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arXiv · 2305.09462

Chabauty--Kim, finite descent, and the Section Conjecture for locally geometric sections

L. Alexander Betts·Theresa Kumpitsch·Martin Lüdtke
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Abstract

Let $X$ be a smooth projective curve of genus $\geq2$ over a number field. A natural variant of Grothendieck's Section Conjecture postulates that every section of the fundamental exact sequence for $X$ which everywhere locally comes from a point of $X$ in fact globally comes from a point of $X$. We show that $X/\mathbb{Q}$ satisfies this version of the Section Conjecture if it satisfies Kim's Conjecture for almost all choices of auxiliary prime $p$, and give the appropriate generalisation to $S$-integral points on hyperbolic curves. This gives a new "computational" strategy for proving instances of this variant of the Section Conjecture, which we carry out for the thrice-punctured line over $\mathbb{Z}[1/2]$.

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L. Alexander Betts, Theresa Kumpitsch, Martin Lüdtke. 2023-05-16. Chabauty--Kim, finite descent, and the Section Conjecture for locally geometric sections. https://arxiv.org/abs/2305.09462

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