Arithmetic Siegel-Weil formula on $X_0(N)$

In this paper, we proved an arithmetic Siegel-Weil formula and the modularity of some arithmetic theta function on the modular curve $X_0(N)$ when $N$ is square free. In the process, we also construct some generalized Delta function for ${\mathrm{\Gamma }}_0(N)$ and proved some explicit Kronecker limit formula for Eisenstein series on $X_0(N)$.