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

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 $g$.

16 pages; v2: refereed version, Philosophical Transactions of the Royal Society A 2015

References in corpus (1)

• Semiample Bertini theorems over finite fields

Cited by in corpus (2)

• How Large is $A_g({\mathbb{F}}_q)$?
• Distributions of traces of Frobenius for smooth plane curves over finite fields