# Analogues of Lehmer's conjecture in positive characteristic

Research paper by **Amilcar Pacheco**

Indexed on: **02 May '03**Published on: **02 May '03**Published in: **Mathematics - Number Theory**

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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
$\phi:A\to\text{End}(\mathbb{G}_a)$ 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.