Heights on Groups and Small Multiplicative Dependencies

TL;DR

This paper extends the concept of Weil height to finitely generated subgroups of algebraic numbers, linking it to geometric volume and deriving bounds on multiplicative dependencies.

Contribution

It introduces a novel generalization of the Weil height for subgroups and establishes a geometric interpretation involving volume, providing new bounds on algebraic dependencies.

Findings

Height of a subgroup equals the volume of a convex symmetric set

Bound on the norm of integer vectors for multiplicative dependencies

Generalization connects algebraic and geometric properties

Abstract

We generalize the absolute logarithmic Weil height from elements of the multiplicative group of algebraic numbers modulo torsion, to finitely generated subgoups. The height of a finitely generated subgroup is shown to equal the volume of a certain naturally occurring, convex, symmetric subset of Euclidean space. This connection leads to a bound on the norm of integer vectors that give multiplicative dependencies among finite sets of algebraic numbers.