New evidence for Rémond's generalisation of Lehmer's conjecture

2025/06/23 by Checcoli, Sara, Dill, Gabriel Andreas · 1 citation
#11G50 #11J95 #11R32 #20K15 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2506.18776

Abstract

In this article, we generalise a result of Pottmeyer from the multiplicative group of the algebraic numbers to almost split semiabelian varieties defined over number fields. This concerns a consequence of Rémond's generalisation of Lehmer's conjecture. Namely, for a finite rank subgroup Γ of an almost split semiabelian variety G, we consider the group of rational points of G over a finite extension of the field generated by the saturated closure of Γ, i.e. the division closure of the subgroup generated by Γ and all its images under geometric endomorphisms of G. We show that this becomes a free group after one quotients out the saturated closure of Γ. The proof uses, amongst other ingredients, a criterion of Pottmeyer, which relies on a result of Pontryagin, together with a result from Kummer theory, of which we reproduce a proof by Rémond.

Citations

• A new way to tackle a conjecture of Rémond
• Location of small points on an elliptic curve by an equidistribution argument
• Galois descent in Galois theories
• Une minoration du minimum essentiel sur les varietes abeliennes
• A Lower Bound for the Canonical Height on Abelian Varieties over Abelian Extensions

Cited by

• Small points in radical extensions of number fields

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