Arithmetic Siegel-Weil formula on ${\mathcal{X}}_0(N)$

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 [SSY22]. This local identity is proved by relating it to an identity in one dimension higher, but at hyperspecial level.