2^∞-Selmer groups, 2^∞-class groups, and Goldfeld's conjecture

2017/02/08 by Alexander Smith, Smith, Alexander · 2 citations
Mathematics · #11G05 #11R45 #Analytic Number Theory Research #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1702.02325

openalex publication_date 2017/02/08 · openalex created_date 2017/02/17 · openalex updated_date 2026/07/28

Abstract

We prove that the 2^∞-class groups of the imaginary quadratic fields have the distribution predicted by the Cohen-Lenstra heuristic. Given an elliptic curve E/Q with full rational 2-torsion and no rational cyclic subgroup of order four, we analogously prove that the 2^∞-Selmer groups of the quadratic twists of E have distribution as predicted by Delaunay's heuristic. In particular, among the twists E^d with |d| < N, the number of curves with rank at least two is o(N).

Citations

• On the 16-rank of class groups of \ℚ(\√(-8p)) for\n p\≡ -1 bmod 4
• Governing fields and statistics for 4-Selmer groups and 8-class groups

Cited by

• On the 2-part of the Birch and Swinnerton-Dyer conjecture for quadratic twists of elliptic curves
• The Birch--Swinnerton-Dyer exact formula for quadratic twists of elliptic curves

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• The distribution of ℓ^∞-Selmer groups in degree ℓ twist families I
• On 2-Selmer groups of twists after quadratic extension
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