A Heuristic approach to the Iwasawa theory of elliptic curves

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 $1$.

Version 2: 20 pages, accepted for publication in Journal of Number theory