Small points and free abelian groups

TL;DR

This paper investigates the relationship between height functions and the structure of free abelian groups in number theory, providing explicit counterexamples to a previously assumed converse statement.

Contribution

It demonstrates that the converse of a known height-group structure relationship does not hold by explicitly constructing counterexamples.

Findings

Height functions do not always attain arbitrarily small positive values.

Groups $F^*$ and $E(F)$ can fail to be free abelian modulo torsion.

Counterexamples show the failure of the converse statement.

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 N\'eron-Tate height, induces a norm on $F^*$ modulo torsion, respectively on $E(F)$ modulo torsion. The groups $F^*$ 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.