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

TL;DR

This paper proves a conjecture about the limiting distribution of the number of g-dimensional principally polarized abelian varieties over finite fields with fixed trace, confirming predictions extending Katz-Sarnak heuristics.

Contribution

It establishes the conjectured limiting distribution for all dimensions g, confirming a key prediction about the behavior of abelian varieties over finite fields.

Findings

Confirmed the conjecture for all g.

Derived distribution results for genus 2 and 3 curves.

Connected distribution of abelian varieties to local factors and Sato-Tate measure.

Abstract

Extending Katz-Sarnak heuristics, Ballini-Lombardo-Verzobio [BLV25] conjectures a limiting distribution as $p \to \infty$ for $\# A_g(\mathbb F_p,t)$, the number of $g$-dimensional PPAVs over $\mathbb F_p$ with trace $t$, as a product of natural local factors $v_\ell(t)$ for non-archimedean places $\ell$ and the Sato-Tate measure $\text{ST}_g$ corresponding to $\infty$. 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 Bergstr\"om-Howe-Garc\'ia-Ritzenthaler [BHLR24] and [BLV25].