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Analogues of Lehmer's conjecture in positive characteristic

Amilcar Pacheco
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Abstract

Let $C$ be a smooth projective irreducible curve defined over a finite field $\mathbb{F}_q$ and $K=\mathbb{F}_q(C)$. Let $A\subset K$ be the ring of functions regular outside a fixed place $\infty$ of $K$. Let $ϕ:A\to\text{End}(\mathbb{G}_a)$ be a Drinfeld $A$-module of rank $r$ defined over a finite extension $L$ of $K$ and $\hat{h}_ϕ$ its canonical height. Given a non-torsion point $α$ of $ϕ$ of degree $d$ over $K$, we prove that $\hat{h}_ϕ(α)\ge 1/d$. A similar statement is proved for the canonical height of a point of infinite order of a non-constant semi-stable elliptic curve defined over $K$, with the absolute constant 1 replaced by a constant depending on the elliptic curve.

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Amilcar Pacheco. 2003-05-02. Analogues of Lehmer's conjecture in positive characteristic. https://arxiv.org/abs/math/0011102

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