  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 [GK93] 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 Fp and the sum of the Fourier coefficients of the
Siegel-Eisenstein series for Sp_4 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.

Number Theory