\(2^k\)-Selmer groups and Goldfeld's conjecture.

Alexander Smith

Take \(E\) to be an elliptic curve over a number field whose four torsion obeys certain technical conditions. In this talk, we will outline a proof that \(100\%\) of the quadratic twists of \(E\) have rank at most one. To do this, we will find the distribution of \(2^k\)-Selmer ranks in this family for every \(k > 1\). Using this framework, we will also find the distribution of the \(2^k\)-class ranks of the imaginary quadratic fields for all \(k > 1\).