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

TL;DR

This paper formulates and proves characteristic p analogues of the Mumford--Tate and André--Oort conjectures for ordinary mod p Shimura varieties of Hodge type, establishing new frameworks and solving related conjectures.

Contribution

It introduces characteristic p analogues of key conjectures, proves them for GSpin Shimura varieties, and connects these results with Ax--Schanuel and Tate-linear conjectures.

Findings

Proved Mumford--Tate and André--Oort analogues for GSpin Shimura varieties.

Solved Chai's Tate-linear conjecture for these varieties.

Established relations among multiple conjectures in the context of mod p Shimura varieties.

Abstract

We formulate characteristic $p$ analogues of the Mumford--Tate and the Andr\'e--Oort conjectures for ordinary mod $p$ Shimura varieties of Hodge type, and set up general frameworks for studying them. We prove the two conjectures for (subvarieties of) arbitrary products of GSpin Shimura varieties, by reducing them, via a notion of linearity for mod $p$ Shimura varieties, to a third conjecture of Ax--Schanuel type. Along the way, we solve Chai's Tate-linear conjecture for products of GSpin Shimura varieties, and reveal an intimate relation among the four conjectures mentioned above. Our proof uses Crew's parabolicity conjecture which is recently proven by D'Addezio.