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# $$2^k$$-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 $$k > 1$$.