Energy integrals over local fields and global height bounds

TL;DR

This paper addresses energy minimization in local fields and applies these results to improve lower bounds on the Weil height in certain algebraic number fields, combining local and archimedean analyses.

Contribution

It introduces new energy minimization techniques for local fields and enhances existing bounds on the Weil height in fields with mixed splitting conditions.

Findings

Improved lower bounds for Weil height in totally p-adic fields.

Unified bounds combining archimedean and non-archimedean places.

New energy minimization solutions for local field problems.

Abstract

We solve an energy minimization problem for local fields. As an application of these results, we improve on lower bounds set by Bombieri and Zannier for the limit infimum of the Weil height in fields of totally p-adic numbers and generalizations thereof. In the case of fields with mixed archimedean and non-archimedean splitting conditions, we are able to combine our bounds with similar bounds at the archimedean places for totally real fields.