Generalized Sato–Tate conjecture for quotients of Fermat curves

Let $E/\mathbb{Q}$ be a subextension of $F/\mathbb{Q}$, let $\mathcal G_E=\operatorname{ST}(\operatorname{Jac}(\mathcal{C}_k)_E)$, and let $\mathcal X_E$ be the set of conjugacy classes of $\mathcal G_E$. For the primes of good reduction $\{\wp_i\}_{i\geq 1}$ of $(\mathcal{C}_k)_E$, ordered by norm, let $x_{\wp_i}\in\mathcal X_E$ be the conjugacy class defined via the isomorphism $\mathcal G_E\simeq \mathcal G_F\rtimes\operatorname{Gal}(F/E)$, and write

$$x_E:=\{x_{\wp_i}\}_{i\geq 1}.$$

Generalized Sato–Tate conjecture. The sequence $x_E$ is equidistributed on $\mathcal X_E$ with respect to the image on $\mathcal X_E$ of the Haar measure of $\mathcal G_E$. This predicts the Sato–Tate distribution of Frobenius conjugacy classes for the Jacobian of the quotient curve over every subextension $E$ of $F$.