Baiqing Zhu

We establish the arithmetic Siegel–Weil formula on the modular curve X0(N) for arbitrary level N, i.e., we relate the arithmetic degrees of special cycles on X_0(N) to the derivatives of Fourier coefficients of a genus-2 Eisenstein series. We prove this formula by a precise identity between the local arithmetic intersection numbers on the Rapoport–Zink space associated to X_0(N) and the derivatives of local representation densities of quadratic forms. When N is odd and square-free, this gives a different proof of the main results in work of Sankaran, Shi and Yang. This local identity is proved by relating it to an identity in one dimension higher, but at hyperspecial level.

For a prime number p>2 and a finite extension F over Q_p, we explain the construction of the difference divisors on the unitary Rapoport–Zink spaces of hyperspecial level over F, and the GSpin Rapoport–Zink spaces of hyperspecial level over associated to a minuscule cocharacter and a basic element b. We prove the regularity of the difference divisors, find the formally smooth locus of both the special cycles and the difference divisors, by a purely deformation-theoretic approach.

For arbitrary level N, we relate the generating series of codimension 2 special cycles on X_0(N) to the derivatives of a genus 2 Eisenstein series, especially the singular terms of both sides. On the analytic side, we use difference formulas of local densities to relate the singular Fourier coefficients of the genus 2 Eisenstein series to the nonsingular Fourier coefficients of a genus 1 Eisenstein series. On the geometric side, we study the reduction of cusps to compute the divisor class of the Hodge bundle and the heights of special divisors. When N is square-free, this gives a different proof of the main results in the works of Du, Yang and Sankaran, Shi, and Yang.

We introduce a “vector valued” version of special cycles on GSpin Rapoport–Zink spaces with almost self-dual level in the context of the Kudla program, with certain linear invariance and local modularity features. They are local analogs of special cycles on GSpin Shimura varieties with almost self-dual parahoric level (e.g. Siegel threefolds with paramodular level). We establish local arithmetic Siegel–Weil formulas relating arithmetic intersection numbers of these special cycles and derivatives of certain local Whittaker functions in any dimension. The proof is based on a reduction formula for cyclic quadratic lattices.

We compute the arithmetic intersections of Hecke correspondences on the product of integral model of modular curve \mathcalX_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 \mathcalX_0(N)^2 and the derivatives of local representation densities of quadratic forms.