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

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 $\mathbb{Q}$.

arXiv comment17 pages

References in corpus (6)

• Independence of rational points on twists of a given curve
• Selmer varieties for curves with CM Jacobians
• Uniform bounds for the number of rational points on curves of small Mordell--Weil rank
• An effective Chabauty-Kim theorem
• Functional transcendence for the unipotent Albanese map
• Unlikely intersections and the Chabauty-Kim method over number fields

Cited by in corpus (2)

• Quadratic Chabauty and rational points I: p-adic heights
• Functional transcendence for the unipotent Albanese map