Canonical periods and congruence formulae
type: Item
instance of: scholarly article
title: Canonical periods and congruence formulae
zbMATH Open document ID: 0979.11027
DOI: 10.1215/S0012-7094-99-09811-3
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: The results in this paper state roughly that congruences between the Fourier coefficients of modular forms give rise to congruences between the critical values for their associated \(L\)-functions. The author shows that periods may be selected for the associated \(L\)-functions so that one may precisely formulate the notion of a congruence between critical values. More precisely, the author proves the following. Let \(p\) be any odd prime and let \(f=\sum a_n q^n\) and \(g=\sum b_nq^n\) be cuspidal newforms of weight 2 on \(\Gamma_1(M)\), such that \(a_n\equiv b_n\pmod{\mathfrak p^r}\), for some prime \(\mathfrak p\) above \(p\) in \(\overline{\mathbb Q}\). Assume that \((M,p)=1\) and that the residual representation attached to \(f\) is irreducible. Fix an isomorphism \(\mathbb C_p \cong \mathbb C\), such that the prime \(p\) of \(\overline{\mathbb Q} \subset \mathbb C\) induces the usual absolute value on \(\mathbb C_p\). Then there exist canonical periods \(\Omega^{\pm}_f\) and \(\Omega^{\pm}_g\) such that the congruence \[ \tau(\overline{\chi}) \frac{L(1,f,\chi)}{-2\pi i \Omega^{\pm}_f} \equiv \tau(\overline{\chi}) \frac{L(1,g,\chi)}{-2\pi i \Omega^{\pm}_g} \pmod{\mathfrak p^r} \] holds for every character \(\chi\). Furthermore, there exists a \(\chi\) such that both sides of the congruence are nonzero modulo \(\mathfrak p\). The author also has some results in the residually reducible case as well. As a corollary, the author shows that most elliptic curves \(E\) with a rational point of order 3 have the property that a positive proportion of its quadratic twists \(E_D\) have rank zero. He does this by employing a theorem of Davenport and Heilbronn on the distribution of 3-torsion in the class groups of quadratic number fields. These results generalize previous examples by \textit{W. Kohnen} [J. Reine Angew. Math. 508, 179-187 (1999; Zbl 0968.11023)] and the reviewer [J. Am. Math. Soc. 11, 635-641 (1998; Zbl 0904.11015)].
Mathematics Subject Classification ID: 11F33
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 11G40
zbMATH DE Number: 1425220
zbMATH Keywords: nonvanishing of \(L\)-series
zbMATH Keywords: congruence of modular forms
zbMATH Keywords: special values of \(L\)-series
author: Vinayak Vatsal
reviewed by: Kevin James
describes a project that uses: ecdata
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1425220