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 positive $k$. We will also show how are techniques may be applied to find the distribution of $2^k$-class groups of quadratic fields.

The pre-talk will focus on the definition of Selmer groups. We will also give some context for the study of the arithmetic statistics of these groups.