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.

11R23, 11G05

arithmetic statistics, noncommutative Iwasawa theory, Selmer groups, Euler characteristics, Akashi series, growth of Mordell-Weil ranks

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Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V6T 1Z2

Département de Mathématiques et de Statistique, Université Laval, Pavillion Alexandre-Vachon, 1045 Avenue de la Médecine, Québec, QC, Canada G1V 0A6

Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V6T 1Z2