Intersections of Hecke correspondences on modular curves
Qiao He, Baiqing Zhu
TL;DR
This work connects arithmetic intersections of Hecke correspondences on the integral model X_0(N) with derivatives of Siegel Eisenstein series by a careful local-to-global analysis. It reduces global intersection problems to precise local identities on Rapoport–Zink spaces and local densities of quadratic lattices, then proves a pair of arithmetic–analytic difference formulas and couples them via a blow-up regularization and a geometric automorphism. The main results verify a Kudla–Rapoport type conjecture in the odd squarefree N setting and relate triple intersections to derivatives of incoherent Eisenstein series, enriching the arithmetic Siegel–Weil program. The techniques yield a systematic framework for comparing derived intersection theory with density derivatives, with applications to CM cycles and to global arithmetic Siegel–Weil formulas, potentially guiding further generalizations to broader Shimura settings.
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.