The fluctuations in the number of points on a hyperelliptic curve over a finite field

TL;DR

This paper investigates the fluctuations in the number of points on hyperelliptic curves over finite fields, revealing different limiting distributions depending on how genus and field size grow, including Gaussian and trinomial-based models.

Contribution

It characterizes the limiting distributions of point count fluctuations for hyperelliptic curves under various growth regimes of genus and field size.

Findings

For fixed genus and growing q, S/√q follows a distribution of traces of random symplectic matrices.

When genus grows with fixed q, S behaves like a sum of independent trinomial variables.

Both genus and q growing lead to S/√q having a Gaussian distribution.

Abstract

The number of points on a hyperelliptic curve over a field of $q$ elements may be expressed as $q+1+S$ where $S$ is a certain character sum. We study fluctuations of $S$ as the curve varies over a large family of hyperelliptic curves of genus $g$. For fixed genus and growing $q$, Katz and Sarnak showed that $S/\sqrt{q}$ is distributed as the trace of a random $2g\times 2g$ unitary symplectic matrix. When the finite field is fixed and the genus grows, we find that the the limiting distribution of $S$ is that of a sum of $q$ independent trinomial random variables taking the values $\pm 1$ with probabilities $1/2(1+q^{-1})$ and the value 0 with probability $1/(q+1)$. When both the genus and the finite field grow, we find that $S/\sqrt{q}$ has a standard Gaussian distribution.