Classification of automorphisms on a deformation family of hyper-Kähler four-folds by p-elementary lattices

We give a classification of all nonsymplectic automorphisms of prime order p acting on irreducible holomorphic symplectic four-folds deformation equivalent to the Hilbert scheme of two points on a K3 surface, for p = 2, 3, and 7 ≤ p ≤ 19. Our classification relates some invariants of the fixed locus to the isometry classes of two natural lattices associated to the action of the automorphism on the second cohomology group with integer coefficients. In several cases we provide explicit examples. As an application, we find new examples of nonnatural nonsymplectic automorphisms of order 3.