Arithmetic statistics and noncommutative Iwasawa theory

Abstract

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 ${\mathrm{Sel}}_{p^{\infty }}(E/{\mathcal{F}}_{\infty })$ 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.