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, and the relationship between the Tate conjecture and the $\ell$-adic monodromy group (Proposition, 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.

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    $\begingroup$ Your statement for the period conjecture is not the right one - it implies that the motivic Galois group of a disjoint union of $n$ points is $GL_n$. The right statement would be to take the smallest Zariski closed subset defined over $\mathbb Q$ containing the period matrix, translating it by any rational point of itself so it contains the identity, and then taking the subgroup it generates - this should be the motivic fundamental group of the Tannakian category of mixed Hodge structures. $\endgroup$ – Will Sawin May 6 at 20:59
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    $\begingroup$ In general, it's wrong to think of these groups as being associated to cohomology theories, but rather to the (Tannakian) categories these cohomology theories end up in - Galois representations, mixed Hodge structures, etc. So which structure you consider to be the de Rham Galois group depends on which category you want de Rham cohomology to lie in. $\endgroup$ – Will Sawin May 6 at 21:04
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    $\begingroup$ If the only thing you want is that it conjecturally agree with the motivic Galois group, you have many options - for instance you could take the Mumford-Tate or $\ell$-adic group, transfer it to de Rham cohomology by a period map, and take a $\mathbb Q$-Zariski closure. $\endgroup$ – Will Sawin May 6 at 21:04

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