(For a formulation of the Mumford–Tate conjecture, see below.)
The question
As far as I know, all non-trivial known cases of the Mumford–Tate conjecture more or less depend on the Mumford–Tate conjecture for Abelian varieties.
That is, we know it for:
- Projective spaces (trivial)
- [edit] Other varieties whose cohomology is generated by algebraic cycles (more or less trivial) [/edit]
- Abelian varieties up to dimension $7$ (I guess, maybe some edge cases left?)
- Curves of genus $\le 7$ (uses the above fact)
- K3-surfaces (using the Kuga–Satake construction, hence AV's)
[edit] I am probably missing out on some results (Pink proved MT-conj for lots of AV's of certain dimensions, etc… The list is to give an idea of the type of results. [/edit]
Q1: Is there an example of a variety $X$ for which the Mumford–Tate conjecture is known, but the proof does not reduce to AV's and projective spaces?
Preciser motivic (up to a precise notion of motive) formulation of the question:
Q2: Is there an example of a motive $M$, such that (i) the Mumford–Tate conjecture is known for $M$, and (ii) the motive $M$ is not in the category of motives generated by AV's and the Tate/Lefschetz motive. (I.e., Milne's $\mathbf{LCM}$ category, in [Milne, 1999b].)
Motivation for the question
Originally the Mumford–Tate conjecture was formulated for Abelian varieties. For Abelian varieties we indeed know some examples where it is true. It seems very natural to generalise the conjecture to arbitrary smooth projective varieties, and I think nowadays most people mean the general version when referring to the conjecture. However, I do not know of any evidence for the more general version apart from undeniable beauty.
Background (formulation of the Mumford–Tate conjecture)
Let $k$ be a finitely generated field of characteristic $0$.
Let $X$ be a projective smooth variety over $k$.
Let $i$ be an integer.
The Betti cohomology $H^{i} = H^{i}(X(\mathbb{C}), \mathbb{Q})$ carries a Hodge structure.
The category of (pure) Hodge structures is Tannakian.
Therefore, $H^{i}$ generates a sub-Tannakian category $\langle H^{i} \rangle^{\otimes}$, and the associated affine group scheme over $\mathbb{Q}$ is called the Mumford–Tate group. We denote it with $\mathrm{MT}_{X}^{i}$.
(Alternatively, it is the Zariski closure over $\mathbb{Q}$ of the image of the Deligne torus in $\mathrm{GL}(H^{i})$.)
The $\ell$-adic étale cohomology $H_{\ell}^{i} = H_{\text{ét}}^{i}(X_{\bar{k}}, \mathbb{Q}_{\ell})$ carries a Galois representation.
The category of (finite-dimensional) Galois representations is Tannakian.
Therefore, $H_{\ell}^{i}$ generates a sub-Tannakian category $\langle H_{\ell}^{i} \rangle^{\otimes}$, and the associated affine group scheme over $\mathbb{Q}_{\ell}$ is called the $\ell$-adic monodromy group. We denote it with $G_{\ell}$.(Alternatively, it is the Zariski closure over $\mathbb{Q}_{\ell}$ of the image of the Galois group in $\mathrm{GL}(H_{\ell}^{i})$.)
The Mumford–Tate conjecture (for $X$ and $i$) is:
$\mathbf{MT}^{i}(X)$: Via the comparison isomorphism of Betti cohomology and $\ell$-adic étale cohomology, the group $\mathrm{MT}_{X}^{i} \times_{\mathbb{Q}} \mathbb{Q}_{\ell}$ is isomorphic to $G_{\ell}^{0}$, the identity component of $G_{\ell}$.
Best Answer
It follows from results of Ribet in "On l-adic representations attached to modular forms" (Invent. Math. 28 (1975), 245–275) that the Mumford-Tate conjecture holds for the motives attached to modular forms for $SL_2(\mathbb{Z})$.
Blasius shows in the article "Modular forms and abelian varieties" (Séminaire de Théorie des Nombres, Paris, 1989–90, 23–29,Progr. Math., 102) that the motives attached to these forms are not in LCM (as defined in the question) if the weight is $>2$.
(Similar results probably hold in much greater generality.)