Some congruence property of modular forms
type: Item
instance of: scholarly article
title: Some congruence property of modular forms
zbMATH Open document ID: 0892.11018
DOI: 10.1007/BF02677851
author: Shoyu Nagaoka
published in: Manuscripta Mathematica
publication date: 1998-08-16T00:00:00Z
full work available at URL: https://eudml.org/doc/156336
review text: The author discusses the existence of a Siegel modular form \(F\) of degree \(n\) and weight \(p-1\) satisfying \(F\equiv 1\pmod p\) componentwise in the Fourier expansion. If \(p>n+2\) is a regular prime, i.e. \(p\) does not divide the class number of \(\mathbb{Q} (e^{2 \pi i/p})\), then the Siegel-Eisenstein series \(E^{(n)}_{p-1}\) possesses this property. The author shows that \(p=16\;843\) and \(p=2\;124\;679\) are the only primes in the range \(3\leq p<4 \cdot 10^6\) satisfying \(E^{(2)}_{p-1} \not\equiv 1\pmod p\). In these two cases he constructs a different Siegel modular form \(F\) of degree 2 and weight \(p-1\) satisfying \(F\equiv 1\pmod p\). These particular \(F\) arise from the Siegel-Eisenstein series as a Hermitian modular form for \(\mathbb{Q} (\sqrt{-3})\) by restricting the domain of definition to the Siegel half-space. The last section refers to applications to mod \(p\) modular forms.
Mathematics Subject Classification ID: 11F46
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 11B68
Mathematics Subject Classification ID: 11F33
zbMATH DE Number: 1135220
zbMATH Keywords: Fourier coefficients
zbMATH Keywords: Bernoulli numbers
zbMATH Keywords: congruence properties
zbMATH Keywords: mod \(p\) modular forms
zbMATH Keywords: Siegel modular form
zbMATH Keywords: Siegel-Eisenstein series
reviewed by: Aloys Krieg
MaRDI profile type: Publication
cites work: Über die Fourierkoeffizienten der Siegelschen Eisensteinreihen
cites work: Irregular Primes and Cyclotomic Invariants to Four Million
cites work: On Siegel Modular Forms of Genus Two
cites work: On the Ring of Modular Forms of Degree Two Over Z
cites work: An explicit formula for the Fourier coefficients of Siegel-Eisenstein series of degree 3
cites work: Dirichletsche Reihe mit Funktionalgleichung in der Theorie der Modulfunktion 2. Grades
cites work: The Maaß space on the Hermitian half-space of degree 2
cites work: Q5341111
cites work: Q4388148
cites work: p-Adic properties of Siegel modular forms of degree 2
cites work: Some congruence properties of Eisenstein series
cites work: On Siegel series for Hermitian forms. II
cites work: Q5341110
cites work: Q5685160
description: scientific article; zbMATH DE number 1135220