A reformulation of the Siegel series and intersection numbers

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 F_p and the sum of the Fourier coefficients of the Siegel-Eisenstein series for Sp ₄/ Q of weight 2, which is independent of p(> 2). In addition, we will explain a description of the local intersection multiplicities of the special cycles over F_p on the supersingular locus of the ‘special fiber’ of the Shimura varieties for GSpin (n, 2) , n≤ 3 in terms of the Siegel series directly.