A Banach space determined by the Weil height

TL;DR

This paper constructs a Banach space from the Weil height on algebraic units modulo roots of unity, revealing its isometric isomorphism to a subspace of L1 on a totally disconnected space, linking number theory and functional analysis.

Contribution

It demonstrates that the completion of the Weil height metric space forms a Banach space and identifies its isometric isomorphism to a subspace of L1 on a specific topological space.

Findings

The metric completion is a Banach space over the reals.

The Banach space is isometrically isomorphic to a subspace of L1.

The space relates to the Galois group invariance.

Abstract

The absolute logarithmic Weil height is well defined on the group of units of the algebraic closure of the rational numbers, modulo roots of unity, and induces a metric topology on this group. We show that the completion of this metric space is a Banach space over the field of real numbers. We further show that this Banach space is isometrically isomorphic to a co-dimension one subspace of L1 of a certain totally disconnected, locally compact space, equipped with a certain measure satisfying an invariance property with respect to the absolute Galois group.