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##### Department of Mathematics, University of California San Diego

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## The Joint Distribution of $\operatorname{Sel}_\phi{(E^d/{\mathbb{Q}})}$ and $\operatorname{Sel}_{\hat\phi}{(E^{\prime d}/{\mathbb{Q}})}$ in Quadratic Twist Families

##### Abstract:

We show that the $\phi$-Selmer ranks of twists of an elliptic curve $E$ with a point of order two are distributed like the ranks of random groups in a manner consistent with the philosophy underlying the Cohen-Lenstra heuristics. If $E$ has a point of order two, then the distribution of $dim_{\mathbb{F}_2} \operatorname{Sel}_\phi{(E^d/{\mathbb{Q}})} - dim_{\mathbb{F}_2} \operatorname{Sel}_{\hat\phi}{(E^{\prime d}/{\mathbb{Q}})}$ tends to the discrete normal distribution $\mathcal{N}(0,\frac{1}{2} \log \log X)$ as $X \rightarrow \infty$. We consider the distribution of $dim_{\mathbb{F}_2} \operatorname{Sel}_\phi{(E^d/{\mathbb{Q}})} - dim_{\mathbb{F}_2} \operatorname{Sel}_{\hat\phi}{(E^{\prime d}/{\mathbb{Q}})}$ has a fixed value $u$. We show that for every $r$, the limiting probability that $dim_{\mathbb{F}_2} \operatorname{Sel}_\phi{(E^d/{\mathbb{Q}})}= r$ is given by an explicit constant $\alpha_{r,u}$ introduced in Cohen and Lenstra's original work on the distribution of class groups.

Host: Kiran Kedlaya

### AP&M 7321

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