Semi-Stable Models of Modular Curves X0(p2) and Some Arithmetic Applications

Abstract

In this paper, we compute the semi-stable models of modular curves X0(p2) for oddprimes p > 3 and compute the Arakelov self-intersection numbers of the relative dualizing sheaves for these models. We give two arithmetic applications of our computations. In particular, we give an effective version of the Bogomolov conjecture following the strategy outlined by Zhang and find the stable Faltings heights of the arithmetic surfaces corresponding to these modular curves.