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We consider the identity component of the Sato–Tate group of the Jacobian of curves of the form [Formula: see text] where [Formula: see text] is the genus of the curve and [Formula: see text] is constant. We approach this problem in three ways. First we use a theorem of Kani-Rosen to determine the splitting of Jacobians for [Formula: see text] 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 [Formula: see text] 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 [Formula: see text], [Formula: see text] and [Formula: see text]. 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 [Formula: see text] for these families of curves.

Each orthogonal group Opnq has a nontrivial GLp1q-extension, which we call GPinpnq. The identity component of GPinpnq is the more familiar GSpinpnq, the general Spin group. We prove that the restriction to GPinpn ´1q of an irreducible admissible representation of GPinpnq over a nonarchimedean local field of characteristic zero is multiplicity free and also prove the analogous theorem for GSpinpnq. The case for GPinpnq is the analogue of the theorem for Opnq proven by Aizenbud, Gourevitch, Rallis and Schiffmann, and the case for GSpinpnq the one for SOpnq by Waldspurger.