Studying the Growth of Mordell-Weil

{} 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 $\Zp$-extension. We describe those $\Zp$-extensions $\Kinf/K$ 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.

2000 Mathematics Subject Classification: Primary 11G05, 11G40; Secondary 11R23, 14G05

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