Arithmetic Siegel–Weil formula on 𝒳0(N)

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 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.