Effectivity in Mochizuki's work on the $abc$-conjecture

TL;DR

This paper provides a constructive proof linking Mochizuki's IUT theory to an effective form of the $abc$-conjecture using computable Belyi maps, thereby advancing the understanding of the conjecture's effectiveness.

Contribution

It introduces a constructive proof that reduces the full $abc$-conjecture to a restricted form via computable Belyi maps, connecting it directly to Mochizuki's IUT theory.

Findings

Effective reduction of the $abc$-conjecture to a restricted form.

Constructive proof utilizing computable Belyi maps.

Implication that an effective $abc$-theorem follows from Mochizuki's Theorem 1.10.

Abstract

This note outlines a constructive proof of a proposition in Mochizuki's paper "Arithmetic elliptic curves in general position," making a direct use of computable non-critical Belyi maps to effectively reduce the full $abc$-conjecture to a restricted form. Such a reduction means that an effective $abc$-theorem is implied by Theorem 1.10 of Mochizuki's final IUT paper ("Inter-universal Teichmuller theory IV: log-volume computations and set-theoretic foundations").