Studying the growth of Mordell-Weil

Abstract

We study the growth of the Mordell-Weil groups $E(K_n)$ of an elliptic curve $E$ as $K_n$ runs through the intermediate fields of a ${\mathbb{Z}}_p$-extension. We describe those ${\mathbb{Z}}_p$-extensions $K_{\infty }/K$ where we expect the ranks to grow to infinity. In the cases where we know or expect the rank to grow, we discuss where we expect to find the submodule of universal norms.