An effective Arakelov-theoretic version of the hyperbolic isogeny theorem

TL;DR

This paper employs Arakelov theory to provide an effective version of Mochizuki's finiteness theorem for hyperbolic curves over algebraic closures of rationals, enhancing the understanding of their isomorphism classes.

Contribution

It introduces an Arakelov-theoretic approach to make Mochizuki's finiteness theorem effective, offering explicit bounds and computational aspects.

Findings

Proves an effective finiteness result for hyperbolic curves with fixed Euler characteristic.

Utilizes Arakelov theory to derive explicit bounds on isomorphism classes.

Extends Mochizuki's original finiteness theorem to an effective setting.

Abstract

For an integer $e$ and hyperbolic curve $X$ over $\overline{\mathbb Q}$, Mochizuki showed that there are only finitely many isomorphism classes of hyperbolic curves $Y$ of Euler characteristic $e$ with the same universal cover as $X$. We use Arakelov theory to prove an effective version of this finiteness statement.