Correspondences between K3. surfaces. With an appendix by Igor Dolgachev.

In this paper we show that there is a correspondence between some $K3$ surfaces with non-isometric transcendental lattices constructed as a twist of the transcendental lattice of the Jacobian of a generic genus 2 curve. Moreover, we show the existence of a correspondence between a general $K3$ surface with $\rho =17$ and a Kummer surface having transcendental lattices $\QQ$-Hodge isomorphic. In the Appendix, I. Dolgachev realizes a geometric correspondence between $X(1,1,1)$ and a Kummer surface, i.e. he gives a geometric realization of the so-called Shioda-Inose structure on $X(1,1,1)$.