Transcendence conjectures about periods of modular forms and rational structures on spaces of modular forms
type: Item
instance of: scholarly article
title: Transcendence conjectures about periods of modular forms and rational structures on spaces of modular forms
zbMATH Open document ID: 0699.10047
DOI: 10.1007/BF02864395
author: Winfried Kohnen
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1989-01-01T00:00:00Z
review text: Let K be an algebraic number field and let \(M^ 0_{2k}\) be the K- vector space of modular forms with respect to SL(2,\({\mathbb{C}})\) whose Fourier coefficients are all in K. Let \(S^ 0_{2k}\) be the subspace of cusp forms. The author conjectures that for any \(f\in M^ 0_{2k}\) there is at least one even n with \(0\leq n\leq 2k-2\) such that \(L(f,n+1)/\pi^{n+1}\not\in K.\) Another conjecture is that for any \(f\in S^ 0_{2k}\) there is at least one odd n with \(0\leq n\leq 2k-2\) such that \(L(f,n+1)/\pi^{n+1}\not\in K.\) These conjectures are supported by a few known properties of periods of modular forms. A consequence would be that \(\zeta (2k-1)/\pi^{2k-1}\not\in \bar Q\) \((k=2,4,6,...)\).
Mathematics Subject Classification ID: 11J81
Mathematics Subject Classification ID: 11F11
zbMATH DE Number: 4146019
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: transcendence
zbMATH Keywords: modular forms
zbMATH Keywords: periods
reviewed by: Frits Beukers
MaRDI profile type: Publication
Wikidata QID: Q110701793
cites work: Q3898582
cites work: Q3754061
cites work: Q3754673
description: scientific article; zbMATH DE number 4146019