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arXiv · 2307.00297

Fields with few small points

Nuno Hultberg
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Abstract

Let $X$ be a projective variety over a number field $K$ endowed with a height function associated to an ample line bundle on $X$. Given an algebraic extension $F$ of $K$ with a sufficiently big Northcott number, we can show that there are finitely many cycles in $X_{\bar{\mathbb{Q}}}$ of bounded degree defined over $F$. Fields $F$ with the required properties were explicitly constructed in arXiv:2107.09027 and arXiv:2204.04446, motivating our investigation. We point out explicit specializations to canonical heights associated to abelian varieties and selfmaps of $\mathbb{P}^n$. We apply similar methods to the study of CM-points. As a crucial tool, we introduce a refinement of Northcott's theorem.

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Nuno Hultberg. 2024-04-05. Fields with few small points. https://arxiv.org/abs/2307.00297

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