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

Let p be a large prime, ℓ⩾2 be a positive integer, m⩾2 be an integer relatively prime to ℓ and P(x)∈Fp[x] be a polynomial which is not a complete ℓ′-th power for any ℓ′ for which GCD(ℓ′,ℓ)=1. Let C be the curve defined by the equation yℓ=P(x), and take the points on C to lie in the rectangle [0,p−1]2. In this paper, we study the distribution of the number of points on C inside small rectangles among residue classes modulo m when we move the rectangle around in [0,p−1]2.