A Lehmer-Type Lower Bound for the Canonical Height on Elliptic Curves Over Function Fields

Abstract

Let $\mathbb{F}$ be the function field of a curve over an algebraically closed field with $\operatorname{char}(\mathbb{F})\ne2,3$, and let $E/\mathbb{F}$ be an elliptic curve. Then for all finite extensions $\mathbb{K}/\mathbb{F}$ and all non-torsion points $P\in{E(\mathbb{K})}$, the $\mathbb{F}$-normalized canonical height of $P$ is bounded below by \[ \hat{h}_E(P) \ge \frac{1}{10500\cdot h_{\mathbb{F}}(j_E)^{2}\cdot [\mathbb{K}:\mathbb{F}]^{2}}. \]