A large empirical study of congruent number elliptic curves up to D < 3×10^6 confirms several Selmer-rank heuristics but does not support the paper's headline claim that Goldfeld's conjecture is rigorously verified.
$2^\infty$-Selmer groups, $2^\infty$-class groups, and Goldfeld's conjecture
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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 $o(N)$.
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Unveiling Arithmetic Statistics of Congruent Number Elliptic Curves via Data Science and Machine Learning
A large empirical study of congruent number elliptic curves up to D < 3×10^6 confirms several Selmer-rank heuristics but does not support the paper's headline claim that Goldfeld's conjecture is rigorously verified.