Rational points on solvable curves over $\mathbb{Q}$ via non-abelian Chabauty

TL;DR

This paper extends non-abelian Chabauty methods to certain solvable curves over nd , proving finiteness of rational points for covers of nd  with solvable Galois groups.

Contribution

It generalizes Kim's non-abelian Chabauty approach to curves dominated by CM Jacobian curves and establishes finiteness results for rational points on solvable covers.

Findings

Kim's non-abelian Chabauty applies to these curves.

Finiteness of rational points on solvable covers of nd .

Superelliptic curves over nd  have finitely many rational points.

Abstract

We study the Selmer varieties of smooth projective curves of genus at least two defined over $\mathbb{Q}$ which geometrically dominate a curve with CM Jacobian. We extend a result of Coates and Kim to show that Kim's non-abelian Chabauty method applies to such a curve. By combining this with results of Bogomolov-Tschinkel and Poonen on unramified correspondences, we deduce that any cover of $\mathbf{P}^1$ with solvable Galois group, and in particular any superelliptic curve over $\mathbb{Q}$, has only finitely many rational points over $\mathbb{Q}$.