Artin conductor

In number theory, 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 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 $\chi$ is a character of $G$, then the Artin conductor of $\chi$ is the number

$f(\chi )=\sum _{i\ge 0}\frac{g_i}{g_0}(\chi (1)-\chi (G_i)),$

where $G_i$ is the $i$-th ramification group (in lower numbering), of order $g_i$, and $\chi (G_i)$ is the average value of $\chi$ on $G_i$. The local conductor is an integer. Heuristically, the Artin conductor measures how far the action of the higher ramification groups is from being trivial. In particular, if $\chi$ is unramified, then its Artin conductor is zero. Thus, if $L$ is unramified over $K$, then the Artin conductors of all $\chi$ are zero.

The wild invariant or Swan conductor of the character is

$f(\chi )-(\chi (1)-\chi (G_0)),$

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

Global Artin conductors

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

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

where the product is over the primes $p$ of $K$, and $f(\chi ,p)$ is the local Artin conductor of the restriction of $\chi$ to the decomposition group of some prime of $L$ lying over $p$. 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

$a_G=\sum _{\chi }f(\chi )\chi$

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 ${\mathbb{\mathbb{Q}}}_{\ell }$, for any prime $\ell$ 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 ${\mathbb{\mathbb{Q}}}_p$, suggesting that there is no easy way to construct the Artin representation explicitly.

Swan representation

The Swan character sw_G is given by

$s{w}_G=a_G-r_G+1$

where r_g is the character of the regular representation and 1 is the character of the trivial representation. 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.

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 define the conductor of an elliptic curve or abelian variety.

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 164: 1–11. doi:10.1515/crll.1931.164.1. https://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, https://www.numdam.org/item?id=MSMF_1971__25__71_0 
• Manin, Yu. I.; Panchishkin, A. A. (2005). Introduction to Modern Number Theory. Encyclopaedia of Mathematical Sciences. 49 (2nd ed.). Springer Berlin, Heidelberg. ISBN 978-3-540-20364-3. 
• Serre, Jean-Pierre (1960). "Sur la rationalité des représentations d'Artin" (in fr). Annals of Mathematics 72 (2): 405–420. doi:10.2307/1970142. 
• Serre, Jean-Pierre (1967). "VI. Local class field theory". in Cassels, J.W.S.; Fröhlich, A.. Algebraic Number Theory. London: Academic Press. pp. 128–161. 
• Snaith, Victor P. (1994). Explicit Brauer Induction With Applications to Algebra and Number Theory. Cambridge Studies in Advanced Mathematics. 40. Cambridge University Press. ISBN 978-0-52146015-6. 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. 
• Weil, André (1946). "L'avenir des mathématiques" (in fr). Boletim da Sociedade de Matemática de São Paulo Paulo 1: 55–68.