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arXiv · 1706.00525

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

Jordan S. Ellenberg·Daniel Rayor Hast
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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 $\mathbb{Q}$.

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Jordan S. Ellenberg, Daniel Rayor Hast. 2017-06-02. Rational points on solvable curves over $\mathbb{Q}$ via non-abelian Chabauty. https://doi.org/10.1093/imrn%2Frnab141

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