Get Artin Representation essential facts below. View Videos or join the Artin Representation discussion. Add Artin Representation to your PopFlock.com topic list for future reference or share this resource on social media.
Suppose that L is a finite Galois extension of the local field K, with Galois group G. If is a character of G, then the Artin conductor of is the number
where Gi is the i-th ramification group (in lower numbering), of order gi, and ?(Gi) is the average value of on Gi. By a result of Artin, 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 ? is unramified, then its Artin conductor is zero. Thus if L is unramified over K, then the Artin conductors of all ? 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.
Global Artin conductors
The global Artin conductor of a representation 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(?,p) is the local Artin conductor of the restriction of 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 characteraG of G is the character
and the Artin representationAG 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.
The Swan characterswG is given by
where rg 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.
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, Zbl0153.07403