prod_p p^{f(\chi,p)} }[/math]
where the product is over the primes p of K, and f(χ,p) is the local Artin conductor of the restriction of [math]\displaystyle{ \chi }[/math] 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
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
- [math]\displaystyle{ a_G=\sum_\chi f(\chi)\chi }[/math]
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
The Swan character swG is given by
- [math]\displaystyle{ sw_G= a_G -r_G+1 }[/math]
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
The 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
- ↑ Serre (1967) p.158
- ↑ 2.0 2.1 Serre (1967) p.159
- ↑ 3.0 3.1 Manin, Yu. I.; Panchishkin, A. A. (2007). Introduction to Modern Number Theory. Encyclopaedia of Mathematical Sciences. 49 (Second ed.). p. 329. ISBN 978-3-540-20364-3.
- ↑ Snaith (1994) p.249
- ↑ 5.0 5.1 Serre (1967) p.160
- ↑ Snaith (1994) p.248
References
- Artin, Emil (1930), "Zur Theorie der L-Reihen mit allgemeinen Gruppencharakteren." (in de), Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 8: 292–306, doi:10.1007/BF02941010
- Artin, Emil (1931), "Die gruppentheoretische Struktur der Diskriminanten algebraischer Zahlkörper." (in de), Journal für die Reine und Angewandte Mathematik 1931 (164): 1–11, doi:10.1515/crll.1931.164.1, ISSN 0075-4102, http://resolver.sub.uni-goettingen.de/purl?GDZPPN002171422
- 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, 25, Paris: Société Mathématique de France, pp. 71–81, http://www.numdam.org/item?id=MSMF_1971__25__71_0
- Serre, Jean-Pierre (1960), "Sur la rationalité des représentations d'Artin", Annals of Mathematics, Second Series 72 (2): 405–420, doi:10.2307/1970142, ISSN 0003-486X
- Serre, Jean-Pierre (1967), "VI. Local class field theory", in Cassels, J.W.S.; Fröhlich, A., 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
- Snaith, V. P. (1994), Explicit Brauer Induction: With Applications to Algebra and Number Theory, Cambridge Studies in Advanced Mathematics, 40, Cambridge University Press, ISBN 0-521-46015-8, https://archive.org/details/explicitbrauerin0000snai
- 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
- Weil, André (1946), "L'avenir des mathématiques", Bol. Soc. Mat. São Paulo 1: 55–68
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