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This research examines Greenberg's μ=0 conjecture related to the Selmer group of elliptic curves over cyclotomic extensions.

July 4, 2026Open Access

A heuristic approach to the Iwasawa theory of elliptic curves

KMKatharina MüllerUniversität der Bundeswehr München ARAnwesh RayChennai Mathematical Institute

Key Points

This research examines Greenberg's μ=0 conjecture related to the Selmer group of elliptic curves over cyclotomic extensions.
Analyzed elliptic curves with good ordinary reduction at an odd prime p.
Extended Poonen and Rains' heuristic methods for statistical perspective.
Utilized a probability measure on pairs of Iwasawa modules to assess their intersection properties.
Demonstrated that the vanishing of the μ-invariant corresponds to the finiteness of the intersection M_1∩M_2.
Showed that this event occurs with probability 1 under the defined probability measures.

Abstract

Let E/ₐ be an elliptic curve and p an odd prime such that E has good ordinary reduction at p and the Galois representation on Ep 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.

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