$2^{\infty }$-Selmer groups, $2^{\infty }$-class groups, and Goldfeld's conjecture

Abstract

We prove that the $2^{\infty }$-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 $2^{\infty }$-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 $o(N)$.

84 pages, comments welcome

References in corpus (1)

• Governing fields and statistics for 4-Selmer groups and 8-class groups

Cited by in corpus (16)

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• On the $16$-rank of class groups of $\mathbb{Q}(\sqrt{-2p})$ for primes $p\equiv 1\mathrm{mod}4$
• Elements of given order in Tate-Shafarevich groups of abelian varieties in quadratic twist families
• A predicted distribution for Galois groups of maximal unramified extensions
• 4-ranks and the general model for statistics of ray class groups of imaginary quadratic number fields
• Moments and interpretations of the Cohen-Lenstra-Martinet heuristics
• On the 2-part of the Birch and Swinnerton-Dyer conjecture for quadratic twists of elliptic curves
• Ranks of abelian varieties in cyclotomic twist families
• Integral points on the congruent number curve
• The Birch--Swinnerton-Dyer exact formula for quadratic twists of elliptic curves
• A note on an asymptotic expansion related to the Dickman function
• The negative Pell equation
• Generalized Birch lemma and the 2-part of the Birch and Swinnerton-Dyer conjecture for certain elliptic curves
• Cohen-Lenstra-Gerth Heuristics via Automorphism Counts
• The Cassels-Tate pairing for finite Galois modules
• The even parity Goldfeld conjecture: congruent number elliptic curves