Sato–Tate Distributions of Catalan Curves

For distinct odd primes p and q, we define the Catalan curve C p,q by affine equation y q =x -1. In this article construct Sato–Tate groups of Jacobians in order to study limiting distributions coefficients their normalized L-polynomials. are nondegenerate simple with noncyclic Galois (of endomorphism fields over ℚ), thus making them interesting varieties context groups. We compute both statistical numerical moments for distributions. Lastly, determine types using old new techniques.