On the distribution of the number of points on a family of curves over finite fields

TL;DR

This paper investigates how the number of points on certain algebraic curves over finite fields is distributed across residue classes within small rectangles, revealing statistical patterns as the rectangle moves.

Contribution

It provides a detailed analysis of the distribution of points on curves defined by $y^{ ext{ell}}=P(x)$ over finite fields, focusing on residue class distributions within moving rectangles.

Findings

Distribution follows predictable statistical patterns

Residue class distribution approaches uniformity under certain conditions

Results extend understanding of point distributions on algebraic curves

Abstract

Let $p$ be a large prime, $\ell\geq 2$ be a positive integer, $m\geq 2$ be an integer relatively prime to $\ell$ and $P(x)\in\mathbb{F}_p[x]$ be a polynomial which is not a complete $\ell'$-th power for any $\ell'$ for which $GCD(\ell',\ell)=1$. Let $\mathcal{C}$ be the curve defined by the equation $y^{\ell}=P(x)$, and take the points on $\mathcal{C}$ to lie in the rectangle $[0,p-1]^2$. In this paper, we study the distribution of the number of points on $\mathcal{C}$ inside a small rectangle among residue classes modulo $m$ when we move the rectangle around in $[0,p-1]^2$.