A heuristic for the distribution of point counts for random curves over a finite field

TL;DR

This paper proposes a heuristic for estimating the number of rational points on large genus random algebraic curves over finite fields, suggesting a Poisson distribution based on Mumford's conjecture.

Contribution

It introduces a new heuristic for point counts on random curves over finite fields, supported by a proven asymptotic case when genus and field size grow.

Findings

Heuristic suggests a Poisson distribution with mean q+1+1/(q-1).

Proves a weaker version for large genus and field size with q much larger than g.

Supports the heuristic with asymptotic analysis.

Abstract

How many rational points are there on a random algebraic curve of large genus $g$ over a given finite field $\mathbb{F}_q$? We propose a heuristic for this question motivated by a (now proven) conjecture of Mumford on the cohomology of moduli spaces of curves; this heuristic suggests a Poisson distribution with mean $q+1+1/(q-1)$. We prove a weaker version of this statement in which $g$ and $q$ tend to infinity, with $q$ much larger than $g$.