Poonen–Rains conjecture on Selmer group distributions of elliptic curves

Fix a global field $F$ and a prime $p$. For an elliptic curve $E$ over $F$, let $\mathrm{Sel}_p(E)$ denote its $p$-Selmer group, and let $\mathscr{D}_p^\mathrm{Ort}(d)$ be the orthogonal-space distribution

$$\mathscr{D}_p^\mathrm{Ort}(d)=\left(\prod_{j\geq 0}\frac{1}{1+p^{-j}}\right)\left(\prod_{j=1}^d\frac{p}{p^j-1}\right).$$

Poonen–Rains conjecture. As $E$ varies over all elliptic curves over $F$ ordered by height,

$$\mathrm{Prob}\left(\dim_{\mathbb{F}_p}\mathrm{Sel}_p(E)=d\right)=\mathscr{D}_p^\mathrm{Ort}(d).$$

Consequently, the average size of $\mathrm{Sel}_p(E)$ for all elliptic curves $E/F$ is $p+1$. This conjecture models Selmer groups as intersections of maximal isotropic subspaces in orthogonal spaces; the source gives no resolution status.