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arXiv · 2401.09396

Curves with prescribed rational points

Katerina Santicola
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Abstract

Given a smooth curve $C/\mathbb{Q}$ with genus $\geq 2$, we know by Faltings' Theorem that $C(\mathbb{Q})$ is finite. Here we ask the reverse question: given a finite set of rational points $S\subseteq \mathbb{P}^n(\mathbb{Q})$, does there exist a smooth curve $C/\mathbb{Q}$ contained in $\mathbb{P}^n$ such that $C(\mathbb{Q})=S$? We answer this question in the affirmative by providing an effective algorithm for constructing such a curve.

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Katerina Santicola. 2024-01-17. Curves with prescribed rational points. https://arxiv.org/abs/2401.09396

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