A heuristic approach to the Iwasawa theory of elliptic curves

Let E/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 $μ=0$ conjecture predicts that the Selmer group of E over the cyclotomic Zₚ-extension of Q is cofinitely generated as a Zₚ-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 $μ$-invariant can be detected by the intersection M₁∩ M₂ of two Iwasawa modules M₁, M₂ 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 (M₁, M₂) respect to which the event that M₁∩ M₂ is finite happens with probability $1$.