Idele class character

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 + mO_v.

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 = m_fm_∞ be a K-modulus, with m_f, the "finite part", being an integral ideal of K and m_∞, the "infinite part", being a (formal) product of real places of K. Let I_m denote the group of fractional ideals of K relatively prime to m_f and let P_m 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 m_f, ord_v(a − 1) is at least as large as the exponent for v in m_f, and a is positive under each real embedding in m_∞. A Hecke character with modulus m is a group homomorphism from I_m into the nonzero complex numbers such that on ideals (a) in P_m its value is equal to the value at a of a continuous homomorphism to the nonzero complex numbers from the product of the multiplicative groups of all archimedean completions of K where each local component of the homomorphism has the same real part (in the exponent). (Here we embed a into the product of archimedean completions of K using embeddings corresponding to the various archimedean places on K.) Thus a Hecke character may be defined on the ray class group modulo m, which is the quotient I_m/P_m.

...
Wikipedia