On the pointwise convergence of the number of abelian varieties over ${\mathbb{F}}_p$ with fixed trace

Extending Katz-Sarnak heuristics, Ballini-Lombardo-Verzobio [BLV25] conjectures a limiting distribution as $p\to \mathrm{\infty }$ for $\mathrm{\#}{A}_g({\mathbb{F}}_p,t)$, the number of $g$-dimensional PPAVs over ${\mathbb{F}}_p$ with trace $t$, as a product of natural local factors $v_{\mathrm{\ell }}(t)$ for non-archimedean places $\mathrm{\ell }$ and the Sato-Tate measure ${\text{ST}}_g$ corresponding to $\mathrm{\infty }$. We prove that their conjecture is true for all $g$. As a consequence, we obtain analogous results on the distribution of curves of genus $2$ and $3$, answering questions of Bergström-Howe-García-Ritzenthaler [BHLR24] and [BLV25].