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

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.