Arithmetic Siegel–Weil formula on X₀ (N)

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.