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$2^k$-Selmer groups and Goldfeld's conjecture, I

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Alexander Smith
Harvard University
Tue, 2019-06-04 09:00 - 10:00
MPIM Lecture Hall

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 $k > 1$.

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