Petersson products of singular and almost singular theta series
type: Item
instance of: scholarly article
title: Petersson products of singular and almost singular theta series
zbMATH Open document ID: 1072.11031
published in: Manuscripta Mathematica
publication date: 2005-02-02T00:00:00Z
review text: Let \(F\), \(G\) be Siegel modular forms of degree \(n\) and weight \(k= {m\over 2}\) with \(n> m\) and \(m\equiv 0\pmod 8\). Then \(F\) and \(G\) are square integrable (according to a theorem of Weissauer), and the authors show that the Petersson product \(\langle F,G\rangle\) is equal to the residue of a Dirichlet series of Rankin-Selberg type at \(s= {n\over 2}\) times some constant depending only on \(m\) and \(n\). The same result holds in the case \(k= {n\over 2}\) whenever \(F\) and \(G\) are linear combinations of theta series. Moreover, if \(n\geq m\) and \(F\) is a Siegel modular form of weight \(k= {m\over 2}\) which is a linear combination of theta series and which is a Hecke eigenform, then the scalar product of \(F\) against a theta series is equal to a known constant times the residue of the standard \(L\)-function of \(F\) at \(s= n+1-k\). Application of these beautiful theorems include a remarkable asymptotic orthogonality relation for theta series and several results on theta liftings. In the final section the authors briefly state how to extend the previous results when lattices of level greater than 1 are dealt with, and they give an inner product formula for Siegel modular forms on the Hecke subgroup of level \(N\). -- The proofs of the main results are based on an inner product formula in the adelic representation theoretic setting due to \textit{J.-S. Li} [Am. J. Math. 119, No. 3, 523--578 (1997; Zbl 0871.11036)].
reviewed by: Jürgen Elstrodt
Mathematics Subject Classification ID: 11F46
Mathematics Subject Classification ID: 11E45
Mathematics Subject Classification ID: 11E10
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 11F67
zbMATH DE Number: 2132054
zbMATH Keywords: Petersson product
zbMATH Keywords: theta series
zbMATH Keywords: \(L\)-function
zbMATH Keywords: Siegel modular form
author: Siegfried Böcherer
author: Francesco Ludovico Chiera
MaRDI profile type: Publication
full work available at URL: s00229-004-0500-y
OpenAlex ID: W1975664613
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DOI: 10.1007/S00229-004-0500-Y
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description: scientific article; zbMATH DE number 2132054