2023/04/21 by Baiqing Zhu, Zhu, Baiqing
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2304.10696
openalex publication_date 2023/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 [SSY22]. This local identity is proved by relating it to an identity in one dimension higher, but at hyperspecial level.