Poonen–Rains conjecture for abelian varieties with real multiplication

Let $K$ be a totally real field with ${\mathcal {O}}$ its ring of integers. Let $p$ be a rational prime and let $\mathfrak{p}$ be a prime ideal of ${\mathcal {O}}$ above $p$ with residue field ${\mathbb {F}}_q$. Let $F$ be an algebraic number field such that $|\mathrm{RM}_{\mathcal {O}}(F)|=\infty$, where $\mathrm{RM}_{\mathcal {O}}(F)$ contains one representative $(A,\iota,\lambda)$ for each relevant isomorphism class of abelian varieties over $F$ with real multiplication by ${\mathcal {O}}$ and an ${\mathcal {O}}$-linear principal polarization. For $p\neq 2$, and also for $p=2$ when $2\nmid d_K$, the associated $\mathfrak{p}$-Selmer group is denoted by $\mathrm{Sel}_{\mathfrak{p}}(A)$.

Poonen–Rains conjecture for real multiplication. If $p$ is odd, then, as $(A,\iota,\lambda)$ varies in $\mathrm{RM}_{\mathcal {O}}(F)$ ordered by conductors,

$$\mathrm{Prob}\left(\mathrm{Sel}_{\mathfrak{p}}(A)=d\right)=\mathscr{D}_q^\mathrm{Ort}(d).$$

Moreover, if $2\nmid d_K$, the same statement holds also for $p=2$. This generalizes the orthogonal random-intersection model for elliptic-curve Selmer groups to families of abelian varieties with real multiplication; the source gives no resolution status.