The distribution of points on curves over finite fields in some small rectangles

TL;DR

This paper investigates how points on certain algebraic curves over finite fields are distributed within small rectangles, revealing Gaussian distribution patterns in long rectangles where traditional bounds are ineffective.

Contribution

It demonstrates that points on specific curves over finite fields follow a Gaussian distribution in large rectangles, even when Weil bounds do not provide useful estimates.

Findings

Distribution of points on curves is Gaussian in long rectangles.

Weil bound fails to give nontrivial information in small rectangles.

Distribution behavior analyzed for curves over finite fields.

Abstract

Let $p$ be a prime. We study the distribution of points on a class of curves $C$ over $\mathbb{F}_p$ inside very small rectangles $B$ for which the Weil bound fails to give nontrivial information. In particular, we show that the distribution of points on $C$ over some long rectangles is Gaussian.