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

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 $F_{cyc}$. Set $F^{(n)}$ be the $n$-th layer of the tower, 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 the growth of the $lambda$-invariant of the Selmer group over $F^{(n)}_{cyc}$ as $nrightarrow infty$. 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.