Hecke Character - Relationship Between The Definitions

Relationship Between The Definitions

The ideal definition is much more complicated than the idelic one, and Hecke's motivation for his definition was to construct L-functions (sometimes referred to as Hecke L-functions) that extend the notion of a Dirichlet L-function from the rationals to other number fields. For a Hecke character χ, its L-function is defined to be the Dirichlet series

carried out over integral ideals relatively prime to the modulus m of the Hecke character. The notation N(I) means the ideal norm. The common real part condition governing the behavior of Hecke characters on the subgroups P_m implies these Dirichlet series are absolutely convergent in some right half-plane. Hecke proved these L-functions have a meromorphic continuation to the whole complex plane, being analytic except for a simple pole of order 1 at s = 1 when the character is trivial. For primitive Hecke characters (defined relative to a modulus in a similar manner to primitive Dirichlet characters), Hecke showed these L-functions satisfy a functional equation relating the values of the L-function of a character and the L-function of its complex conjugate character.

Consider a character ψ of the idele class group, taken to be a map into the unit circle which is 1 on principal ideles and on an exceptional finite set S containing all infinite places. Then ψ generates a character χ of the ideal group IS, the free abelian group on the prime ideals not in S. Take a uniformising element π for each prime p not in S and define a map Π from IS to idele classes by mapping each p to the class of the idele which is π in the p coordinate and 1 everywhere else. Let χ be the composite of Π and ψ. Then χ is well-defined as a character on the ideal group.

In the opposite direction, given an admissible character χ of IS there corresponds a unique idele class character ψ. Here admissible refers to the existence of a modulus m based on the set S such that the character χ is 1 on the ideals which are 1 mod m.

The characters are 'big' in the sense that the infinity-type when present non-trivially means these characters are not of finite order. The finite-order Hecke characters are all, in a sense, accounted for by class field theory: their L-functions are Artin L-functions, as Artin reciprocity shows. But even a field as simple as the Gaussian field has Hecke characters that go beyond finite order in a serious way (see the example below). Later developments in complex multiplication theory indicated that the proper place of the 'big' characters was to provide the Hasse-Weil L-functions for an important class of algebraic varieties (or even motives).