A Lehmer-type lower bound for the canonical height on elliptic curves over function fields (Q6556228)

In this article the author gives an explicit lower bound for the height of any non-torsion point on an elliptic curve over a function field. The bound is in terms of the height of the \(j\)-invariant of the curve, and the degree of the field of definition of the point over the field of definition of the curve. Specifically, let \(F/k\) is an extension of an algebraic closed field \(k\) of characteristic different from \(2\) and \(3\) of transcendence degree \(1\), let \(E/F\) be an elliptic curve which is not isotrivial, and let \(P\in E(K)\) be a point of infinite order, where \(K/F\) is a finite extension. Then Silverman proves\N\[\N\widehat{h}_{E/F}(P)\geq \frac{1}{10500\cdot h_{F}(j_E )^2 \cdot [K:F]^2}.\N\]\N\NThe broader context of this is the elliptic analogue of Lehmer's conjecture, which posits a similar lower bound for the Néron-Tate height of an algebraic point of infinite order on an elliptic curve defined over a number field. Specifically, the conjecture is that for \(E/\mathbb{Q} \) an elliptic curve, there is a constant \(C(E)\) such that for all \(P\in E(K)\) of infinite order defined over a finite extension \(K/\mathbb{Q}\),\N\[\N\widehat{h}_E (P)\geq \frac{C(E)}{[K:\mathbb{Q}]}.\N\]\NSilverman's theorem is proved by a detailed analysis of the local heights of \(P\).