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

TL;DR

This paper investigates the statistical distribution of L-polynomials for a family of algebraic curves, determining their Sato-Tate groups and point counts over finite fields, revealing new insights into their arithmetic properties.

Contribution

It computes the Sato-Tate groups and distributions for the curves $y^ ext{ell}=x(x^ ext{ell}-1)$, providing explicit formulas for point counts over finite fields.

Findings

Determined the Sato-Tate groups for the curves.

Established the limiting distributions of normalized L-polynomials.

Derived formulas for point counts over finite fields with specific congruence conditions.

Abstract

Let $C_\ell/\mathbb Q$ denote the 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 $q\equiv 1 \pmod{\ell^2}$.