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The term degenerate is used to describe abelian varieties whose Hodge rings contain exceptional cycles -- Hodge cycles that are not generated by divisor classes. We can see the effect of the exceptional cycles on the structure of an abelian variety through its Mumford-Tate group, Hodge group, and Sato-Tate group. In this article we examine degeneracy through these different but related lenses. We specialize to a family of abelian varieties of Fermat type, namely Jacobians of hyperelliptic curves of the form $y^2=x^m-1$. We prove that the Jacobian of the curve is degenerate whenever $$m$$ is an odd, composite integer. We explore the various forms of degeneracy for several examples, each illustrating different phenomena that can occur. 

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 and y2 = 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. 

We consider the identity component of the Sato–Tate group of the Jacobian of curves of the form C1 : y2 = x2g+2 + c, C2 : y2 = x2g+1 + cx, C3 : y2 = x2g+1 + c, where g is the genus of the curve and c in Q* is constant. We approach this problem in three ways. First we use a theorem of Kani-Rosen to determine the splitting of Jacobians for C1 curves of genus 4 and 5 and prove what the identity component of the Sato–Tate group is in each case. We then determine the splitting of Jacobians of higher genus C1 curves by finding maps to lower genus curves and then computing pullbacks of differential 1-forms. In using this method, we are able to relate the Jacobians of curves of the form C1, C2 and C3. Finally, we develop a new method for computing the identity component of the Sato–Tate groups of the Jacobians of the three families of curves. We use this method to compute many explicit examples, and find surprising patterns in the shapes of the identity components ST^0(C) for these families of curves.