Small points and free abelian groups

Abstract

Let $F$ be an algebraic extension of the rational numbers and $E$ an elliptic curve defined over some number field contained in $F$. The absolute logarithmic Weil height, respectively the Néron-Tate height, induces a norm on $F^{\ast }$ modulo torsion, respectively on $E(F)$ modulo torsion. The groups $F^{\ast }$ and $E(F)$ are free abelian modulo torsion if the height function does not attain arbitrarily small positive values. In this paper we prove the failure of the converse to this statement by explicitly constructing counterexamples.

References in corpus (3)

• Small Height and Infinite Non-Abelian Extensions
• Global discrepancy and small points on elliptic curves
• Heights of points with bounded ramification

Cited by in corpus (2)

• Mordell--Weil groups over large algebraic extensions of fields of characteristic zero
• Fields Generated by Finite Rank Subgroups of ${\overline{\mathbb{Q}}}^{\ast }$