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Let $E$ be a motive over $\mathbb Q$. (I should precise, that by a motive I mean here a pure motive over $\mathbb Q$, with coefficients in $\mathbb Q$, that I see here as a conjectural object which exists in a world were standard conjectures and anything else you can expect is true, in the spirit of Grothendieck, and like in Serre's paper in Motives, PSPM 55, volume 1 - a paper which unfortunately does not seem to be online except partially on Amazom).

Recall that the motivic galois group of $E$, denoted $G_{M(E)}$, is defined as the tensor-automorphic group of the functor "Betty realization" from the category $M(E)$, defined as the smallest Tanakkian sub-category of the category of motives containing $E$, to the category of $\mathbb Q$-vector spaces.

An equivalent (under the conjectures of Tate and Hodge) and more concrete (at least for me) definition is as follows (see the same paper of Serre): let $H_b(E)$ be the Betty realization of $E$ (a finite-dimensional $\mathbb Q$-vector space), and let $\rho_\ell$ be the $\ell$-adic realization of $E$, which is a continuous semi-simple representation $\rho_\ell$ of $\Gamma_{\mathbb Q}$ (the absolute Galois group of $\mathbb Q$) over $H_b(E) \otimes \mathbb Q_\ell$. Then $G_{M(E)}$ is the algebraic subgroup $Gl(H_b(E))$ such that $G_{M(E)} \otimes \mathbb Q_\ell$ is the Zariski-closure of the image of $\rho_\ell$, for every prime number $\ell$.

In any case, $G_{M(E)}$ is a reductive group. My question is

What reductive groups over $\mathbb Q$ arise as $G_{M(E)}$ for some pure motive $E$?

This question is asked by Serre in the section 8 of his paper. At that time there was not much known on it apparently (it was 20 years ago): Serre notices that such a group $G$ must have an element $\gamma \in G(\mathbb R)$ such that $\gamma^2=1$ and whose centralizer is a maximal compact subgroup of $G(\mathbb R)$. This necessary conditions implies that $Sl_2$ is not a motivic Galois group. On the positive side, I believe that symplectic group $GSp_{2n}$ are known to be motivic Galois group (attached to $E=$ an abelian variety over $\mathbb Q$ of dimension $n$, sufficiently generic), and I know a few other examples like those attached to elliptic curves with complex multiplication.

There has been a lot of work and progresses on motives since then, and I am asking to know what progresses have been made in the direction of this question. I would be very surprised if it had been solved completely (even assuming the standard conjectures + Hodge conjecture + Tate conjecture), but I would be interested in any partial results:

For example, what if we just try to classify the connected reductive group which arise as neutral component of a $G_{M(E)}$? those groups up to isogeny ? those groups after extension to the algebraic closure of $\mathbb Q$? Conjectural answers (conjectural in the sense that they are not proved even assuming the conjectures assumed above) are also more than welcome.

Even a list of example of groups that are provably or possibly motivic Galois group which goes beyond the small list I have given would be very useful. And of course, any references more recent than Serre's could be very useful.

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It's rather explicitly known which reductive groups arise as the motivic Galois groups of abelian motives (i.e., those in the Tannakian category generated by abelian varieties). See Deligne's Corvallis talk and Milne's second Seattle talk (same conference as Serre's article). Beyond that, not much more is known (see Rabelais's answer). It's hoped that all the groups defining Shimura varieties with rational weight arise as motivic Galois groups, but this is a (very interesting) open question. –  anon Apr 15 '13 at 18:16

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up vote 10 down vote accepted

The first question (applied to $\mathrm{GL}(2)$-abelian varieties over $\mathbf{Q}$) seems to include the following problem: what totally real fields $F$ occur as the field of coefficients of a classical weight $2$ modular form? This seems a totally impossible question to answer. For example, it includes the question of which Hilbert modular surfaces $X_F$ have rational points; since $X_F$ is of general type for $F$ of suitable large discriminant, the answer seems hard to predict in advance (especially because fields $F$ of arbitrarily large degree do actually occur).

For the second, surely the work of Zhiwei Yun (http://arxiv.org/pdf/1112.2434v1.pdf) is relevant here.

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