Refined Sato–Tate conjecture for genus-2 abelian surfaces

Let $A$ be an abelian surface over a number field $k$, and let $\operatorname{ST}_{A}\subset\operatorname{USp}(4)$ be its Sato–Tate group. For each prime $\mathfrak{p}$ of good reduction, the normalized local $L$-polynomial determines a conjugacy class $s(\mathfrak{p})\in\operatorname{Conj}(\operatorname{ST}_{A})$. Define $\mu_{\operatorname{ST}_{A}}$ to be the pushforward to $\operatorname{Conj}(\operatorname{ST}_{A})$ of normalized Haar measure on $\operatorname{ST}_{A}$. Refined Sato–Tate conjecture for genus 2. The classes $s(\mathfrak{p})$ are equidistributed with respect to $\mu_{\operatorname{ST}_{A}}$. This refines the general Sato–Tate equidistribution statement by identifying the measure through the Sato–Tate group, and is presented in the source following the formulation of Fité, Kedlaya, Rotger, and Sutherland. The source uses it to study Frobenius distributions for genus-2 curves and their Jacobians.