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Rational Points on Solvable Curves over ℚ via Non-Abelian Chabauty
International Mathematics Research Notices ( IF 0.9 ) Pub Date : 2021-05-07 , DOI: 10.1093/imrn/rnab141
Jordan S Ellenberg 1 , Daniel Rayor Hast 2
Affiliation  

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 P$^1$ with solvable Galois group, and in particular any superelliptic curve over $\mathbb{Q}$, has only finitely many rational points over $\mathbb{Q}$.

中文翻译:

ℚ 上可解曲线上的有理点通过非阿贝尔 Chabauty

我们研究了在 $\mathbb{Q}$ 上定义的至少两个属的平滑射影曲线的 Selmer 变体,它们在几何上以 CM Jacobian 曲线为主。我们扩展 Coates 和 Kim 的结果以表明 Kim 的非阿贝尔 Chabauty 方法适用于这样的曲线。通过将其与 Bogomolov–Tschinkel 和 Poonen 在无分支对应上的结果相结合,我们推断 P$^1$ 与可解 Galois 群的任何覆盖,特别是 $\mathbb{Q}$ 上的任何超椭圆曲线,只有有限多个$\mathbb{Q}$ 上的理性点。
更新日期:2021-05-07
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