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

TL;DR

This paper confirms predictions about the distribution of $2^\infty$-class groups of imaginary quadratic fields and $2^\infty$-Selmer groups of certain elliptic curves, supporting Cohen-Lenstra and Delaunay heuristics.

Contribution

It proves the distribution of $2^\infty$-class groups and Selmer groups aligns with heuristic predictions, extending understanding of these algebraic structures.

Findings

Distribution of $2^\infty$-class groups matches Cohen-Lenstra heuristic.

Distribution of $2^\infty$-Selmer groups matches Delaunay's heuristic.

Number of quadratic twists with rank at least two is negligible among all twists.

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 $o(N)$.