Generalized Lehmer conjecture for abelian varieties

Let $A$ be an abelian variety over a number field $K$, and let ${\mathcal L}$ be a symmetric ample line bundle on $A$. For a point $P\in A(\overline{K})$, write $D=[K(P):K]$ and let $g_0(P)$ denote the dimension of the smallest algebraic subgroup of $A$ containing $P$.

Generalized Lehmer conjecture. There exists a positive constant $c=c(A,K,\mathcal L)$ such that

$$\widehat h(P)\geq cD^{-1/g_0(P)}$$

for all nontorsion points $P\in A(\overline{K})$.

This is a height lower-bound problem for nontorsion points on abelian varieties, generalizing Lehmer-type conjectures for other algebraic groups. The source identifies it as an open problem; the theorem proved in the paper establishes a different uniform positive lower bound for points over the maximal abelian extension.