The intersection matrices of $X_0(p^r)$ and some applications

TL;DR

This paper computes intersection matrices for certain modular curves and derives asymptotic formulas for their Arakelov self-intersection numbers, aiding in understanding effective bounds related to the Bogomolov conjecture.

Contribution

It provides explicit intersection matrices for $X_0(p^r)$ with r=3,4 and an asymptotic expression for their Arakelov self-intersection numbers, advancing the study of modular curves.

Findings

Computed intersection matrices for $X_0(p^r)$ with r=3,4.

Derived asymptotic formulas for Arakelov self-intersection numbers.

Facilitated bounds on the stable Faltings height of modular curves.

Abstract

We compute intersection matrices for modular curves of the form $X_0(p^r)$ with $r \in \{3,4\}$ and as an application, we compute an asymptotic expression for the Arakelov self-intersection number of the relative dualizing sheaf of Edixhoven's minimal regular model for the modular curve $X_0(p^r)$ over $\qq$ with $r$ as above. This computation will be useful to understand an effective version of the Bogolomov conjecture for the stable models of modular curves $X_0(p^r)$ with $r \in \{3,4\}$ and obtain a bound on the stable Faltings height for those curves.