A reformulation of the Siegel series and intersection numbers

TL;DR

This paper introduces a new conceptual reformulation of the Siegel series, revealing deep structural connections with local intersection multiplicities and deriving identities linking intersection numbers with Fourier coefficients of Siegel-Eisenstein series.

Contribution

It provides a novel reformulation and inductive formula for the Siegel series, establishing structural parallels with intersection multiplicities and deriving new identities involving modular correspondences.

Findings

Established a new identity between intersection numbers and Fourier coefficients.

Provided a structural explanation linking Siegel series and intersection multiplicities.

Described local intersection multiplicities on supersingular loci in terms of Siegel series.

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 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.