Title: On the Joint Distribution Of $\mathrm{Sel}_ϕ(E/\mathbb{Q})$ and $\mathrm{Sel}_{\hatϕ}(E^\prime/\mathbb{Q})$ in Quadratic Twist Families,
Abstract: 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.