Hecke character

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In mathematics, 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 (often written Grössencharakter, Grossencharakter, Grössencharacter, Grossencharacter et cetera).

Contents 1 Definition using ideles 2 Definition using ideals 3 Special cases 4 Examples 5 Tate's thesis 6 References

[edit] Definition using ideles

A Hecke character is a character of the idele class group of a number field or function field.

Some authors define a Hecke character to be a quasicharacter rather than a character. The difference is that a quasicharacter is a homomorphism to the non-zero complex numbers, while a character is a homomorphism to the unit circle. Any quasicharacter (of the idele class group) can be written uniquely as a 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 vanishes on ideles whose infinite part is 1 and whose finite part is integral and congruent to 1 mod m.

[edit] Definition using ideals

The role of the Hecke characters was clarified gradually. Initially a Hecke character χ was defined on fractional ideals in a number field K, in such a way that

was a Dirichlet series with good analytic properties. Here I runs over ideals of the ring of integers of K that have no factor in common with a given ideal, the conductor f. The notation N(I) means the norm of an ideal. Further χ was supposed to take a particular form on principal ideals (α) with α subject to a congruence modulo f: it was a certain product of powers of the conjugates of α, specified by what would now be called an infinity-type.

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).

[edit] Special cases

• A Dirichlet character is a Hecke character of finite order.
• 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; more precisely, class field theory identifies the Hilbert characters with the characters of the class group.

[edit] Examples

• For the field of rational numbers, the idele class group is isomorphic to the product of the 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.

[edit] Tate's thesis

Hecke's original proof of the functional equation for L(s,χ) used an explicit theta-function. John Tate's celebrated doctoral dissertation, written under the supervision of Emil Artin, applied Pontryagin duality systematically, to remove the need for any special functions. A later reformulation in a Bourbaki seminar by Weil (Fonctions zetas et distributions, Séminaire Bourbaki 312, 1966) showed that the essence 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.

[edit] References

• J. Tate, Fourier analysis in number fields and Hecke's zeta functions (Tate's 1950 thesis), reprinted in Algebraic Number Theory by J. W. S. Cassels, A. Frohlich ISBN 0-12-163251-2
• J. Neukirch, Algebraic number theory, ISBN 3-540-65399-6