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

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.