Hecke 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.

Definition using ideles

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, or as a homomorphism to the unit circle in C. Any quasicharacter 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 is trivial on the group of finite ideles whose every v-adic component lies in 1 + mOv.

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 = mfm∞ be a
K-modulus, with mf, the "finite part", being an integral ideal of K and m∞, the "infinite part", being a 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
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 is at least as large as the exponent for v in mf, and a is positive under each real embedding in m∞. A Hecke character with modulus m
is a group homomorphism from Im into the nonzero complex numbers
such that on ideals in Pm 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. Thus a Hecke character may be defined on the ray class group modulo m, which is the quotient Im/Pm.
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.

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 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 means the ideal norm. The common real part condition governing the behavior of Hecke characters on the subgroups Pm 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, 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. 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.

Special cases

Hecke's original proof of the functional equation for L 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 showed that parts of Tate's proof could be expressed by distribution theory: the space of distributions on the adele group of K transforming under the action of the ideles by a given χ has dimension 1.

Algebraic Hecke characters

An algebraic Hecke character is a Hecke character taking algebraic values: they were introduced by Weil in 1947 under the name type A0. Such characters occur in class field theory and the theory of complex multiplication.
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 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.