Sato-Tate distributions of y2 = xp − 1 and y2 = x2p − 1

Abstract

We determine the Sato-Tate groups and prove the generalized Sato-Tate conjecture for the Jacobians of curves of the form y2=xp−1 and y2=x2p−1, where p is an odd prime. Our results rely on the fact the Jacobians 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 of the generalized Sato-Tate conjecture by comparing them to moment statistics obtained for the traces in the normalized L-polynomials of the curves.