Isogeny class and Frobenius root statistics for abelian varieties over finite fields

Let $I(g,q,N)$ be the number of isogeny classes of $g$-dimensional abelian varieties over a finite field  $\mathbb F$ having a fixed number $N$ of $\mathbb F$-rational points. We describe the asymptotic (for $q\to\infty$) distribution of $I(g,q,N)$ over possible values of $N$. We also prove an analogue of the Sato–Tate conjecture for isogeny classes of $g$-dimensional abelian varieties.