A p-adic criterion for Lehmer's conjecture

TL;DR

This paper develops a p-adic analogue of a known complex height distribution result, providing lower bounds for algebraic numbers based on the distribution of their conjugates in p-adic fields, and proves Lehmer's conjecture for certain algebraic numbers.

Contribution

It introduces a p-adic criterion linking the height of algebraic numbers to the distribution of their conjugates, advancing the understanding of Lehmer's conjecture.

Findings

Established a p-adic analogue of conjugate distribution near the unit circle.

Derived lower bounds for the height based on conjugate distribution in p-adic fields.

Proved Lehmer's conjecture for algebraic numbers with many conjugates in a p-adic extension.

Abstract

For a non-zero algebraic number $\alpha$ of degree $d$, let $h(\alpha)$ denote its logarithmic Weil height. It is known that when $h(\alpha)$ is small, and $d$ is large, the conjugates of $\alpha$ are clustered near the unit circle and have angular equidistribution in the complex plane about the origin. In this paper, we establish a $p$-adic analogue of this result by obtaining lower bounds for $h(\alpha)$ in terms of the number of its conjugates that lie in a finite extension of $\mathbb{Q}_p$, for some prime $p$. As a consequence, we prove Lehmer's conjecture for all $\alpha$ such that $\gg \sqrt{d\log d}$ many of its conjugates lie in a finite extension of $\mathbb{Q}_p$.