Some Applications of the Hales-Jewett Theorem to Field Arithmetic

TL;DR

This paper applies the Hales-Jewett theorem to field arithmetic, establishing infinite rational points on certain hyperelliptic and elliptic curves over fields with specific Galois group properties.

Contribution

It introduces novel applications of the Hales-Jewett theorem to prove results about rational points on hyperelliptic and elliptic curves over fields with finitely generated Galois groups.

Findings

Hyperelliptic curves over such fields have infinitely many rational points.

Elliptic curves with full 2-torsion over these fields have infinite rank.

Results depend on the field not being finite or of characteristic 2.

Abstract

Let $K$ be a field whose absolute Galois group is finitely generated. If $K$ neither finite nor of characteristic 2, then every hyperelliptic curve over $K$ with all of its Weierstrass points defined over $K$ has infinitely many $K$-points. If, in addition, $K$ is not locally finite, then every elliptic curve over $K$ with all of its 2-torsion rational has infinite rank over $K$. These and similar results are deduced from the Hales-Jewett theorem.