# grössencharacter

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

 $A_{K}={\prod_{\nu}}^{\prime}K_{\nu}^{\ast}$

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:

 $\psi:A_{K}\longrightarrow\mathbb{C}^{\ast}$

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

Let $\mathcal{O}_{K}$ 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 $\wp$ be a prime of $K$, let $\pi$ be a uniformizer of $K_{\wp}$ and let $\alpha_{\wp}\in A_{K}$ be the element which is $\pi$ at the place $\wp$ and $1$ at all other places. We define:

 $\psi(\wp)=\begin{cases}0,\text{ if }\psi\text{ is ramified at }\wp;\\ \psi(\alpha_{\wp}),\text{ otherwise}.\end{cases}$
###### Definition 2.

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

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

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).

Title grössencharacter Grossencharacter 2013-03-22 15:45:19 2013-03-22 15:45:19 alozano (2414) alozano (2414) 5 alozano (2414) Definition msc 11R56 GrossencharacterAssociatedToACMEllipticCurve grossencharacter