Intersections of Hecke correspondences on modular curves

TL;DR

This paper computes arithmetic intersections of Hecke correspondences on modular curves and relates them to derivatives of Siegel Eisenstein series, establishing a link between geometric intersection theory and automorphic forms.

Contribution

It provides a new identity connecting arithmetic intersection numbers on Rapoport--Zink spaces with derivatives of local representation densities of quadratic forms.

Findings

Derived explicit formulas for intersection numbers on modular curves.

Linked intersection theory with derivatives of Siegel Eisenstein series.

Established a precise identity involving local representation densities.

Abstract

We compute the arithmetic intersections of Hecke correspondences on the product of integral model of modular curve $\mathcal{X}_0(N)$ and relate it to the derivatives of certain Siegel Eisenstein series when $N$ is odd and squarefree. We prove this by establishing a precise identity between the arithmetic intersection numbers on the Rapoport--Zink space associated to $\mathcal{X}_0(N)^{2}$ and the derivatives of local representation densities of quadratic forms.