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arXiv · 2304.10696

Arithmetic Siegel-Weil formula on $\mathcal{X}_{0}(N)$

Baiqing Zhu
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Abstract

We establish the arithmetic Siegel-Weil formula on the modular curve $\mathcal{X}_{0}(N)$ for arbitrary level $N$, i.e., we relate the arithmetic degrees of special cycles on $\mathcal{X}_{0}(N)$ 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 $\mathcal{X}_{0}(N)$ 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 [SSY22]. This local identity is proved by relating it to an identity in one dimension higher, but at hyperspecial level.

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Baiqing Zhu. 2023-04-21. Arithmetic Siegel-Weil formula on $\mathcal{X}_{0}(N)$. https://doi.org/10.2140/ant.2025.19.1771

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