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

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 $$\mathcal{B }$$ for which the Weil bound fails to give nontrivial information. In particular, we show that the distribution of points on $$C$$ over long rectangles is Gaussian.