Higher Siegel-Weil formulas over function fields

Abstract: The Siegel-Weil formula relates the integral of a theta function along a classical group H to a special value of a Siegel-Eisenstein series on another group G. Kudla proposed an "arithmetic analogue" of the Siegel-Weil formula, relating intersection numbers of special cycles on Shimura varieties for H to the first derivative at a special value of a Siegel-Eisenstein series on G. We study a function field analogue of this problem in joint work with Zhiwei Yun and Wei Zhang. We define special cycles on moduli stacks of unitary shtukas, construct associated virtual fundamental classes, and relate their degrees to the derivatives to *all* orders of Siegel-Eisenstein series. The results can be seen as “higher derivative” analogues of the Kudla-Rapoport Conjecture. A key to the proof is a categorification of local density formulas for Fourier coefficients of Eisenstein series, and a parallel categorification of the degrees of virtual fundamental classes of special cycles, in terms of a global variant of Springer theory.

algebraic geometrynumber theory

Audience: researchers in the topic

UCSB Seminar on Geometry and Arithmetic