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.
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
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 "infinite part", being a (formal) product of real places of
denote the subgroup of principal fractional ideals
of a continuous homomorphism to the nonzero complex numbers from the product of the multiplicative groups of all Archimedean completions of
where each local component of the homomorphism has the same real part (in the exponent).
Thus a Größencharakter may be defined on the ray class group modulo
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.
A Hecke character and a Größencharakter are essentially the same notion with a one-to-one correspondence[how?].
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)[1] that extend the notion of a Dirichlet L-function from the rationals to other number fields.
For a Größencharakter χ, its L-function is defined to be the Dirichlet series carried out over integral ideals relatively prime to the modulus
The common real part condition governing the behavior of Größencharakter on the subgroups
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 '
For primitive Größencharakter (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.
, which is the free abelian group on the prime ideals not in
there corresponds a unique idele class character
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.
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).
Hecke's original proof of the functional equation for L(s,χ) used an explicit theta-function.
John Tate's 1950 Princeton doctoral dissertation, written under the supervision of Emil Artin, applied Pontryagin duality systematically, to remove the need for any special functions.
A similar theory was independently developed by Kenkichi Iwasawa which was the subject of his 1950 ICM talk.
A later reformulation in a Bourbaki seminar by Weil 1966 showed that parts of Tate's proof could be expressed by distribution theory: the space of distributions (for Schwartz–Bruhat test functions) on the adele group of K transforming under the action of the ideles by a given χ has dimension 1.
An algebraic Hecke character is a Hecke character taking algebraic values: they were introduced by Weil in 1947 under the name type A0.
[6] Indeed let E be an elliptic curve defined over a number field F with complex multiplication by the imaginary quadratic field K, and suppose that K is contained in F. Then there is an algebraic Hecke character χ for F, with exceptional set S the set of primes of bad reduction of E together with the infinite places.
This character has the property that for a prime ideal p of good reduction, the value χ(p) is a root of the characteristic polynomial of the Frobenius endomorphism.
As a consequence, the Hasse–Weil zeta function for E is a product of two Dirichlet series, for χ and its complex conjugate.