Small points in radical extensions of number fields

2026/07/31 by Andrea Conti, Ilaria Del Corso, Arnaud Plessis +1
Mathematics · #math.NT #msc:11G50 #msc:11R32 #msc:11S15 #msc:11S20 #msc:12F10

paper · pdf

18 pages, comments are welcome!

arxiv created 2026/07/31 · arxiv updated 2026/08/03

Abstract

We study small points in radical extensions of algebraic fields. Given an algebraic extension \mathbbF of ℚ, a finitely generated subgroup Γ⊆ \mathbbF^×, and a rational prime p, we give a general criterion ensuring that \mathbbF(Γ^p-div)∖ Γ^div has the Bogomolov property. This problem is motivated by a conjecture of Rémond, formulated when \mathbbF is a number field, predicting that such radical extensions contain no unexpected small points outside the divisible hull of the group used to generate them. As applications, we obtain new cases of Rémond's conjecture for radical extensions generated by division points with respect to a finite set of primes, recovering and extending previous results of Amoroso and the third author. Our argument is based on a recent result on small points in p-adic Lie extensions.

Citations

• New evidence for Rémond's generalisation of Lehmer's conjecture
• Supersingular primes and Bogomolov property
• On points of small height in infinite extensions
• Bogomolov property and Galois representations
• A new way to tackle a conjecture of Rémond
• Bogomolov Property of some infinite nonabelian extensions of a totally v-adic field
• Small Heights in Large Non-Abelian Extensions
• Minoration de la hauteur de Weil dans un compositum de corps de rayon
• Explicit Small Heights in Infinite Non-Abelian Extensions
• Equidistribution and the heights of totally real and totally p-adic\n numbers