K3 surfaces of Picard rank one which are double covers of the projective plane

Andreas-Stephan ELSENHANS a and Jörg JAHNEL a a Universität Göttingen, Mathematisches Institut, Bunsenstraße 3–5, D-37073 Göttingen, Germany 1 Abstract. We construct explicit examples of K3 surfaces over Q which are of degree 2 and the geometric Picard rank of which is equal to 1. We construct, particularly, examples in the form w2 = det M where M is a symmetric (3× 3)-matrix of ternary quadratic forms or a symmetric (6× 6)-matrix of ternary linear forms. Our method is based on reduction modulo p for p = 3 and 5.