Algebra and Geometry Seminar

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 on the moduli space of abelian 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...