Generic generalized Sato–Tate conjecture for abelian varieties

Let $A$ be an abelian variety of dimension $g\geq 1$ over a number field $K$. For each prime ideal $\mathfrak{q}$ of good reduction, let $\overline{P}_{\mathfrak{q}}(T)$ be the renormalized Frobenius polynomial, and let $\operatorname{Conj}(\operatorname{USp}(2g))$ denote the space of conjugacy classes in the unitary symplectic group. Generalized Sato–Tate conjecture in the generic case. If $A$ has “no extra structure,” then the sequence of conjugacy classes in $\operatorname{Conj}(\operatorname{USp}(2g))$ corresponding to the polynomials $\overline{P}_{\mathfrak{q}}(T)$ is equidistributed with respect to the image of Haar measure. This is the higher-dimensional analogue of the non-CM elliptic-curve case; the phrase “no extra structure” is not further formalized in the supplied context.