An Algorithm to Compute Automorphism Groups of K3 Surfaces and an Application to Singular K3 Surfaces

Let |X| be a complex algebraic |$K3$| surface or a supersingular |$K3$| surface in odd characteristic. We present an algorithm by which, under certain assumptions on |X|⁠, we can calculate a finite set of generators of the image of the natural homomorphism from the automorphism group of |X| to the orthogonal group of the Néron–Severi lattice of |X|⁠. We then apply this algorithm to certain complex |$K3$| surfaces, among which are singular |$K3$| surfaces whose transcendental lattices are of small discriminants.