Arithmetic Statistics and noncommutative Iwasawa Theory

TL;DR

This paper investigates the algebraic structure and growth patterns of Selmer groups associated with elliptic curves over noncommutative $p$-adic Lie extensions, offering new insights into their asymptotic behavior.

Contribution

It introduces new arithmetic statistical methods to analyze the structure and growth of Selmer groups in noncommutative Iwasawa theory.

Findings

Provides asymptotic formulas for Mordell--Weil rank growth

Characterizes the algebraic structure of Selmer groups in noncommutative towers

Offers new statistical tools for 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 $Sel_{p^\infty}(E/\mathcal{F}_\infty)$ 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.