Lang–Silverman conjecture for canonical heights on abelian varieties

Let $g\geq 1$, let $k$ be a number field, let $A/k$ be an abelian variety of dimension $g$, let $\mathcal{D}\in\operatorname{Div}(A)$ be an ample divisor, and let $P\in A(k)$ such that

$$\mathbb{Z}\cdot P=\{mP\mid m\in\mathbb{Z}\}$$

is Zariski-dense in $A$. Write $\widehat{h}_{A,\mathcal{D}}$ for the Néron–Tate height associated with $\mathcal{D}$ and $h_{\mathrm{F}}(A/k)$ for the relative Faltings height. Lang–Silverman's conjecture. For every number field $k$ and every $g\geq 1$, there is a positive constant $c(k,g)$ such that

$$\widehat{h}_{A,\mathcal{D}}(P) \geq c(k,g)\,\max\Big\{1,h_{\mathrm{F}}(A/k)\Big\}.$$

This generalizes Lang's elliptic-curve conjecture to higher-dimensional abelian varieties. The paper studies results toward the assertion, including lower bounds for families in which suitable auxiliary heights remain bounded, but does not establish the conjecture in general.