Let p be a prime. In this paper, we use techniques from Iwasawa theory to study questions about rank jump of elliptic curves in cyclic extensions of degree p. We also study growth of the p-primary Selmer group and the Shafarevich–Tate group in cyclic degree-p extensions and improve upon previously known results in this direction.

Let E be an elliptic curve defined over a number field F with good ordinary reduction at all primes above p, and let $F_\infty $ be a finitely ramified uniform pro-p extension of F containing the cyclotomic $\mathbb {Z}_p$ -extension $F_{\operatorname {cyc}}$ . Set $F^{(n)}$ be the nth layer of the tower, and $F^{(n)}_{\operatorname {cyc}}$ the cyclotomic $\mathbb {Z}_p$ -extension of $F^{(n)}$ . We study the growth of the rank of $E(F^{(n)})$ by analyzing the growth of the $\lambda $ -invariant of the Selmer group over $F^{(n)}_{ \operatorname {cyc}}$ as $n\rightarrow \infty $ . This method has its origins in work of A. Cuoco, who studied $\mathbb {Z}_p^2$ -extensions. Refined estimates for growth are proved that are close to conjectured estimates. The results are illustrated in special cases.