A Heuristic approach to the Iwasawa theory of elliptic curves

TL;DR

This paper investigates Greenberg's $$-conjecture for elliptic curves over $7$ with good ordinary reduction at an odd prime, using statistical heuristics and intersection properties of Iwasawa modules.

Contribution

It extends Poonen and Rains' heuristics to provide probabilistic evidence supporting Greenberg's conjecture via intersection analysis of Iwasawa modules.

Findings

Probability that the intersection of two Iwasawa modules is finite is 1.

Supports the conjecture that the $$-invariant vanishes for a broad class of elliptic curves.

Provides a heuristic framework linking module intersections to the vanishing of the $$-invariant.

Abstract

Let $E_{/\mathbb{Q}}$ be an elliptic curve and $p$ an odd prime such that $E$ has good ordinary reduction at $p$ and the Galois representation on $E[p]$ is irreducible. Then Greenberg's $\mu=0$ conjecture predicts that the Selmer group of $E$ over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ is cofinitely generated as a $\mathbb{Z}_p$-module. In this article we study this conjecture from a statistical perspective. We extend the heuristics of Poonen and Rains to obtain further evidence for Greenberg's conjecture. The key idea is that the vanishing of the $\mu$-invariant can be detected by the intersection $M_1\cap M_2$ of two Iwasawa modules $M_1, M_2$ with additional properties in a given inner product space. The heuristic is based on showing that there is a probability measure on the space of pairs $(M_1, M_2)$ respect to which the event that $M_1\cap M_2$ is finite happens with probability $1$.