Title: Arithmetic Siegel-Weil formula on $X_{0}(N)$,
Abstract: 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 $\Gamma_0(N)$ and proved some explicit Kronecker limit formula for
Eisenstein series on $X_0(N)$.