On the Joint Distribution Of $\mathrm{Sel}_\phi(E/\mathbb{Q})$ and $\mathrm{Sel}_{\hat\phi}(E^\prime/\mathbb{Q})$ in Quadratic Twist Families

If $E$ is an elliptic curve with a point of order two, then work of Klagsbrun and Lemke Oliver shows that the distribution of $\dim_{\mathbb{F}_2}\mathrm{Sel}_\phi(E^d/\mathbb{Q}) - \dim_{\mathbb{F}_2} \mathrm{Sel}_{\hat\phi}(E^{\prime d}/\mathbb{Q})$ within the quadratic twist family tends to the discrete normal distribution $\mathcal{N}(0,\frac{1}{2} \log \log X)$ as $X \rightarrow \infty$. We consider the distribution of $\mathrm{dim}_{\mathbb{F}_2} \mathrm{Sel}_\phi(E^d/\mathbb{Q})$ within such a quadratic twist family when $\dim_{\mathbb{F}_2} \mathrm{Sel}_\phi(E^d/\mathbb{Q}) - \dim_{\mathbb{F}_2} \mathrm{Sel}_{\hat\phi}(E^{\prime d}/\mathbb{Q})$ has a fixed value $u$. Specifically, we show that for every $r$, the limiting probability that $\dim_{\mathbb{F}_2} \mathrm{Sel}_\phi(E^d/\mathbb{Q}) = r$ is given by an explicit constant $\alpha_{r,u}$. The constants $\alpha_{r,u}$ are closely related to the $u$-probabilities introduced in Cohen and Lenstra's work on the distribution of class groups, and thus provide a connection between the distribution of Selmer groups of elliptic curves and random abelian groups. Our analysis of this problem has two steps. The first step uses algebraic and combinatorial methods to directly relate the ranks of the Selmer groups in question to the dimensions of the kernels of random $\mathbb{F}_2$-matrices. This proves that the density of twists with a given $\phi$-Selmer rank $r$ is given by $\alpha_{r,u}$ for an unusual notion of density. The second step of the analysis utilizes techniques from analytic number theory to show that this result implies the correct asymptotics in terms of the natural notion of density.