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Questions tagged [hecke-characters]

Generalization of a Dirichlet character to construct a class of L-functions.

In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of L-functions larger than Dirichlet L-functions, and a natural setting for the Dedekind zeta-functions and certain others which have functional equations analogous to that of the Riemann zeta-function.

A name sometimes used for Hecke character is the German term Größencharakter. (Wikipedia)

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What does the German word "Zerlegungsautomorphismus" translate to?

I would like to know if any of our German friends can translate that word for me. Zerlegung is factorisation, isn't it? So what is factorisation automorphism? This is taken from Deuring's paper “Die Zetafunktion einer algebraischen Kurve vom…
BlackAdder
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Understanding Hecke Characters as Extension of Dirichlet Characters

I understand the concept of a Dirichlet character, and am interested in its generalizations to arbitrary number fields. I have heard that this generalization is called a Hecke character. However, I am not familiar with adeles or ideles, so I don't…
Math Rules
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Hecke characters and Conductors

Motivation: Let $\ell$ be an odd prime. There is a conductor-preserving correspondence between primitive Dirichlet characters of order $\ell$ and cyclic, degree $\ell$ number fields $K/\mathbb{Q}$. The proof of this correspondence can be found in…
Jackson Morrow
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Characters of a quadratic extension and convergence

(I follow up this question from MO, since it appears to get no real interest in there) Let $F$ be a non-archimedean local field and $E$ a quadratic extension on $F$, $\chi$ a quasi-character of $E^\star$ and $\psi$ a positive character of $E^\star$.…
Wolker
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Ireland-Rosen Hecke Character for $y^2=x^3-Dx$

I would like to refer you to page $310$ of Ireland-Rosen: A Classical Intro to Modern Number Theory. Firstly, to construct the Hecke character, it is enough to specify $\chi(P)$ for prime ideals $P$ in $\mathbb{Z}[i]$. If $P$ divides $2D$, then…
BlackAdder
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understanding Hecke characters

How do I understand Hecke characters? For example, is there a bijection between Hecke characters and something? For example, if a Hecke character factors through a ray class group, by Artin reciprocity, we would have a 1-1 correspondence with…
user471019
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Special values of Hecke $L$-function on imaginary quadratic fields

Let $K$ be an imaginary quadratic number field and $\mathcal{O}_K$ its ring of integers. Let $\chi$ be an algebraic Hecke character on $K$ with conductor $\mathfrak{f}$ and infinity type $(a,b)$, i.e. $$ \chi (\mathfrak{a}) =…
LeLoupSolitaire
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Restriction of Hecke Characters

Let $L$ be a number field and let $\Psi$ be a finite order Hecke character on $L$ with (finite part) of conductor as $\mathfrak{f}$. Suppose we define $$\psi(m) :=\Psi(m\mathcal{O}_L)$$ for all integers $m\in\mathbb{Z}$. Does $\psi$ define a…
Krishnarjun
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Fourier expansion of multiplicative analogue of Hecke operator

Let $f$ be a modular form of weight $2k$, that is $f$ is a holomorphic on $\mathbb{H}$, $f$ extends holomorphically at infinity and satisfies $$ f\left(\frac{az+b}{cz+d}\right) = (cz+d)^{2k}f(z), \hspace{8mm} \begin{pmatrix} a & b \\ c & d…
user114158
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Bibliography on Hecke characters

Do you know of any book, lecture note or survey article that deals in detail with Grössencharaktere and Hecke characters, this is, in both "ideal" and "idèle" interpretations? With "in detail" I mean something like Neukirch's "Algebraic Number…
efs
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Complex Galois Representaions

I'm trying to understand 1 and 2 dimensional complex representations, induced representations and associated Artin L-functions for a project. I'm finding it hard to find appropriate material to help me get into this on the internet. Can anyone…
user200419
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Dirichlet L-series and Hecke L-series

I'm working on L-series (reading Rosen's book Number Theory in Function fields) and i read that Dirichlet $L$-series are supposed to be a special case of Hecke $L$-series, and i can't understand why ?
hyuno
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