Lehmer problem and Drinfeld modules

TL;DR

This paper establishes a new lower bound estimate for the canonical height on a family of Drinfeld modules, extending previous results and focusing on precision related to key parameters like height, degree, and rank.

Contribution

It introduces a Dobrowolski-style lower bound for canonical heights of Drinfeld modules, including complex multiplication cases, with improved parameter precision.

Findings

Derived a lower bound estimate for canonical height on Drinfeld modules

Extended previous results to modules with complex multiplication

Provided estimates considering both separable and inseparable degrees

Abstract

We propose a lower bound estimate in Dobrowolski's style of the canonical height on a certain family of Drinfeld modules of characteristic 0, including under some hypothesis on their degree and their base field, the complex multiplication case, extending so a previous result of L. Denis on Carlitz modules. Our study is focused on the highest possible level of precision on the parameters involved with rapport to the main values which characterize the Drinfeld module (height, base field degree and rank) and it provides an estimate in function of both separable and inseparable degree of the algebraic points.