3D mirror symmetry in positive characteristic

Via the formulation of (quantum) Hikita conjecture with coefficients in a characteristic $p$ field, we explain an arithmetic aspect of the theory of 3D mirror symmetry. Namely, we propose that the action of Steenrod-type operations and Frobenius-constant quantizations intertwine under the (quantum) Hikita isomorphism for 3D mirror pairs, and verify this for the Springer resolutions and hypertoric varieties.