Modified André–Pink–Zannier conjecture over a curve

Let $K$ be a field of characteristic $0$, let $S$ be a base curve, and let $\pi\colon \mathcal{A}\to S$ be an abelian scheme that is not isotrivial. Let $\mathcal{A}_{\Gamma}$ denote the specified isogeny orbit, let $\mathcal{V}\subset \mathcal{A}$ be an irreducible subvariety, and let $\mathcal{A}_s$ and $\mathcal{A}_\xi$ denote the fiber over $s\in S$ and the geometric generic fiber, respectively. Write $(\mathcal{A}_\xi)_{\operatorname{tors}}$ for the torsion subgroup and $\operatorname{Tr}(\mathcal{A}_\xi^{\overline{K(S)}/\bar{K}}(\bar{K}))$ for the relevant $\overline{K(S)}/\bar{K}$-trace.

Modified André–Pink–Zannier conjecture. If $\mathcal{A}_{\Gamma}\cap\mathcal{V}$ is Zariski dense in $\mathcal{V}$, then one of the following holds:

1. $\mathcal{V}$ is a translate of an abelian subvariety of $\mathcal{A}_s$ by a point of $\mathcal{A}_{\Gamma}\cap\mathcal{A}_s$ for some $s\in S$.
2. $\pi(\mathcal{V})=S$ and, over $\overline{K(S)}$, every irreducible component of $\mathcal{V}_\xi$ is a translate of an abelian subvariety of $\mathcal{A}_\xi$ by a point in

$$(\mathcal{A}_\xi)_{\operatorname{tors}}+\operatorname{Tr}\left(\mathcal{A}_\xi^{\overline{K(S)}/\bar{K}}(\bar{K})\right).$$

This is a modified version of the André–Pink–Zannier conjecture for a base curve and generalizes the setting of the theorem proved in the paper. The conjecture allows arbitrary characteristic-zero $K$, with no restrictions on the abelian scheme or the fixed abelian variety; the paper notes that the required intersection bound is not known in full generality.