Toward a generalization of Lehmer's problem to adelic curves

TL;DR

This paper explores extending Lehmer's problem, originally formulated for algebraic numbers, to the broader context of finitely generated fields over the rational numbers, aiming to understand its implications in adelic curves.

Contribution

It introduces a framework for generalizing Lehmer's problem to adelic curves and finitely generated fields over , expanding the scope of the original problem.

Findings

Proposes a new formulation of Lehmer's problem for adelic curves.

Analyzes the properties of heights in finitely generated fields.

Suggests potential avenues for future research in number theory and algebraic geometry.

Abstract

In this short note, we investigate the generalization of Lehmer's problem to finitely generated fields over $\mathbb{Q}$.