Intersection theory on Shimura surfaces

TL;DR

This paper extends Kudla's program by analyzing intersection multiplicities on Shimura surfaces, relating them to Fourier coefficients of Hilbert modular forms, and providing new results in higher-dimensional cases.

Contribution

It proves new results on intersections of special cycles on Shimura surfaces, linking these to Fourier coefficients of modular forms, advancing Kudla's conjectural framework.

Findings

Intersection multiplicities correspond to Fourier coefficients of Hilbert modular forms.

Established relations between special cycle intersections and automorphic forms.

Extended Kudla's program to higher-dimensional Shimura varieties.

Abstract

Kudla has proposed a general program to relate arithmetic intersection multiplicities of special cycles on Shimura varieties to Fourier coefficients of Eisenstein series. The lowest dimensional case, in which one intersects two codimension one cycles on the integral model of a Shimura curve, has been completed by Kudla-Rapoport-Yang. In the present paper we prove results in a higher dimensional setting. On the integral model of a Shimura surface we consider the intersection of a Shimura curve with a codimension two cycle of complex multiplication points, and relate the intersection to certain cycles classes constructed by Kudla-Rapoport-Yang. As a corollary we deduce that our intersection multiplicities appear as Fourier coefficients of a Hilbert modular form of half-integral weight.