p-adic monodromy and mod p unlikely intersections

We introduce a mod p analogue of the Mumford—Tate conjecture, which governs the p-adic monodromy of families of mod p abelian varieties.  It turns out that the conjecture is closely related to a notion of formal linearity of mod p Shimura varieties. Surprisingly, the conjecture can be reduced to an unlikely intersection problem of Ax—Schanuel type —— a phenomenon that is unique to positive characteristic. This gives rise to new perspectives for attacking the Mumford—Tate problem: say, algebraization and p-adic O-minimal theory ……