Skip to main content

Questions tagged [hecke-characters]

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

Filter by
Sorted by
Tagged with
1 vote
0 answers
32 views

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 ...
Logwan27's user avatar
2 votes
0 answers
49 views

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^{\...
Sardines's user avatar
  • 769
2 votes
0 answers
96 views

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{\...
user682141's user avatar
  • 1,016
2 votes
1 answer
86 views

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,...
user682141's user avatar
  • 1,016
4 votes
0 answers
53 views

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 ...
Mario's user avatar
  • 739
1 vote
0 answers
69 views

Classifying all Hecke Characters of a given field and a given conductor [closed]

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 $\...
blueinfinity's user avatar
2 votes
0 answers
464 views

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 ...
user avatar
0 votes
0 answers
213 views

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's user avatar
  • 1
1 vote
0 answers
74 views

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 ...
Krishnarjun's user avatar
4 votes
2 answers
481 views

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 ...
Math Rules's user avatar
2 votes
1 answer
251 views

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. ...
LeLoupSolitaire's user avatar
1 vote
0 answers
148 views

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), \...
user114158's user avatar
1 vote
0 answers
86 views

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 ...
efs's user avatar
  • 284
3 votes
1 answer
139 views

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^\...
Wolker's user avatar
  • 1,077
4 votes
0 answers
203 views

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 $K/\mathbb{Q}$. ...
Jackson Morrow's user avatar
1 vote
1 answer
146 views

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 ...
user200419's user avatar
2 votes
1 answer
160 views

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 $P$ ...
BlackAdder's user avatar
  • 4,029
10 votes
1 answer
424 views

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 ...
BlackAdder's user avatar
  • 4,029