Energy integrals and small points for the Arakelov height

TL;DR

This paper investigates the minimal positive values of the Arakelov height on the projective line, characterizes equality cases, and solves energy minimization problems to establish lower bounds in specific number fields.

Contribution

It identifies the smallest positive Arakelov height, characterizes equality cases, and solves energy minimization problems to derive new lower bounds in number fields.

Findings

Identified the minimal positive Arakelov height value.

Characterized all cases where the minimum is attained.

Established lower bounds for the Arakelov height in certain number fields.

Abstract

We study small points for the Arakelov height on the projective line. First, we identify the smallest positive value taken by the Arakelov height, and we characterize all cases of equality. Next we solve several archimedean energy minimization problems with respect to the chordal metric on the projective line, and as an application, we obtain lower bounds on the Arakelov height in fields of totally real and totally p-adic numbers.