The sparse small points conjecture for principally polarised abelian schemes

Let $S/K$ be a variety, let $A/S$ be a principally polarised abelian scheme, and let $\sigma\in A(S)$ be a section of infinite order. For an integer $d>0$ and $\epsilon\geq0$, define

$${\mathbf{T}}_\epsilon(d)=\{p\in S(\overline{K})\mid[\kappa(p):\mathbb{Q}]\leq d\text{ and }\hat{\operatorname{h}}(\sigma(p))\leq\epsilon\}.$$

Say that $\sigma$ has sparse $\delta$-small $d$-points if for every $0\leq\epsilon<\delta$, the set ${\mathbf{T}}_\epsilon(d)$ is not Zariski dense in $S$. Sparse small points conjecture. For every principally polarised abelian scheme $A/S$, every infinite-order section $\sigma\in A(S)$, and every $d>0$, there exists $\delta>0$ such that $\sigma$ has $\delta$-small $d$-points. This is presented as a strengthening of the torsion-sparsity conjecture and is not proved in general; the paper establishes special cases.