Analogues of Lehmer's conjecture in positive characteristic

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 $\phi:A\to\text{End}(\mathbb{G}_a)$ be a Drinfeld $A$-module of rank $r$ defined over a finite extension $L$ of $K$ and $\hat{h}_{\phi}$ its canonical height. Given a non-torsion point $\alpha$ of $\phi$ of degree $d$ over $K$, we prove that $\hat{h}_{\phi}(\alpha)\ge 1/d$. 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.