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Let $p$ be an odd prime. We state characteristic $p$ analogues of theMumford--Tate conjecture and the Andr\'e--Oort conjecture for ordinary strataof mod $p$ Shimura varieties. We prove the conjectures in the case of GSpinShimura varieties and products of modular curves. The two conjectures are bothrelated to a notion of linearity for mod $p$ Shimura varieties, about whichChai has formulated the Tate-linear conjecture. We will first treat theTate-linear conjecture, above which we then build the proof of thecharacteristic $p$ analogue of the Mumford--Tate conjecture. Finally, we usethe Tate-linear conjecture and the characteristic $p$ analogue of theMumford--Tate conjecture to prove the characteristic $p$ analogue of theAndr\'e--Oort conjecture. The proofs of these conjectures use Chai's results onmonodromy of $p$-divisible groups and rigidity theorems for formal tori, aswell as Crew's parabolicity conjecture which is recently proven by D'Addezio.

Characteristic $p$ analogues of the Mumford--Tate and Andr\'e--Oort conjectures for ordinary GSpin Shimura varieties