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

Take \(E/Q\) to be an elliptic curve with full rational 2-torsion (satisfying some extra technical assumptions). In this talk, we will show that 100% of the quadratic twists of \(E\) have rank less than two, thus proving that the BSD conjecture implies Goldfeld's conjecture in these families. To do this, we will extend Kane's distributional results on the 2-Selmer groups in these families to \(2^k\)-Selmer groups for any \(k > 1\). In addition, using the close analogy between \(2^k\)-Selmer groups and \(2^{k+1}\)-class groups, we will prove that the \(2^{k+1}\)-class groups of the quadratic imaginary fields are distributed as predicted by the Cohen-Lenstra heuristics for all \(k > 1\).