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We determine the Sato-Tate groups and prove the generalized Sato-Tateconjecture for the Jacobians of curves of the form $$ y^2=x^p-1 \text{ and }y^2=x^{2p}-1,$$ where $p$ is an odd prime. Our results rely on the fact theJacobians of these curves are nondegenerate, a fact that we prove in the paper.Furthermore, we compute moment statistics associated to the Sato-Tate groups.These moment statistics can be used to verify the equidistribution statement ofthe generalized Sato-Tate conjecture by comparing them to moment statisticsobtained for the traces in the normalized $L$-polynomials of the curves.

Sato-Tate Distributions of $y^2=x^p-1$ and $y^2=x^{2p}-1$