PlanetMath: grössencharacter

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grössencharacter

(Definition)

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

$\displaystyle A_K={\prod_\nu}' K_\nu^\ast$

where the product is a restricted direct product running over all places (infinite and finite) of

(see entry on ideles). Recall that $K^\ast$ embeds into

diagonally:

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

where $x_\nu$ is the image of

under the embedding of

into its completion at the place $\nu$ , $K_\nu$ .

Definition 1 A Grössencharacter $\psi$ on is a continuous homomorphism:

$\displaystyle \psi:A_K \longrightarrow \mathbb{C}^\ast$

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 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 . We may define a homomorphism on the (multiplicative) group of non-zero fractional ideals of as follows. Let $\wp$ be a prime of , 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 at all other places. We define:

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

Definition 2 The Hecke L-series attached to a Grössencharacter $\psi$ of is given by the Euler product over all primes of :

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

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 (what is usually called Tate's thesis).

"grössencharacter" is owned by alozano.

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grossencharacter

This object's parent.

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Cross-references: ring, functional equation, analytic continuation, Euler product, uniformizer, fractional ideals, group, multiplicative, ring of integers, prime, unramified, continuous, completion, embedding, image, finite, infinite, places, running, restricted direct product, product, idele group, number field
There are 4 references to this entry.

This is version 2 of grössencharacter, born on 2006-03-10, modified 2006-03-10.
Object id is 7709, canonical name is Grossencharacter.
Accessed 1541 times total.

Classification:

AMS MSC:

11R56 (Number theory :: Algebraic number theory: global fields :: Adèle rings and groups)

Pending Errata and Addenda

None.

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