The isogeny-class size estimate for abelian varieties of Lubin–Tate type

Let $A\in \mathcal{A}_{g,1}({\mathbb F}_{q})$ be a geometrically simple principally polarised abelian variety with Weil $q$-number $\alpha_A$, and suppose that its $p$-divisible group is of Lubin–Tate type, meaning that the dimension of its local-local part or that of its dual is at most one. Write $K={\mathbb Q}(\alpha_A)$, assume that $K\otimes {\mathbb Q}_p$ is a direct sum of two unramified field extensions of ${\mathbb Q}_p$ of degree $g$, and let $N(A,{\mathbb F}_{q^n})$ denote the size of the isogeny class of $A$ over ${\mathbb F}_{q^n}$. The isogeny-class size estimate. For a set of positive integers $n$ with positive natural density, the estimate

$$N(A,{\mathbb F}_{q^n})=q^{n\left(\frac{(g+1)(g-2)}{4}+1+o(1)\right)}$$

holds. This estimate describes the asymptotic size of isogeny classes in the Lubin–Tate Newton stratum; the supplied text does not establish whether the asserted equality is proved or conjectural.