Periods of modular forms of weight one
type: Item
instance of: scholarly article
title: Periods of modular forms of weight one
zbMATH Open document ID: 1099.11027
author: François Martin
published in: Journal of Number Theory
publication date: 2006-03-24T00:00:00Z
review text: The author develops a theory of periods for modular forms of weight one, in analogy with the classic works of M. Eichler and G. Shimura. To cite one of his results, let \(G:=\text{SL}_2(\mathbb{Z})\), let \(\Gamma\) be a subgroup of finite index of \(G\), and let \(M_1(\Gamma)\) stand for the complex vector space of \(\Gamma\)-modular forms of weight one. Further, let \(V:=\left(\begin{smallmatrix} 1 & 0\\ 0 & -1\end{smallmatrix}\right)\), let \(\Gamma':=V\Gamma V\), and let \({\mathfrak M}(\Gamma): =M_1 (\Gamma)\oplus M_1(\Gamma')\). The author defines an injective map \(r: {\mathfrak M}(\Gamma)\hookrightarrow{\mathfrak F}(\Gamma)\) of the vector space \({\mathfrak M})\) into \({\mathfrak F}(\Gamma)\), the space of functions \(f:(\mathbb{R}\setminus\{0\})\times (\Gamma\setminus G)\to\mathbb{C}\) analytic in the first variable, and characterizes the image \(r({\mathfrak M}(\Gamma) )\) of that map as the subspace of those functions \(f\) in \({\mathfrak F}( \Gamma)\) which satisfy the so-called ``Manin relations'' and certain growth conditions at infinity.
reviewed by: B. Z. Moroz
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F25
zbMATH DE Number: 5015059
zbMATH Keywords: periods of modular forms
zbMATH Keywords: Petersson scalar product
zbMATH Keywords: Manin relations
zbMATH Keywords: modular forms of weight one
zbMATH Keywords: Hecke operators
zbMATH Keywords: L-functions
MaRDI profile type: Publication
full work available at URL: j.jnt.2005.05.007
OpenAlex ID: W1967393292
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DOI: 10.1016/J.JNT.2005.05.007
description: scientific article; zbMATH DE number 5015059