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In this paper, we proved an arithmetic Siegel-Weil formula and the modularityof some arithmetic theta function on the modular curve $X_0(N)$ when $N$ issquare free. In the process, we also construct some generalized Delta functionfor $\Gamma_0(N)$ and proved some explicit Kronecker limit formula forEisenstein series on $X_0(N)$.

Arithmetic Siegel-Weil formula on $X_{0}(N)$