A reformulation of the Siegel series and intersection numbers

Abstract

In this paper, we will explain a conceptual reformulation and inductive formula of the Siegel series. Using this, we will explain that both sides of the local intersection multiplicities of Gross and Keating (Invent Math 112(225–245):2051, 1993) and the Siegel series have the same inherent structures, beyond matching values. As an application, we will prove a new identity between the intersection number of two modular correspondences over $$\mathbb {F}_p$$ and the sum of the Fourier coefficients of the Siegel-Eisenstein series for $$\mathrm {Sp}_4/\mathbb {Q}$$ of weight 2, which is independent of $$p \left( > 2\right) $$ . In addition, we will explain a description of the local intersection multiplicities of the special cycles over $$\mathbb {F}_p$$ on the supersingular locus of the ‘special fiber’ of the Shimura varieties for $$\mathrm {GSpin}(n,2), n\le 3$$ in terms of the Siegel series directly.