Gaussian point count statistics for families of curves over a fixed finite field

TL;DR

This paper constructs families of algebraic curves over a fixed finite field where the distribution of the number of points over the field converges to a Gaussian distribution as the family size grows.

Contribution

It introduces specific families of curves with point count statistics that become Gaussian over a fixed finite field, expanding understanding of statistical behavior in algebraic geometry.

Findings

Point count statistics tend to Gaussian distribution for these families.

Average number of points on curves in these families increases without bound.

Distribution convergence holds as the family size increases.

Abstract

We produce a collection of families of curves, whose point count statistics over F_p becomes Gaussian for p fixed. In particular, the average number of F_p points on curves in these families tends to infinity.