On the regularized Siegel-Weil formula
type: Item
instance of: scholarly article
title: On the regularized Siegel-Weil formula
zbMATH Open document ID: 1022.11020
author: Atsushi Ichino
publication date: 2001-10-07T00:00:00Z
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11F46
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 1653375
zbMATH Keywords: Weil representation
zbMATH Keywords: quadratic forms
MaRDI profile type: Publication
full work available at URL: crll.2001.076
OpenAlex ID: W2084484005
cites work: Q4011189
DOI: 10.1515/CRLL.2001.076
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published in: Journal für die Reine und Angewandte Mathematik
review text: The Siegel-Weil formula is an identity between an integral of a theta function over a certain orthogonal group and the special value of Eisenstein series defined on a symplectic group. In this form it is due to \textit{A. Weil} [Acta Math. 113, 1-87 (1965; Zbl 0161.02304); ibid. 111, 143-211 (1964; Zbl 0203.03305)]. It specializes to several theorems of Siegel's concerning the representation of integers or forms by quadratic or Hermitian forms.NEWLINENEWLINENEWLINELet \(k\) be an algebraic number field. Suppose that the quadratic space on which the orthogonal group is defined is of dimension \(m\) and \(r\) is its Witt index. Suppose the symplectic group is \(Sp_{2n}\). Then the Eisenstein series is convergent if \(m>2n+2\). The integral of the theta function is convergent if \(m-r>n +1\). The Eisenstein series can be analytically continued and so given a meaning where it is not directly defined, except where there are poles. Also the integral of the theta series can be regularized by a method due to Kudla and Rallis in which the isotropic part of the orthogonal group is kept under control. NEWLINENEWLINENEWLINEThe author then proves that a Siegel-Weil formula holds if \(n+1<m\leq 2n+2\) and \(m-r\leq n+1\) using these extensions of the usual concepts. The proof makes use of delicate refinements of techniques used in a series of papers by Ikeda (here especially Fourier-Jacobi coefficients) and by Kudla and Rallis. There is no restriction on the number field \(k\).
reviewed by: Samuel James Patterson
description: scientific article; zbMATH DE number 1653375