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

TL;DR

This paper computes semi-stable models of modular curves $X_0(p^2)$ for odd primes and applies these results to arithmetic problems, including an effective Bogomolov conjecture and Faltings heights.

Contribution

It provides explicit semi-stable models for $X_0(p^2)$ and derives new arithmetic applications, such as effective bounds and height calculations.

Findings

Explicit semi-stable models for $X_0(p^2)$ are constructed.

Arakelov self-intersection numbers are computed.

Effective bounds for the Bogomolov conjecture are established.

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.