On differential operators and congruences for Siegel modular forms of degree two
type: Item
instance of: scholarly article
title: On differential operators and congruences for Siegel modular forms of degree two
zbMATH Open document ID: 0571.10026
author: Takakazu Satoh
published in: Proceedings of the Japan Academy. Series A
publication date: 1984-01-01T00:00:00Z
review text: Let r be a positive integer. Let f and g be Siegel modular forms of degree two and weight w and \(w+2r\) respectively which are nonzero common eigen functions of all Hecke operators. We denote by \(\lambda\) (m,f) the eigenvalue of the m-th Hecke operator on f. Under some conditions, e.g. all the Fourier coefficients of f and g belong to an algebraic number field K, the author proves the congruences \(m^{2r} \lambda (m,f)\equiv \lambda (m,g) mod {\mathfrak p}\) for all \(m\geq 1\), where \({\mathfrak p}\) is a certain prime ideal of K. The proof uses the Maass operator and Sturm's holomorphic projection to avoid treating the Eisenstein series of weight two. The key point is an integrability property of a \(C^{\infty}\)-map (differentiation followed by holomorphic projection). \{The full details will be published in J. Math. Soc. Japan (to appear).\}
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11F33
zbMATH DE Number: 3910487
zbMATH Keywords: differential operator
zbMATH Keywords: different weight
zbMATH Keywords: Siegel modular forms
zbMATH Keywords: degree two
zbMATH Keywords: eigenvalue
zbMATH Keywords: Hecke operator
zbMATH Keywords: congruences
zbMATH Keywords: Maass operator
zbMATH Keywords: holomorphic projection
MaRDI profile type: Publication
full work available at URL: pjaa.60.339
OpenAlex ID: W2086774167
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DOI: 10.3792/PJAA.60.339
description: scientific article; zbMATH DE number 3910487