A Heuristic approach to the Iwasawa theory of elliptic curves

2024/09/23 by Müller, Katharina, Ray, Anwesh
#11G05 #11R23 #11R45 #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)

paper · doi:10.48550/arxiv.2409.15056

Abstract

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 ℤ_p-extension of ℚ is cofinitely generated as a 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 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.

Related

• The Iwasawa μ-invariants of Elliptic Curves over ℚ
• The Iwasawa μ-invariant of certain elliptic curves of analytic rank zero
• Remarks on Greenberg's conjecture for Galois representations associated to elliptic curves
• Ranks of GL2 Iwasawa modules of elliptic curves
• Statistics for Iwasawa invariants of elliptic curves, \rmIII