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Arithmetic statistics and noncommutative Iwasawa theory. (English) Zbl 1497.11271

Summary: 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.

MSC:

11R23 Iwasawa theory
11G05 Elliptic curves over global fields
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