| Let k be a field of finite type and E/k(t) a non-isotrivial elliptic curve. We will show how to construct one-parameter families over k of quadratic twists of E such that, for almost all k-fibers, the Mordell-Weil rank of the twist is zero. If we regard the fibers of the family as elliptic surfaces, then our methods show something stronger: for almost all k-fibers, the Tate conjectures are true. If time permits, we will show how one can construct other one-parameter families over k of varieties such that the Tate conjectures are true for almost all k-fibers. |