Arakelov Self-intersection numbers of minimal regular models of modular curves $X_0(p^2)$

TL;DR

This paper derives an asymptotic formula for the Arakelov self-intersection number of the dualizing sheaf on minimal regular models of modular curves $X_0(p^2)$, aiding in effective bounds related to the Bogomolov conjecture.

Contribution

It provides the first asymptotic expression for these intersection numbers and applies this to prove an effective Bogomolov conjecture for certain modular curves.

Findings

Asymptotic formula for self-intersection numbers derived

Effective bounds for Bogomolov conjecture established

Bound on stable Faltings height obtained in related work

Abstract

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 $X_0(p^2)$ over $\mathbb{Q}$. The computation of the self-intersection numbers are used to prove effective Bogolomov conjecture for the semi-stable models of modular curves $X_0(p^2)$ and obtain a bound on the stable Faltings height for those curves in a companion article arXiv:1802.06968.