Siegel Eisenstein series of arbitrary level and theta series
type: Item
instance of: scholarly article
title: Siegel Eisenstein series of arbitrary level and theta series
zbMATH Open document ID: 0870.11028
DOI: 10.1007/BF02940806
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1997-02-13T00:00:00Z
review text: Let \(f\) be a Siegel modular form of genus \(n\) and level \(q\) which does not vanish at all zero-dimensional cusps. Generalizing an observation of the reviewer, the author shows that \(f\) is uniquely determined by its values at the zero-dimensional cusps if the weight of \(f\) is large enough and if \(f\) is an eigenfunction of the Hecke operator \((T(p))^m\) for some \(m\geq 1\) and for at least one prime number \(p\equiv\pm 1\bmod q\). This implies that each \(f\) of the type above is a linear combination of the transformed Eisenstein series \(E_r\mid L\) \((L\in\text{Sp} (n,\mathbb{Z})\); \(r\) sufficiently large). In particular, the Siegel mean value \(f\) of the theta series of the genus of a positive definite even integral quadratic form satisfies the assumptions of the above theorem, and this yields a new proof of (a restricted version) of the analytic form of Siegel's Hauptsatz. The author also investigates the converse problem to express Eisenstein series as linear combinations of theta series and proves; A linear combination \(E\) of Eisenstein series can be expressed as a linear combination of suitable theta series if and only if the values of \(E\) at the zero-dimensional cusps satisfy the same relations which are a priori known to hold for the theta functions.
Mathematics Subject Classification ID: 11F46
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11F25
Mathematics Subject Classification ID: 11E45
zbMATH DE Number: 977868
zbMATH Keywords: Siegel Eisenstein series
zbMATH Keywords: Siegel modular form
zbMATH Keywords: zero-dimensional cusps
zbMATH Keywords: eigenfunction of the Hecke operator
zbMATH Keywords: Siegel mean value
zbMATH Keywords: theta series
zbMATH Keywords: genus
zbMATH Keywords: even integral quadratic form
zbMATH Keywords: Siegel's Hauptsatz
author: Eberhard Freitag
reviewed by: Jürgen Elstrodt
MaRDI profile type: Publication
cites work: Action of Hecke operator \(T(p)\) on theta series
cites work: BEHAVIOR OF THETA SERIES OF DEGREE $ N$ UNDER MODULAR SUBSTITUTIONS
cites work: Eine Charakterisierung der Eisenstein-Reihe zur Siegelschen Modulgruppe. (A characterization of the Eisenstein series for the Siegel modular group)
cites work: Q3916637
cites work: Q3963086
cites work: Singular modular forms and theta relations
description: scientific article; zbMATH DE number 977868