On the Distribution of the Number of Points on Algebraic Curves in Extensions of Finite Fields

TL;DR

This paper investigates the distribution of the number of rational points on algebraic curves over finite field extensions, deriving an asymptotic formula under certain independence conditions, and relates it to a generalized Sato-Tate distribution.

Contribution

It provides a new asymptotic formula for point counts on algebraic curves over finite fields under a multiplicative independence condition, extending the understanding of their distribution.

Findings

Asymptotic formula for the distribution of normalized point counts

Condition satisfied by ordinary elliptic curves and genus 2 curves

Relation to a generalized Sato-Tate distribution

Abstract

Let $\cC$ be a smooth absolutely irreducible curve of genus $g \ge 1$ defined over $\F_q$, the finite field of $q$ elements. Let $# \cC(\F_{q^n})$ be the number of $\F_{q^n}$-rational points on $\cC$. Under a certain multiplicative independence condition on the roots of the zeta-function of $\cC$, we derive an asymptotic formula for the number of $n =1, ..., N$ such that $(# \cC(\F_{q^n}) - q^n -1)/2gq^{n/2}$ belongs to a given interval $\cI \subseteq [-1,1]$. This can be considered as an analogue of the Sato-Tate distribution which covers the case when the curve $\E$ is defined over $\Q$ and considered modulo consecutive primes $p$, although in our scenario the distribution function is different. The above multiplicative independence condition has, recently, been considered by E. Kowalski in statistical settings. It is trivially satisfied for ordinary elliptic curves and we also establish it for a natural family of curves of genus $g=2$.