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arXiv · 2601.20824

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

Zhao Yu Ma·Jit Wu Yap·Jeff Achter·Julia Gordon
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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].

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Zhao Yu Ma, Jit Wu Yap, Jeff Achter, Julia Gordon. 2026-01-28. On the pointwise convergence of the number of abelian varieties over $\mathbb{F}_p$ with fixed trace. https://arxiv.org/abs/2601.20824

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