On Koecher-Maass series of Siegel modular forms
type: Item
instance of: scholarly article
title: On Koecher-Maass series of Siegel modular forms
zbMATH Open document ID: 1041.11036
author: Winfried Kohnen
author: J. Sengupta
published in: Mathematische Zeitschrift
publication date: 2004-06-17T00:00:00Z
review text: Here we see a new proof that the Koecher-Maass series of a Siegel modular form \(f\) (for the full integral symplectic group) gives rise to a function \(D^*_f(s)\) having an analytic continuation to the whole complex plane and obeying a functional equation. This is a classical result, but the proof offered here is by constructing a holomorphic kernel function for \(f\mapsto D^*_f(s)\). More precisely the authors exhibit a function \(F_k(Z;s)\) (for \(s\in{\mathbb C}\) and \(Z\) in the Siegel upper half-plane) such that in a suitable region \[ \left\langle F_k(\cdot;s), f^*\right\rangle=C_{n,k}D^*_f(s), \] where \(f\) is a weight \(k\) form for \(\text{ Sp}_n\), \(f^*\) is obtained by taking the complex conjugate of the Fourier coefficients and \(\langle\cdot,\cdot\rangle\) is the Petersson scalar product. The results are deduced from analogous properties of \(F_k\), which are fairly easy consequences of its definition by a series. The proof seems on the face of it to be very different from the classical proof of these results. There is a mild restriction: at least as given here, the new proof is only valid for \(k>3(n+1)\) (and even).
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11F46
zbMATH DE Number: 2075722
zbMATH Keywords: modular form
zbMATH Keywords: Dirichlet series
zbMATH Keywords: functional equation
reviewed by: G. K. Sankaran
MaRDI profile type: Publication
full work available at URL: s002090100311
OpenAlex ID: W1996426135
DOI: 10.1007/S002090100311
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description: scientific article; zbMATH DE number 2075722