Arithmetic Siegel–Weil formula on 𝒳0(N)

Zhu, Baiqing

doi:10.2140/ant.2025.19.1771

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Arithmetic Siegel–Weil formula on $\mathcal X_0(N)$

Baiqing Zhu

Vol. 19 (2025), No. 9, 1771–1822

DOI: 10.2140/ant.2025.19.1771

Abstract

We establish the arithmetic Siegel–Weil formula on the modular curve $𝒳_{0} (N)$ for arbitrary level $N$ , i.e., we relate the arithmetic degrees of special cycles on $𝒳_{0} (N)$ to the derivatives of Fourier coefficients of a genus-2 Eisenstein series. We prove this formula by a precise identity between the local arithmetic intersection numbers on the Rapoport–Zink space associated to $𝒳_{0} (N)$ and the derivatives of local representation densities of quadratic forms. When $N$ is odd and square-free, this gives a different proof of the main results in work of Sankaran, Shi and Yang. This local identity is proved by relating it to an identity in one dimension higher, but at hyperspecial level.

Keywords

modular curves, arithmetic intersection, local densities, Eisenstein series

Mathematical Subject Classification

Primary: 11F27, 11F30, 11F32, 11F41, 14G40

Milestones

Received: 15 December 2023

Revised: 22 June 2024

Accepted: 3 September 2024

Published: 21 July 2025

Authors

Baiqing Zhu
Columbia University New York, NY United States

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Vol. 19, No. 9, 2025