Let $ X$ be a smooth projective variety over $ \mathbb{Q}$ . The theory of motives predicts that for each cohomology theory, there should be a distinguished Zariski closed subgroup of $ GL(H^k_{\bullet}(X))$ , the motivic Galois group. This group has conjectural descriptions for Betti and $ \ell$ -adic cohomology, and I am wondering if there is any literature on this group for de Rham cohomology.

The Betti cohomology $ H^k_B(X)$ is a rational Hodge structure, hence there is a representation $ \mathrm{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}_m\to GL(H^k_B(X)\otimes\mathbb{R})$ . The Mumford-Tate group $ MT^k(X)\leq GL(H^k_B(X))$ is defined to be the smallest Zariski-closed subgroup containing the image of the representation. The Hodge conjecture would imply that $ MT^k(X)$ has finite index in the Betti motivic Galois group.

The $ \ell$ -adic cohomology $ H^k(X;\mathbb{Q}_\ell)$ is a representation of the absolute Galois group, and the $ \ell$ -adic monodromy group $ G_\ell^k(X)\leq GL(H^k(X;\mathbb{Q}_\ell))$ is defined to be the smallest Zariski-closed subgroup containing the image of the representation. The Tate conjecture would imply that $ G_\ell^k(X)$ is the full $ \ell$ -adic motivic Galois group.

Is there any literature on an analogue of the Mumford-Tate group or the $ \ell$ -adic monodromy group inside $ GL(H^k_{dR}(X))$ ?

I believe that a conjecture of Ogus would imply that the de Rham motivic Galois group is smallest Zariski closed subgroup whose $ \mathbb{Q}_p$ points contain $ F_p$ for all sufficiently large $ p$ (where $ F_p\in GL(H^k_{dR}(X))(\mathbb{Q}_p)$ is the crystalline Frobenius). However I have some doubt about this, because André’s book on motives states the relationship between the Hodge conjecture and the Mumford-Tate group (Proposition 7.2.2.1), and the relationship between the Tate conjecture and the $ \ell$ -adic monodromy group (Proposition 7.3.2.1), but does not give an analogous statement for the Ogus conjecture.

I also believe that the period conjecture of Grothendieck would imply that the de Rham motivic Galois group is the smallest Zariski closed subgroup containing all elements $ \varphi\in GL(H^k_{dR}(X))(\mathbb{Q})$ whose image in $ GL(H^k_{B}(X))(\mathbb{C})$ under the Betti-de Rham comparison isomorphism is contained in $ GL(H^k_{B}(X))(\mathbb{Q})$ . However, I also cannot find a statement like this in André’s book.