2^k-Selmer groups, the Cassels-Tate pairing, and Goldfeld's conjecture by 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.