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For a K3 surface over an algebraically closed field of odd characteristic, the representation of the automorphism group on the global two forms is finite. If the K3 surface is supersingular, it is isomorphic to the representation on the discriminant group of the Neron-Severi group. If the K3 surface is of finite height, the representation on the (etale or crystalline) transcendental cycles is also finite and there is a canonical projection from the representation on the transcendental cycles to the representation on the two forms. We prove that, if the base field is an algebraic closure of a finite field, this projection is an isomorphism and the rank of the transcendental cycles is a multiple of the order of the representation on the two forms. They are analogous to the classical results on complex K3 surfaces. From these results, we deduce that the height and the Artin invariant of a K3 surface with a purely non-symplectic automorphism of higher order are determined by a congruence class of the base characteristic.

The representations of the automorphism groups and the Frobenius invariants of K3 surfaces