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

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arxiv 1802.06968 v3 pith:RAQSAOXH submitted 2018-02-20 math.NT

classification math.NT
keywords arithmeticcurvesmodelsmodularapplicationscomputegivesemi-stable
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abstract

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 of the relative dualising 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.

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