$p$-adic monodromy and mod $p$ unlikely intersections, II

We study ordinary abelian schemes in characteristic $p$ and their moduli spaces from the perspective of char $p$ Mumford–Tate, log Ax–Lindemann, and geometric André–Oort conjectures (abbreviated as ${\text{MTT}}_p$, ${\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{A}\mathrm{L}}_p$ and geoAO${}_p$). In this paper, we achieve multiple goals: ($\mathbf{\text{A}}$) establish the implication ${\mathrm{M}\mathrm{T}}_p\iff {\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{A}\mathrm{L}}_p\Rightarrow \mathrm{g}\mathrm{e}\mathrm{o}\mathrm{A}{\mathrm{O}}_{\mathrm{p}}$, and show that they all follow from the Tate conjecture for abelian varieties. The equivalence ${\mathrm{M}\mathrm{T}}_p\iff {\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{A}\mathrm{L}}_p$ is exploited from both sides, which enables us to \noindent($\mathbf{\text{B}}$) develop a representation theory approach to ${\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{A}\mathrm{L}}_p$ and $\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{A}{\mathrm{O}}_{\mathrm{p}}$ by first establishing many cases of MT${}_p$ via classical techniques, and ($\mathbf{\text{C}}$) develop an algebraization approach to ${\text{MTT}}_p$ that transcends the limitation of classical methods. In particular, we introduce ``crystalline Hodge loci’', a rigid analytic geometric object that encodes the essential information needed for proving ${\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{A}\mathrm{L}}_p$, while being very approachable via (integral and relative) $p$-adic Hodge theory. This enables us to prove ${\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{A}\mathrm{L}}_p$ for compact Tate-linear curves with unramified $p$-adic monodromy. As an application, we establish ${\text{MTT}}_p$ for many abelian fourfolds of $p$-adic Mumford type.