Artin conductor

Suppose that L is a finite Galois extension of the local field K, with Galois group G. If ${\displaystyle \chi }$ is a character of G, then the Artin conductor of ${\displaystyle \chi }$ is the number

${\displaystyle f(\chi )=\sum _{i\geq 0}{\frac {g_{i}}{g_{0}}}(\chi (1)-\chi (G_{i}))}$

where G_i is the i-th ramification group (in lower numbering), of order g_i, and χ(G_i) is the average value of ${\displaystyle \chi }$ on G_i.[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

${\displaystyle f(\chi )-(\chi (1)-\chi (G_{0})),}$

in other words, the sum of the higher order terms with i > 0.

The global Artin conductor of a representation ${\displaystyle \chi }$ of the Galois group G of a finite extension L/K of global fields is an ideal of K, defined to be

${\displaystyle {\mathfrak {f}}(\chi )=\prod _{p}p^{f(\chi ,p)}}$

where the product is over the primes p of K, and f(χ,p) is the local Artin conductor of the restriction of ${\displaystyle \chi }$ 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 character a_G of G is the character

${\displaystyle a_{G}=\sum _{\chi }f(\chi )\chi }$

and the Artin representation A_G 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 Q_l, 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 Q_p, suggesting that there is no easy way to construct the Artin representation explicitly.[5]

The Swan character sw_G is given by

${\displaystyle sw_{G}=a_{G}-r_{G}+1}$

where r_g 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.

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.

• Artin, Emil (1930), "Zur Theorie der L-Reihen mit allgemeinen Gruppencharakteren.", Abhandlungen 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, 25, Paris: Société Mathématique de France, pp. 71–81, MR 0374106
• 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, JSTOR 1970142, MR 0171775
• 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, 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, André (1946), "L'avenir des mathématiques", Bol. Soc. Mat. São Paulo, 1: 55–68, MR 0020961