Hecke Character - Definition Using Ideals

Definition Using Ideals

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.

Strictly speaking, Hecke made the stipulation about behavior on principal ideals for those admitting a totally positive generator. So, in terms of the definition given above, he really only worked with moduli where all real places appeared. The role of the infinite part m_∞ is now subsumed under the notion of an infinity-type.