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

Abstract

Let $p$ be a large prime, $\ell \ge 2$ be a positive integer, $m\ge 2$ be an integer relatively prime to $\ell$ and $P(x)\in {\mathbb{F}}_p[x]$ be a polynomial which is not a complete ${\ell }^{\prime }$-th power for any ${\ell }^{\prime }$ for which $GCD({\ell }^{\prime },\ell )=1$. 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 $[0,p-1{]}^2$.

23 pages