  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 $[0,p-1]^2$.
