Fields with few small points

TL;DR

This paper investigates finiteness properties of algebraic cycles over fields with large Northcott numbers, using refined height bounds and applying to canonical heights, CM-points, and specializations.

Contribution

It introduces a refinement of Northcott's theorem and demonstrates finiteness results for cycles over fields with sufficiently large Northcott numbers, with explicit constructions and applications.

Findings

Finiteness of cycles with bounded degree over certain fields.

Explicit constructions of fields with large Northcott numbers.

Applications to canonical heights and CM-points.

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.