  We produce explicit elliptic curves over \Bbb F_p(t) whose Mordell-Weil
groups have arbitrarily large rank. Our method is to prove the conjecture of
Birch and Swinnerton-Dyer for these curves (or rather the Tate conjecture for
related elliptic surfaces) and then use zeta functions to determine the rank.
In contrast to earlier examples of Shafarevitch and Tate, our curves are not
isotrivial. Asymptotically these curves have maximal rank for their conductor.
Motivated by this fact, we make a conjecture about the growth of ranks of
elliptic curves over number fields.
