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

TL;DR

This paper investigates the asymptotic growth of Mordell-Weil ranks of elliptic curves over noncommutative p-adic Lie extensions, providing refined estimates and illustrating the results in specific cases.

Contribution

It introduces new methods to analyze the growth of ranks in noncommutative towers, extending prior work on abelian extensions and offering refined asymptotic estimates.

Findings

Established refined growth estimates for Mordell-Weil ranks

Extended analysis to noncommutative p-adic Lie extensions

Provided illustrative examples of the growth behavior

Abstract

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 $n\rightarrow \infty$. 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.