The generalized Sato–Tate conjecture for Jacobians

Let $C$ be a smooth projective curve over a number field $k$. Let $X_k$ be the set of conjugacy classes of $\operatorname{ST}(\operatorname{Jac}(C)_k)$, and let $\{p_i\}_{i\geq 1}$ be the primes of good reduction for $C$ over $k$, ordered by norm. For each $p_i$, let $x_{p_i}\in X_k$ be the corresponding conjugacy class. Generalized Sato–Tate conjecture. The sequence $\{x_{p_i}\}_{i\geq 1}$ is equidistributed on $X_k$ with respect to the image on $X_k$ of the Haar measure of $\operatorname{ST}(\operatorname{Jac}(C)_k)$. This generalizes the classical Sato–Tate conjecture to Jacobians and higher-dimensional abelian varieties. The paper states that the conjecture is not fully proven, although many low-dimensional cases are known.