Appears in collection : Rational points on irrational varieties

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 $k>1$.