On the regularized Siegel-Weil formula
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 \(k\).
scientific article; zbMATH DE number 1653375