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Concept# Hecke character

Summary

In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of
L-functions larger than Dirichlet L-functions, and a natural setting for the Dedekind zeta-functions and certain others which have functional equations analogous to that of the Riemann zeta-function.
A name sometimes used for Hecke character is the German term Größencharakter (often written Grössencharakter, Grossencharacter, etc.).
A Hecke character is a character of the idele class group of a number field or global function field. It corresponds uniquely to a character of the idele group which is trivial on principal ideles, via composition with the projection map.
This definition depends on the definition of a character, which varies slightly between authors: It may be defined as a homomorphism to the non-zero complex numbers (also called a "quasicharacter"), or as a homomorphism to the unit circle in C ("unitary"). Any quasicharacter (of the idele class group) can be written uniquely as a unitary character times a real power of the norm, so there is no big difference between the two definitions.
The conductor of a Hecke character χ is the largest ideal m such that χ is a Hecke character mod m. Here we say that χ is a Hecke character mod m if χ (considered as a character on the idele group) is trivial on the group of finite ideles whose every v-adic component lies in 1 + mOv.
The original definition of a Hecke character, going back to Hecke, was in terms of
a character on fractional ideals. For a number field K, let
m = mfm∞ be a
K-modulus, with mf, the "finite part", being an integral ideal of K and m∞, the "infinite part", being a (formal) product of real places of K. Let Im
denote the group of fractional ideals of K relatively prime to mf and
let Pm denote the subgroup of principal fractional ideals (a)
where a is near 1 at each place of m in accordance with the multiplicities of
its factors: for each finite place v in mf, ordv(a − 1) is at least as large as the exponent for v in mf, and a is positive under each real embedding in m∞.

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