Trace operator and theta series
type: Item
instance of: scholarly article
title: Trace operator and theta series
zbMATH Open document ID: 0936.11029
author: Jens Funke
author: Rainer Schulze-Pillot
author: Siegfried Böcherer
published in: Journal of Number Theory
publication date: 2000-05-24T00:00:00Z
full work available at URL: 12407.pdf
review text: The authors investigate the space \(M_n^k (M,\chi)\) of Siegel modular forms for \(\Gamma_0^{(n)}(M)\) and appropriate character \(\chi\) and the subspace \(\theta^{(n)} (V,M)\) spanned by the theta series of degree \(n\) of lattices \(L\) of level dividing \(M\) in the quadratic space \((V,q)\). The main question is whether \[ \theta^{(n)} (V,M)\cap M_n^k (N,\chi)= \theta^{(n)} (V,N) \] holds for \(N|M\), where mostly \(M/N\) is a prime. The question is reformulated by means of the trace operator from \(\Gamma_0^{(n)}(M)\) into \(\Gamma_0^{(n)}(N)\): Can the image of a theta series of a lattice of level \(M\) be expressed as a linear combination of theta series of lattices of level \(N\) in the same space? The question is dealt with by two different methods. At first they deal with singular modular forms and use commutation relations with the Siegel \(\varphi\)-operator. The other approach starts from a direct calculation of the Fourier expansion. Both ways lead to the same results.
Mathematics Subject Classification ID: 11F46
Mathematics Subject Classification ID: 11F27
zbMATH DE Number: 1347934
zbMATH Keywords: basis problem
zbMATH Keywords: Siegel modular forms
zbMATH Keywords: theta series
zbMATH Keywords: trace operator
zbMATH Keywords: commutation relations
zbMATH Keywords: Siegel \(\phi\)-operator
reviewed by: Aloys Krieg
MaRDI profile type: Publication
OpenAlex ID: W2012077469
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DOI: 10.1006/JNTH.1999.2404
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description: scientific article; zbMATH DE number 1347934