Let $p$ be an odd prime. Associated to a pair $(E, \mathcal{ F}_\infty)$ consisting of a rational elliptic curve $E$ and a $p$-adic Lie extension $\mathcal{ F}_\infty$ of $\mathbb{ Q}$, is the $p$-primary Selmer group $Sel_{ p^\infty}(E/\mathcal{ F}_\infty)$ of $E$ over $\mathcal{ F}_\infty$. 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.
Categories: math.NT