Nondegeneracy and Sato-Tate Distributions of Two Families of Jacobian Varieties

Abstract

We consider the curves $y^2=x^{2^m} -1$ and $y^2=x^{2^{d}+1}-x$ over the rationals. These curves are related via their associated Jacobian varieties in that the Jacobians of the latter appear as factors of the Jacobians of the former. One of the principal aims of this paper is to fully describe their Sato-Tate groups and distributions by determining generators of the component groups. In order to do this, we first prove the nondegeneracy of the two families of Jacobian varieties via their Hodge groups. We then use results relating Sato-Tate groups and twisted Lefschetz groups of nondegenerate abelian varieties to determine the generators of the associated Sato-Tate groups. The results of this paper add new examples to the literature of families of nondegenerate Jacobian varieties and of noncyclic component groups of Sato-Tate groups. Furthermore, we compute moment statistics associated to the Sato-Tate groups which 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.