In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of L-functionss 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.

Definition

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 ideless, 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 \mathbb{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.

Größencharakter

A Größencharakter (often written Grössencharakter, Grossencharacter, etc.), origin of a Hecke character, going back to Hecke, is defined in terms of a character on the group of fractional ideals. For a number field-modulus, the "infinite part", being a (formal) product of real places.) Thus a Größencharakter may be defined on the ray class group.

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 Größencharakter and Hecke character

A Hecke character and a Größencharakter are essentially the same notion with a one-to-one correspondence. 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 Größencharakter , its L-function is defined to be the Dirichlet series

denotes the ideal norm implies these Dirichlet series are absolutely convergent in some right half-plane. Hecke proved these L when the character is trivial. 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.

. is well-defined as a character on the ideal group.

In the opposite direction, given an admissible..

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-functionss, 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-functionss for an important class of algebraic varieties (or even motivess).

Special cases

• A Dirichlet character is a Hecke character of finite order. It is determined by values on the set of totally positive principal ideals which are 1 with respect to some modulus m.

• A Hilbert character is a Dirichlet character of conductor 1. The number of Hilbert characters is the order of the class group of the field. Class field theory identifies the Hilbert characters with the characters of the Galois group of the Hilbert class field.

Examples

• For the field of rational numbers, the idele class group is isomorphic to the product of positive reals with all the unit groups of the p-adic integers. So a quasicharacter can be written as product of a power of the norm with a Dirichlet character.

• A Hecke character χ of the Gaussian integers of conductor 1 is of the form

• χ((a)) = |a|s(a/|a|)4n

  for s imaginary and n an integer, where a is a generator of the ideal (a). The only units are powers of i, so the factor of 4 in the exponent ensures that the character is well defined on ideals.

Tate's thesis

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 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.

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(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.

Notes

References

• J. Tate, Fourier analysis in number fields and Hecke's zeta functions (Tate's 1950 thesis), reprinted in Algebraic Number Theory edd J. W. S. Cassels, A. Fröhlich (1967) pp.305–347.