  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.
