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Let $E$ be an elliptic curve defined over a number field $F$ with goodordinary reduction at all primes above $p$, and let $F_\infty$ be a finitelyramified uniform pro-$p$ extension of $F$ containing the cyclotomic$\mathbb{Z}_p$-extension $F_{cyc}$. Set $F^{(n)}$ be the $n$-th layer of thetower, and $F^{(n)}_{cyc}$ the cyclotomic $\mathbb{Z}_p$-extension of$F^{(n)}$. We study the growth of the rank of $E(F^{(n)})$ by analyzing thegrowth of the $\lambda$-invariant of the Selmer group over $F^{(n)}_{cyc}$ as$n\rightarrow \infty$. This method has its origins in work of A.Cuoco, whostudied $\mathbb{Z}_p^2$-extensions. Refined estimates for growth are provedthat are close to conjectured estimates. The results are illustrated in specialcases.

Asymptotic growth of Mordell-Weil ranks of elliptic curves in noncommutative towers