Unlikely Intersections with Bruhat Strata

Let $\mathcal{A}_{g}$ be the moduli space of $g$-dimensional principally polarized abelian varieties over $\mathbb{Z}$, and let $\mathcal{T} \subset \mathcal{A}_{g}$ be a closed locus, also defined over $\mathbb{Z}$. Motivated by unlikely intersection conjectures, we study the intersection of $\mathcal{T}_{\mathbb{F}_{p}}$ with the Bruhat strata in $\mathcal{A}_{g,\mathbb{F}_{p}}$ as $p$-varies; these are strata characterized by the existence of certain subgroup schemes inside the $p$-torsion of the fibres. We find that, away from a finite set of primes, positive-dimensional ``unlikely'' intersections of $\mathcal{T}_{\mathbb{F}_{p}}$ with such strata are all accounted for by intersections of $\mathcal{T}$ with special loci inside $\mathcal{A}_{g}$. This result generalizes to all abelian-type Shimura varieties, and variations of Hodge structures equipped with certain motivic data. It moreover gives another example of how functional transcendence principles in characteristic zero can be used to study unlikely intersections in positive characteristic, building on recent work by the author.