Isometries of indefinite binary forms and applications to K3-surfaces

Abstract: Classical number theory of binary quadratic forms tells us that the  isometry group of the associated lattice is essentially either infinite cyclic or infinite dihedral. This can be used to determine the full group of automorphisms of K3-surfaces with Picard number two. In particular one finds back a well known criterion for finiteness but it also yields an explicit algorithm to  find  the automorphism group  once one knows  the Picard lattice. I shall illustrate this for K3's that are in two ways double covers of the plane. This is a report of joint work with F. Galluzzi and G. Lombardo from Turin.