K3 surfaces with a non-symplectic automorphism and product-quotient surfaces

We classify all the K3 surfaces which are minimal models of the quotient of the product of two curves C 1 ×C 2 by the diagonal action of either the group \Z/p\Z or the group \Z/2p\Z. These K3 surfaces admit a non-symplectic automorphism of order p induced by an automorphism of one of the curves C_1 or C_2 . We prove that most of the K3 surfaces admitting a non-symplectic automorphism of order p (and in fact a maximal irreducible component of the moduli space of K3 surfaces with a non-symplectic automorphism of order p ) are obtained in this way.Inaddition, we show that one can obtain the same set of K3 surfaces under more restrictive assumptions namely one of the two curves, say C_2 , is isomorphic to a rigid hyperelliptic curve with an automorphism \delta_p of order p and the automorphism of the K3 surface is induced by \delta_p. Finally, we describe the variation of the Hodge structures of the surfaces constructed and we give an equation for some of them.