In algebraic number theory, the conductor of a finite abelian extension of local or global fields provides a quantitative measure of the ramification in the extension.
Let L/K be a finite abelian extension of non-archimedean local fields.
, is the smallest non-negative integer n such that the higher unit group is contained in NL/K(L×), where NL/K is field norm map and
is the maximal ideal of K.[1] Equivalently, n is the smallest integer such that the local Artin map is trivial on
[4] More precisely, the conductor computes the non-triviality of higher ramification groups: if s is the largest integer for which the "lower numbering" higher ramification group Gs is non-trivial, then
Specifically,[6] where χ varies over all multiplicative complex characters of Gal(L/K),
is the Artin conductor of χ, and lcm is the least common multiple.
The conductor can be defined in the same way for L/K a not necessarily abelian finite Galois extension of local fields.
[7] However, it only depends on Lab/K, the maximal abelian extension of K in L, because of the "norm limitation theorem", which states that, in this situation,[8][9] Additionally, the conductor can be defined when L and K are allowed to be slightly more general than local, namely if they are complete valued fields with quasi-finite residue field.
[11] The conductor of an abelian extension L/K of number fields can be defined, similarly to the local case, using the Artin map.
Specifically, let θ : Im → Gal(L/K) be the global Artin map where the modulus m is a defining modulus for L/K; we say that Artin reciprocity holds for m if θ factors through the ray class group modulo m. We define the conductor of L/K, denoted
[18] An infinite prime v occurs in the conductor if, and only if, v is real and becomes complex in L.