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.

中文翻译：

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模块化曲线X 0（p 2）的半稳定模型和一些算术应用

在本文中，我们为奇数素数p > 3计算了模块化曲线X ₀（p ²）的半稳定模型，并为这些模型计算了相对二元滑轮的Arakelov自相交数。我们给出了计算的两个算术应用。尤其是，我们按照Zhang概述的策略给出了Bogomolov猜想的有效版本，并找到了与这些模块化曲线相对应的算术曲面的稳定Faltings高度。