Sato-Tate Groups and Distributions of $y^{\ell }=x(x^{\ell }-1)$

Abstract

Let $C_{\ell }/\mathbb{Q}$ denote the curve with affine model $y^{\ell }=x(x^{\ell }-1)$, where $\ell \ge 3$ 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 $q\equiv 1(\mathrm{mod}{\ell }^2)$.

Final version. Accepted to Acta Arithmetica