  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 $g$.
