Diophantine decidability for curves and Grothendieck's section conjecture

TL;DR

This paper presents an algorithm to determine the existence of rational points on certain algebraic curves, contingent on the truth of Grothendieck's section conjecture, linking Diophantine problems with deep conjectures in algebraic geometry.

Contribution

It establishes a conditional algorithm for deciding rational points on curves of genus at least two based on Grothendieck's section conjecture.

Findings

Algorithm exists under the assumption of the section conjecture

Connects Diophantine decidability with a major conjecture in algebraic geometry

Provides a new approach to rational point problems on curves

Abstract

Let $X$ be a smooth, projective, geometrically irreducible curve of genus at least two defined over a number field $K$. We prove that there is an algorithm that determines whether $X$ has a $K$-rational point if Grothendieck's section conjecture holds for $X$.