Zeta zeros of function fields heuristically equidistribute as genus goes to infinity

Julian Shah

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arXiv · 2608.07804

Zeta zeros of function fields heuristically equidistribute as genus goes to infinity

Julian Shah

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Abstract

We propose a heuristic for the growth rate of the average class number of curves in $\mathcal M_g(\mathbb F_q)$ as $g \to \infty$. We show that this heuristic implies that the zeros of the zeta functions of such curves become equidistributed on the circle as $g\to\infty$. Our heuristic is based on the assumption that the stable cohomology of the universal Jacobian dominates a point count using the trace formula, following the method of Achter et al.

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Julian Shah. 2026-08-07. Zeta zeros of function fields heuristically equidistribute as genus goes to infinity. https://arxiv.org/abs/2608.07804

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