Banaszak–Kedlaya's algebraic Sato–Tate conjecture

Let $A/F$ be an algebraic variety of dimension $g$ over a number field. Let $G_{A,\ell}$ be the Zariski closure of the image of its $\ell$-adic representation, 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, such that $\operatorname{AST}^0(A)$ is reductive and, for every prime $\ell$,

$$G^1_{A,\ell}=\operatorname{AST}(A)\otimes_{\mathbb Q}\mathbb Q_\ell.$$

This conjecture was proposed as a refinement related to the Mumford–Tate conjecture. It provides an algebraic group whose base changes describe the $ell$-adic monodromy groups and underlies the construction of the Sato–Tate group.