Analogues of Lehmer's conjecture in positive characteristic

2000/11/15 by Pacheco, Amilcar
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.math/0011102

Abstract

Let C be a smooth projective irreducible curve defined over a finite field \mathbbF_q and K=\mathbbF_q(C). Let A⊂ K be the ring of functions regular outside a fixed place ∞ of K. Let ϕ:A\toEnd(\mathbbG_a) be a Drinfeld A-module of rank r defined over a finite extension L of K and h_ϕ its canonical height. Given a non-torsion point α of ϕ of degree d over K, we prove that h_ϕ(α)≥ 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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