Arithmetic Statistics and noncommutative Iwasawa Theory

Let $p$ be an odd prime. Associated to a pair $(E,{\mathcal{F}}_{\mathrm{\infty }})$ consisting of a rational elliptic curve $E$ and a $p$-adic Lie extension ${\mathcal{F}}_{\mathrm{\infty }}$ of $\mathbb{Q}$, is the $p$-primary Selmer group $Se{l}_{p^{\mathrm{\infty }}}(E\mathrm{/}{\mathcal{F}}_{\mathrm{\infty }})$ of $E$ over ${\mathcal{F}}_{\mathrm{\infty }}$. In this paper, we study the arithmetic statistics for the algebraic structure of this Selmer group. The results provide insights into the asymptotics for the growth of Mordell–Weil ranks of elliptic curves in noncommutative towers.