\(2^\infty\)-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 $k > 1$.