Sato–Tate groups and distributions of $y^\ell =x(x^\ell -1)$

Let $C_\ell /\mathbb Q$ denote the nonsingular curve with affine model $y^\ell =x(x^\ell -1)$, where $\ell \geq 3$ is prime. In this paper we study the limiting distributions of the normalized $L$-polynomials of the curves by computing their Sato–Tate groups and distributions. We also provide results for the number of points on the curves over finite fields, including a formula in terms of Jacobi sums when the field $\mathbb F_q$ satisfies $q\equiv 1\,(\mathrm {mod}\,\ell ^2)$.