Petersson products of modular forms associated with the values of \(L\)-functions
type: Item
instance of: scholarly article
title: Petersson products of modular forms associated with the values of \(L\)-functions
zbMATH Open document ID: 1021.11016
author: Lionel Fourquaux
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2003-06-30T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/225413
review text: Let \(p\) be a prime, \(\chi \) a Dirichlet character \(\pmod{p}\), \(f(z)=\sum_{n=1}^{\infty }a_ne^{2\pi inz}\) a cusp form of weight 2 for the congruence group \(\Gamma _0(p)\), and \(L(f,\chi ,s) = \sum_{n=1}^{\infty }\chi (n)a_nn^{-s}\) the corresponding \(L\)-function twisted with the character \(\chi \). The mapping \(f\mapsto L(f,\chi ,1)\), which is a linear form in the space \(S_2(\Gamma _0(p))\), can be represented as the inner product against a certain unique form \(h_{\chi }\in S_2(\Gamma _0(p))\), thus \(L(f, \chi ,1)=(f, h_{\chi })\). The main result in this paper is an identity for the inner product \((h_{\chi }, h_{\chi '})\) if \(\chi \) and \(\chi '\) are non-trivial characters of different parity. This identity involves Gauss sums and values of Dirichlet \(L\)-functions at \(s=1\).
Mathematics Subject Classification ID: 11F66
zbMATH DE Number: 1943333
zbMATH Keywords: modular forms
zbMATH Keywords: Petersson inner product
zbMATH Keywords: \(L\)-functions
reviewed by: Matti Jutila
MaRDI profile type: Publication
OpenAlex ID: W2334865868
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DOI: 10.5802/JTNB.352
description: scientific article; zbMATH DE number 1943333