Let K be a number fieldMathworldPlanetmath and let AK be idele group of K, i.e.


where the product is a restricted direct productPlanetmathPlanetmathPlanetmath running over all places (infinite and finite) of K (see entry on Recall that K embeds into AK diagonally:


where xν is the image of x under the embedding of K into its completion at the place ν, Kν.

Definition 1.

A Grössencharacter ψ on K is a continuous homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:


which is trivial on K, i.e. if xK then ψ((xν)ν)=1. We say that ψ is unramified at a prime of K if ψ(O)=1, where O is the ring of integersMathworldPlanetmath inside K. Otherwise we say that ψ is ramified at .

Let 𝒪K be the ring of integers in K. We may define a homomorphism on the (multiplicative) group of non-zero fractional idealsMathworldPlanetmathPlanetmath of K as follows. Let be a prime of K, let π be a uniformizer of K and let αAK be the element which is π at the place and 1 at all other places. We define:

ψ()={0, if ψ is ramified at ;ψ(α), otherwise.
Definition 2.

The Hecke L-series attached to a Grössencharacter ψ of K is given by the Euler productMathworldPlanetmath over all primes of K:


Hecke L-series of this form have an analytic continuation and satisfy a certain functional equation. This fact was first proved by Hecke himself but later was vastly generalized by Tate using Fourier analysis on the ring AK (what is usually called Tate’s thesis).

Title grössencharacter
Canonical name Grossencharacter
Date of creation 2013-03-22 15:45:19
Last modified on 2013-03-22 15:45:19
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 5
Author alozano (2414)
Entry type Definition
Classification msc 11R56
Related topic GrossencharacterAssociatedToACMEllipticCurve
Defines grossencharacter