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In this paper, we compute the semi-stable models of modular curves $X_0(p^2)$for odd primes $p > 3$ and compute the Arakelov self-intersection numbers ofthe relative dualising sheaves for these models. We give two arithmeticapplications of our computations. In particular, we give an effective versionof the Bogomolov conjecture following the strategy outlined by Zhang and findthe stable Faltings heights of the arithmetic surfaces corresponding to thesemodular curves.

Semi-stable models of modular Curves $X_0(p^2)$ and some arithmetic applications