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

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