# Is there literature on a de Rham analogue of the Mumford-Tate group or ell-adic monodromy group?

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.

• 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. – Will Sawin May 6 at 20:59
• 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. – Will Sawin May 6 at 21:04
• 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. – Will Sawin May 6 at 21:04