Determination of modular forms by twists of critical \(L\)-values
type: Item
instance of: scholarly article
title: Determination of modular forms by twists of critical \(L\)-values
zbMATH Open document ID: 0905.11024
author: Wenzhi Luo
author: Dinakar Ramakrishnan
published in: Inventiones Mathematicae
publication date: 1997-12-10T00:00:00Z
review text: Let \(f\) be a normalized holomorphic newform of weight \(2k\), level \(N\), and trivial character. Let \(\chi\) be a primitive Dirichlet character of conductor \(M_\chi\), and let \(L(f,\chi,s)\) denoted the twisted \(L\)-function for \(f\), normalized to have functional equation under \(s\mapsto 1-s\). Then the authors show that for any \(s_0\in{\mathbb C}\) the \(L\)-values \(L(f,\chi,s_0)\) as \(\chi\) ranges over the characters whose conductor and order are powers of a fixed odd prime \(p\), \((p,N)=1\), determine \(f\). They also show that the central \(L\)-values \(L(f,\chi,1/2)\) as \(\chi\) ranges over the quadratic primitive Dirichlet characters with \((M_\chi,N)=1\) determine \(f\). These results have many implications. The first result implies that the fundamental periods of \(f\) may be recovered by taking a limit of averages of the algebraic parts of the central critical values for twists of \(f\), the twists being of order a power of the fixed prime \(p\), \((p,N)=1\). Next, the authors show that the field \(K_f\) generated over \({\mathbb Q}\) by the Fourier coefficients of \(f\) is also generated over \({\mathbb Q}\) by the set of ratios \(g(\chi_0)g(\chi)^{-1}L(f,\chi,1/2) /L(f,\chi_0,1/2)\) where \(\chi_0\) is a fixed quadratic character such that \(L(f,\chi_0,1/2)\neq 0\), \(g(\nu)\) denotes the Gauss sum associated to the character \(\nu\), and \(\chi\) ranges over the quadratic characters of the same parity as \(\chi_0\). A result of \textit{G. Shimura} [Commun. Pure Appl. Math. 29, 783-804 (1976; Zbl 0348.10015)] implies that \(K_f\) contains this field; the authors apply Shimura's reciprocity law to obtain equality. The authors also give an application to \(p\)-adic \(L\)-functions. One says that a finite order \(p\)-adic character \(\eta\) is of tame type \(i\pmod {p-1}\) if \(\eta=\omega^i\nu\) where \(\omega\) denotes the Teichmüller character and \(\nu\) is a wild character. Then the authors show that for any such \(\eta\), \(0\leq i\leq …
reviewed by: Solomon Friedberg
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 1096056
zbMATH Keywords: modular form
zbMATH Keywords: critical \(L\)-value
zbMATH Keywords: twisted \(L\)-function
zbMATH Keywords: \(p\)-adic \(L\)-function
zbMATH Keywords: Kohnen subspace
zbMATH Keywords: Shimura correspondence
zbMATH Keywords: fundamental periods
zbMATH Keywords: Fourier coefficients
zbMATH Keywords: modular elliptic curve
MaRDI profile type: Publication
full work available at URL: s002220050189
OpenAlex ID: W1983492906
DOI: 10.1007/S002220050189
description: scientific article; zbMATH DE number 1096056