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

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We classify all the K3 surfaces which are minimal models of the quotient of the product of two curves C₁× C₂ by the diagonal action of either the group /p or the group /2p. These K3 surfaces admit a non-symplectic automorphism of order p induced by an automorphism of one of the curves C₁ or C₂. 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. In addition, we show that one can obtain the same set of K3 surfaces under more restrictive assumptions namely one of the two curves, say C₂, is isomorphic to a rigid hyperelliptic curve with an automorphism δₚ of order p and the automorphism of the K3 surface is induced by δₚ. Finally, we describe the variation of the Hodge structures of the surfaces constructed and we give an equation for some of them.