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.