### Elliptic curves with large rank over function fields, by Douglas Ulmer

We produce explicit elliptic curves over Fp(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.

Douglas Ulmer <ulmer@math.arizona.edu>