grössencharacter

Let $K$ be a number field and let $A_K$ be idele group of $K$, i.e.

$$A_K=\prod _{\nu }{K}_{\nu }^{\ast }$$

where the product is a restricted direct product running over all places (infinite and finite) of $K$ (see entry on http://planetmath.org/node/Ideleideles). Recall that $K^{\ast }$ embeds into $A_K$ diagonally:

$$x\in K^{\ast }\mapsto {(x_{\nu })}_{\nu }$$

where $x_{\nu }$ is the image of $x$ under the embedding of $K$ into its completion at the place $\nu$, $K_{\nu }$.

Definition 1.

A Grössencharacter $\psi$ on $K$ is a continuous homomorphism:

$$\psi :A_K⟶{ℂ}^{\ast }$$

which is trivial on $K^{\mathrm{\ast }}$, i.e. if $x\mathrm{\in }{K}^{\mathrm{\ast }}$ then $\psi \mathit{}\mathrm{(}{\mathrm{(}{x}_{\nu }\mathrm{)}}_{\nu }\mathrm{)}\mathrm{=}\mathrm{1}$. We say that $\psi$ is unramified at a prime $\mathrm{\wp }$ of $K$ if $\psi \mathit{}\mathrm{(}{\mathrm{O}}_{\mathrm{\wp }}^{\mathrm{\ast }}\mathrm{)}\mathrm{=}\mathrm{1}$, where ${\mathrm{O}}_{\mathrm{\wp }}$ is the ring of integers inside $K_{\mathrm{\wp }}$. Otherwise we say that $\psi$ is ramified at $\mathrm{\wp }$.

Let ${𝒪}_K$ be the ring of integers in $K$. We may define a homomorphism on the (multiplicative) group of non-zero fractional ideals of $K$ as follows. Let $\mathrm{\wp }$ be a prime of $K$, let $\pi$ be a uniformizer of $K_{\mathrm{\wp }}$ and let ${\alpha }_{\mathrm{\wp }}\in A_K$ be the element which is $\pi$ at the place $\mathrm{\wp }$ and $1$ at all other places. We define:

$$\psi (\mathrm{\wp })=\{\begin{array}{cc}0,\text{ if }\psi \text{ is ramified at }\mathrm{\wp };\hfill & \\ \psi ({\alpha }_{\mathrm{\wp }}),\text{ otherwise}.\hfill & \end{array}$$

Definition 2.

The Hecke L-series attached to a Grössencharacter $\psi$ of $K$ is given by the Euler product over all primes of $K$:

$$L(\psi ,s)=\prod _{\mathrm{\wp }}{\left(1-\frac{\psi (\mathrm{\wp })}{{(N_{\mathbb{Q}}^K(\mathrm{\wp }))}^s}\right)}^{-1}.$$

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 $A_K$ (what is usually called Tate’s thesis).