K3 surfaces, cyclotomic polynomials and orthogonal groups

Abstract Let X be a complex projective K 3 surface and let TX T X be its transcendental lattice; the characteristic polynomials of isometries of TX T X induced by automorphisms of X are powers of cyclotomic polynomials. Which powers of cyclotomic polynomials occur? The aim of this note is to answer this question, as well as related ones, and give an alternative approach to some results of Kondō, Machida, Oguiso, Vorontsov, Xiao and Zhang; this leads to questions and results concerning orthogonal groups of lattices.