# 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 χ is a character of *G*, then the Artin conductor of χ is the number

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 χ on *G*_{i}.^{[1]} By a result of Artin, the local conductor is an integer.^{[2]}^{[3]} If χ is unramified, then its Artin conductor is zero. 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 conductors

The **global Artin conductor** of a representation χ 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 χ 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** *a*_{G} of *G* is the character

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]}

## Swan representation

The Swan character *sw*_{G} is given by

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.

## 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

## References

- 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 - Artin, Emil (1931), "Die gruppentheoretische Struktur der Diskriminanten algebraischer Zahlkörper.",
*Journal für Reine und Angewandte Mathematik*(in German),**164**: 1–11, doi:10.1515/crll.1931.164.1, ISSN 0075-4102, 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",
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*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",
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