The Mumford--Tate conjecture for abelian varieties

Let $A$ be an abelian variety of dimension $g$ over a number field, and fix a prime $\ell$. Let $G_\ell^{\rm zar}\subseteq \operatorname{GSp}_{2g}$ and $G_\ell^{1,\rm zar}\subseteq \operatorname{Sp}_{2g}$ be the Zariski closures of the corresponding $d9\ell$-adic Galois representations. Let $\operatorname{MT}(A)$ and $\operatorname{Hg}(A)$ denote the Mumford--Tate and Hodge groups of $A$. Mumford--Tate conjecture. The identity component of $G_\ell^{\rm zar}$ equals

$$\operatorname{MT}(A)\otimes_{\mathbf{Q}}\mathbf{Q}_\ell;$$

equivalently, the identity component of $G_\ell^{1,\rm zar}$ equals

$$\operatorname{Hg}(A)\otimes_{\mathbf{Q}}\mathbf{Q}_\ell.$$

Deligne proved the corresponding inclusions, and the equalities are known for abelian varieties of dimension $g\leq 3$. In general the conjecture remains open; when true, it determines the identity component of the Sato--Tate group up to conjugacy.