A reformulation of the Siegel series and intersection numbers

2018/05/04 by Cho, Sungmun, Yamauchi, Takuya
#11F30 #11F46 #11G18 #14C17 #14G35 #14J15 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1805.01666

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 [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 Sp₄ 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.

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