Kudla's modularity conjecture on integral models of orthogonal Shimura varieties

TL;DR

This paper constructs special cycle classes on integral models of orthogonal Shimura varieties and shows they correspond to Fourier coefficients of Siegel modular forms, confirming Kudla's conjecture in this setting.

Contribution

It provides a new construction of special cycle classes on integral models and proves their relation to modular forms, advancing Kudla's modularity conjecture.

Findings

Cycle classes are realized as Fourier coefficients of Siegel modular forms.

The construction confirms Kudla's conjecture for integral models.

Recovers known results on the generic fiber of Shimura varieties.

Abstract

We construct a family of special cycle classes on the regular integral model of an orthogonal Shimura variety, and show that these cycle classes appear as Fourier coefficients of a Siegel modular form. Passing to the generic fiber of the Shimura variety recovers a result of Bruinier and Raum, originally conjectured by Kudla.