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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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Abelian extension over imaginary quadratic field

Notation: For a finite abelian extension $F / K$, let $\mathfrak{f}_{F / K} \subset \mathcal{O}_K$ denotes its conductor such that $F$ is contained in the ray class field $K(\mathfrak{f})$. In particular, the set of primes dividing $\mathfrak{f}_{F…
Mario
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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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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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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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Reconciling different ideal-theoretic definitions of Hecke Characters

I'm reading Chapter 7 of Neukirch's book on algebraic number theory, where the author defines a Größencharakter (6.1) as: Let $\mathfrak{m}$ be an integral ideal of the number field $K$, and let $J^{\mathfrak{m}}$ be the group of all ideals of $K$…
Sardines
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Functional equation of the Hecke L function in ideal term and "ideal number" term (Neukirch Chapter VII)

$\def\A{\mathbb{A}} \def\B{\mathbb{B}} \def\C{\mathbb{C}} \newcommand{\Cx}{\mathbb{C}^{\times}} \def\F{\mathbb{F}} \def\G{\mathbb{G}} \def\H{\mathbb{H}} \def\K{\mathbb{K}} \def\M{\mathbb{M}} \def\N{\mathbb{N}} \def\O{\mathcal{O}} \def\P{\mathbb{P}} \…
user682141
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Complete proof for the shape of quasicharacters of $\mathbb{R}^{\times}, \mathbb{C}^{\times}$

Quasicharacters (:=continuous group homomorphism to $\mathbb{C}^{\times}$) of $\mathbb{R}^{\times}, \mathbb{C}^{\times}$ seems to be known to be following forms (the following is quoted from [Raghuram, p.8]): (1) Quasicharacters of…
user682141
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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…
user1091543
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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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Analytic continuation for $L$-series with real character in Murty

In Problems in Algebraic Number Theory from M. Ram Murty, he mentions on page 148 how one show non-vanishing at 1 for a certain $L$-series. Here, he works with a generalization of Dirichlet $L$-series (a special case of Hecke $L$-series), that is…
Logwan27
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Classifying all Hecke Characters of a given field and a given conductor

I'm rather very new to this topics and in the hopes of understanding Tate's Thesis I have come to the issue of Hecke Character. Given the following definition: Let $F$ be a number field and let $\mathbb{I}_F$ be the idele group of $F$. A Hecke…
blueinfinity
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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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