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

Baiqing Zhu

Vol. 19 (2025), No. 9, 1771–1822
Abstract

We establish the arithmetic Siegel–Weil formula on the modular curve 𝒳0(N) for arbitrary level N, i.e., we relate the arithmetic degrees of special cycles on 𝒳0(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 𝒳0(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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