# Mumford-Tate conjecture for mixed Tate motives

Let $$X$$ be a (not necessarily smooth or proper) variety over a number field $$k$$. Suppose we are given

1. A subquotient $$V_{dR}$$ of the algebraic de Rham cohomology $$H_{dR}^i(X)$$ (defined in the non-smooth case via a thickening, as in e.g. Hartshorne's algebraic de Rham cohomology paper),

2. For each embedding $$\iota: k\to \mathbb{C}$$, a subquotient $$V_\iota$$ of $$H^i(X_{\mathbb{C}}^{an}, \mathbb{Q})$$ respecting the $$\mathbb{Q}$$-mixed Hodge structure, and

3. For some fixed prime $$\ell$$ and algebraic closure $$\bar k$$ of $$k$$, a subquotient $$V_\ell$$ of $$H^i(X_{\bar k, \text{ét}}, \mathbb{Q}_\ell)$$ respecting the action of $$\text{Gal}(\bar k/k)$$, such that

4. For each $$\iota$$, the corresponding comparison isomorphism $$H^i_{dR}(X)\otimes_k \mathbb{C} \to H^i(X_{\mathbb{C}}^{an}, \mathbb{Q})\otimes \mathbb{C}$$ sends $$V_{dR}$$ to $$V_\iota$$, and

5. For each $$\iota$$, the corresponding comparison isomorphism $$H^i(X_{\mathbb{C}}^{an}, \mathbb{Q})\otimes \mathbb{Q}_\ell \to H^i(X_{\bar k, \text{ét}}, \mathbb{Q}_\ell)$$ sends $$V_\iota$$ to $$V_\ell$$, and

6. The $$V_\iota$$ is of mixed Tate type (i.e. all $$h^{p,q}$$'s vanish for $$p\not=q$$). Or equivalently $$V_\ell$$ is an iterated extension of powers of the cyclotomic character.

(If you'd like, you can think of $$V$$ as the realization of a mixed Tate motive over $$k$$ under any of the various formalisms for mixed Tate motives over number fields -- I think that a priori the conditions above are slightly weaker but expected to be equivalent.)

I'd like to know -- is the analogue of the Mumford-Tate conjecture known to be true for such $$V$$? Explicitly, one expects that the Lie algebra of the image of $$\text{Gal}(\bar k/k)\to GL(V_\ell)$$ is the extension of scalars to $$\mathbb{Q}_\ell$$ of the Lie algebra of a $$\mathbb{Q}$$-group, given as the Tannaka dual of the subcategory of mixed Tate Hodge structures over $$k$$, generated by the Hodge realization $$(V_{dR}, V_{\iota_1}, V_{\iota_2}, \cdots)$$.

My feeling is that one can probably extract this from the literature by comparing ranks of various Ext groups (in one's favorite category of mixed Tate motives and in the $$\ell$$-adic and Hodge settings) but I am hoping it is written explicitly somewhere.

EDIT: Fixed some errors in the formulation of the Hodge realization.

• Hi Daniel. I'm not sure it is written down anywhere, so it would be nice if you did. Dec 30, 2020 at 18:18
• @DonuArapura It seems like I might have to, since I have an application in mind! Dec 30, 2020 at 18:44