TITLE: Are there motives which do not, or should not, show up in the cohomology of any Shimura variety?
QUESTION [42 upvotes]: Let $F$ be a real quadratic field and let $E/F$ be an elliptic curve with conductor 1 (i.e. with good reduction everywhere; these things can and do exist) (perhaps also I should assume E has no CM, even over F-bar, just to avoid some counterexamples to things I'll say later on). Let me assume that $E$ is modular. Then there will be some level 1 Hilbert modular form over $F$ corresponding to $E$. But my understanding is that the cohomology of $E$ will not show up in any of the "usual suspect" Shimura varieties associated to this situation (the level 1 Hilbert modular surface, or any Shimura curve [the reason it can't show up here is that a quaternion algebra ramified at precisely one infinite place must also ramify at one finite place]).
If you want a more concrete assertion, I am saying that the Tate module of $E$, or any twist of this, shouldn't show up as a subquotient of the etale cohomology of the Shimura varieties attached to $GL(2)$ or any of its inner forms over $F$ (my knowledge of the cohomology of Hilbert modular surfaces is poor though; I hope I have this right).
But here's the question. I have it in my head that someone once told me that $E$ (or perhaps more precisely the motive attached to $E$) should not show up in the cohomology of any Shimura variety. This is kind of interesting, because here is a programme for meromorphically continuing the L-function of an arbitrary smooth projective variety over a number field to the complex plane:
1) Observe that automorphic forms for GL_n have very well-behaved L-functions; prove that they extend to the whole complex plane. [standard stuff].
2) Prove the same for automorphic forms on any connected reductive algebraic group over a number field [i.e. prove Langlands functoriality]
3) Prove that the L-functions attached to the cohomology of Shimura varieties can be interpreted in terms of automorphic forms [i.e. prove conjectures of Langlands, known in many cases]
4) Prove that the cohomology of any algebraic variety at all (over a number field) shows up in the cohomology of a Shimura variety. [huge generalisation of Taniyama-Shimura-Weil modularity conjecture]
My understanding is that this programme, nice though it looks, is expected to fail because (4) is expected not to be true. And I believe I was once assured by an expert that the kind of variety for which problems might occur is the elliptic curve over $F$ mentioned above. At the time I did not understand the reasons given to me for why this should be the case, so of course now I can't reproduce them.
Have I got this right or have I got my wires crossed?
EDIT (more precisely, "addition"): Milne's comment below seems to indicate that I did misremember, and that in fact I was probably only told what Milne mentions below. So in fact I probably need to modify the question: the question I'd like to ask now is "is (4) a reasonable statement?".

REPLY [4 votes]: In their 2017 preprint
On subquotients of the étale cohomology of Shimura varieties
C.Johannsson and J.Thorne have made substantial progresses towards question 4. In particular, they show that under reasonable conjectures, the motive attached to an elliptic curve over an imaginary CM field does not appear as subquotient of the intersection cohomology of the minimal compactification of a Shimura variety, and hence should also not appear in the ordinary cohomology of open Shimura varieties.
Very roughly, they show that the conjectures of Arthur on $A$-packets and of Kottwitz on the intersection cohomology of Shimura varieties imply that any Galois representation of the absolute Galois group of a CM field which shows up as a subquotient of the intersection cohomology of Shimura varieties and which is strongly irreducible, that is to say which remains irreducible after restriction to the absolute Galois group of any finite extension, is conjugate self-dual up to a twist. As there are motivic strongly irreducible Galois representations which are not conjugate self-dual up to a twist, this gives a negative answer to statement 4.
It seems to me, by the way, that their work is a concrete implementation of the strategy suggested in the answer of Laurent F. (whoever that guy is).