The Siegel-Weil formula in geometry and arithmetic

Abstract

The present paper is an extended version of the lecture notes of a course given by the first author at the summer school on Formulas of Siegel and Weil (Bielefeld, September 2025). We survey three perspectives on the Siegel-Weil formula: classical, geometric, and arithmetic. We first recall the Siegel-Weil formula for elliptic theta series arising from positive definite lattices. We next discuss the higher genus case for lattices of arbitrary signature from an adelic viewpoint. After introducing orthogonal Shimura varieties, we present the geometric Siegel-Weil formula, in which the generating series of volumes of special cycles is shown to be an Eisenstein series. We conclude with the (partly conjectural) arithmetic Siegel-Weil formula, relating degrees of special cycles in arithmetic Chow groups to central derivatives of Eisenstein series.