### 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 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.

Original language | English |
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Pages (from-to) | 1757-1826 |

Number of pages | 70 |

Journal | Mathematische Annalen |

Volume | 377 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - 2020 Aug 1 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Mathematische Annalen*,

*377*(3-4), 1757-1826. https://doi.org/10.1007/s00208-020-01999-2