The algebraic Sato–Tate conjecture

Let $A$ be a nonsingular projective $g$-dimensional variety over $\mathbb C$. For each prime $\ell$, let $G_{A,\ell}$ be the Zariski closure of the image of the $\ell$-adic representation on the rational Tate module, and set $G^{1}_{A,\ell}:=G_{A,\ell}\cap \operatorname{Sp}_{2g,\mathbb Q_\ell}$. Algebraic Sato–Tate conjecture. There is an algebraic subgroup $\operatorname{AST}(A)$ of $\operatorname{GSp}_{2g}$ over $\mathbb Q$, called the algebraic Sato–Tate group of $A$, such that the connected component of the identity $\operatorname{AST}^0(A)$ is reductive and, for each prime $\ell$, $G^{1}_{A,\ell}=\operatorname{AST}(A)\otimes_{\mathbb Q}\mathbb Q_\ell$. This conjecture provides an algebraic group underlying the Sato–Tate construction and relates all $\ell$-adic monodromy groups. Its resolution status is not specified in the supplied text.