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.

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. By a result of Artin, the local conductor is an integer. 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 or Swan conductor of the character is

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

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. 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

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.

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