A heuristic for the distribution of point counts for random curves over   a finite field
  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 $g$. 
Jeffrey D. Achter, Daniel Erman, Kiran S. Kedlaya, Melanie Matchett   Wood, and David Zureick-Brown
math.NT math.AG