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

Abstract

We determine the Sato-Tate groups and prove the generalized Sato-Tate conjecture 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 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.

21 pages. Final version; to appear in Journal of Algebra

References in corpus (1)

• Sato-Tate groups of abelian threefolds: a preview of the classification