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(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 $G_{\ell}$.

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    $\begingroup$ Ben Moonen (which if I understand correctly is your advisor; this is a comment for onlookers) gave a great talk recently on the Mumford-Tate conjecture for divisors which he can prove for many surfaces with h^{2,0}=1 beyond the K3 case. It is still based on Kuga-Satake, hence at the end of the day Hodge theory for abelian varieties (plus the machinery of Deligne and André on deformation of absolute Hodge/motivated classes). It is still a rather non-trivial variation on the K3 case. $\endgroup$
    – Simon Pepin Lehalleur
    Commented Nov 24, 2014 at 14:58
  • $\begingroup$ @SimonPepinLehalleur — Ben Moonen is indeed my advisor. I indeed know of this result, but since there is no reference yet I didn't mention it. And, like you say, it is still based on Kuga–Satake. I totally agree with you that it is a non-trivial variation. However, since I was able to overlook the results of Zhao on AV's of dimension 4 until last week, I wondered if there might be any results that do not go back to AV's. (Btw, the MT-conj is true for the generic variety; since the MT-grp and the Gal-grp are generically both the full symplectic (resp. orthogonal) grp for $i$ odd (resp. even). $\endgroup$
    – jmc
    Commented Nov 24, 2014 at 15:37
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    $\begingroup$ I think the work of Ribet on the image of the Galois representation associated to modular forms implies the Mumford-Tate conjecture for the corresponding motives. Blasius has shown that not all these motives are in LCM. $\endgroup$
    – naf
    Commented Nov 25, 2014 at 3:58
  • $\begingroup$ @ulrich – Thanks for your comment! Would you mind posting it as an answer (together with references to Ribet and Blasius)? Then I can accept it, and therewith mark this question as “Solved”. $\endgroup$
    – jmc
    Commented Nov 25, 2014 at 12:08

2 Answers 2

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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.)

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With regards to Q1, I believe one can easily prove that the Mumford-Tate conjecture holds whenever all the cohomology is generated by algebraic cycles. This occurs for example for quadric hypersurfaces, cubic surfaces, toric varieties, or more generally "celluar" varieties.

I would not say that this proof "reduces to the case of projective spaces or abelian varieties". If you want a non-rational example, Enriques surfaces should work.

Of course these don't work for Q2, since the associated motives are built out of Tate motives.

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    $\begingroup$ Thank you for your answer! I agree that it does not reduce to "projective spaces or abelian varieties", but it does fall in the class of "trivial cases" for me; exactly because their motives are built out of Tate motives. I should have been more clear about this in my question. $\endgroup$
    – jmc
    Commented Nov 24, 2014 at 17:14
  • $\begingroup$ This is helpful for me too! I realize this is a response to an old answer, but when does this also apply to products of curves $C \times C$? For example, is it enough to have the genus restriction mentioned above? It looks like trying to understand the Mumford-Tate group can get difficult even in some simple-looking cases. $\endgroup$
    – modnar
    Commented Dec 5, 2017 at 6:18
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    $\begingroup$ @modnar, Sorry for a very late reply. The Mumford-Tate group of $C^n$ is isomorphic to the Mumford-Tate group of $C$. The Mumford-Tate conjecture for $C^n$ is equivalent to MTC for $C$. For products $C_1 \times \ldots \times C_n$ of curves (or abelian varieties), see arxiv.org/abs/1804.06840 $\endgroup$
    – jmc
    Commented May 13, 2019 at 11:53

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