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grössencharacter
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(Definition)
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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 ideles). 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: $$\psi:A_K \longrightarrow \Complex^\ast$$ 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:
Definition 2 The Hecke L-series attached to a Grössencharacter $\psi$ of $K$ is given by the Euler product over all primes of $K$ : $$L(\psi,s)=\prod_\wp\left(1-\frac{\psi(\wp)}{(N_{\Rats}^K(\wp))^s}\right)^{-1}.$$
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).
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"grössencharacter" is owned by alozano.
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Cross-references: ring, analysis, functional equation, analytic continuation, Euler product, uniformizer, fractional ideals, group, multiplicative, ring of integers, prime, unramified, homomorphism, 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 2700 times total.
Classification:
| AMS MSC: | 11R56 (Number theory :: Algebraic number theory: global fields :: Adèle rings and groups) |
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Pending Errata and Addenda
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