Mumford–Tate conjecture for étale cohomology

Let $X$ be a smooth projective variety defined over a number field $k$, let $\ell$ be a prime, and set

$$H=\operatorname{H}^{2i}(X(\mathbb{C}),\mathbb{Q}(i)).$$

Let $G_\ell$ be the generic fiber of the Zariski closure of the image of the $\ell$-adic Galois representation on $\operatorname{H}^{2i}_{\operatorname{et}}(\overline{X},\mathbb{Q}_\ell(i))$, and write $G_\ell^\circ$ for its identity component. Mumford–Tate conjecture. Under the comparison isomorphism

$$\operatorname{H}^{2i}_{\operatorname{et}}(\overline{X},\mathbb{Q}_\ell(i))\cong H\otimes\mathbb{Q}_\ell,$$

the Mumford–Tate group $\operatorname{MT}(H)\times_{\mathbb{Q}}\mathbb{Q}_\ell$ is isomorphic, as an algebraic group, to $G_\ell^\circ$. This conjecture predicts that the Hodge-theoretic Mumford–Tate group and the arithmetic $\ell$-adic algebraic monodromy group agree after passage to $\mathbb{Q}_\ell$; its status is not resolved in the supplied text.