On Profinite Hyperbolicity and Diophantine Geometry

TL;DR

This paper investigates hyperbolicity in profinite groups and its implications for diophantine geometry, linking Galois group properties to solutions of hyperbolic curves over global fields.

Contribution

It introduces a new perspective connecting hyperbolic profinite groups with diophantine problems and explores implications of Galois group structures on rational solutions.

Findings

Grothendieck's section conjecture and Shafarevich's freeness conjecture imply infinitely many solutions for hyperbolic curves.

Proposes reformulating diophantine problems in terms of hyperbolic profinite groups.

Highlights the significance of Galois group properties in diophantine geometry.

Abstract

In this note, we explore the notion of hyperbolicity of topologically finitely generated profinite groups. Some applications to diophantine geometry are suggested and we try to reformulate certain problems in diophantine geometry in terms of hyperbolic profinite groups. Then, we introduce many occasions in which Galois groups are free profinite and try to explore implications of this condition in the world of diophantine geometry. In particular, we prove that, Grothendieck's "section conjecture" plus Shafarevich's "freeness conjecture" imply that hyperbolic curves have infinitely many solutions over the maximal abelian extension of a global field. This makes Mordell's conjecture, which was proved by Faltings, more interesting.