Intersections of arithmetic Hirzebruch-Zagier cycles

TL;DR

This paper links the intersection multiplicity of arithmetic Hirzebruch-Zagier cycles to the Fourier coefficients of a derivative of a Siegel-Eisenstein series, confirming a conjecture by Kudla and Rapoport.

Contribution

It proves a conjecture relating intersection theory of special cycles to automorphic forms, specifically connecting intersection multiplicities with Fourier coefficients of derivatives of Eisenstein series.

Findings

Established a precise relation between cycle intersections and Fourier coefficients.

Proved Kudla and Rapoport's conjecture in this context.

Enhanced understanding of the arithmetic geometry of special cycles.

Abstract

We establish a close connection between the intersection multiplicity of three arithmetic Hirzebruch-Zagier cycles and the Fourier coefficients of the derivative of a certain Siegel-Eisenstein series at its center of symmetry. Our main result proves a conjecture of Kudla and Rapoport.