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Department of Mathematics pres.

Colloquium Series Seminar

NT RTG Lectures I: $2^k$-Selmer groups and Goldfeld's conjecture.

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 $k > 1$. Speaker(s): Alexander Smith (Harvard University)

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