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?".
Best Answer
The question may be precised depending on what you call "show up". More precisely:
Concerning the cohomology of Shimura varieties there are two points of view: intersection cohomology or ordinary cohomology (this is of course the same for compact Shimura varieties).
Concerning the fact that the cohomology shows up in something we can add an option: it may show up potentially.
Then we can add another option: to proove it shows up in the Tannakian sub category of motives generated by the motives of Shimura varieties (even weaker (?): the class in the K_0 of motives of your variety is a virtual combination of the classes of motives showing up in Shimura varieties).
Let's first say we look at intersection cohomology. As stated your hope 4) is false for trivial reasons: the only Artin motives showing up in the intersection cohomology of Shimura varieties are the abelian one...In fact by purity they show up only in the H^0 that has been computed by Deligne and is abelian.
Of course if you put option (2) in my list this counterexample disappears.
Now you may say: yes but we can twist an Artin motive by a CM character and ask the same question. This is where I come to the following point: you're saying that because the twisting operation that is a particular case of Langlands functoriality is a known Langlands functoriality. Where I want to come is that in fact if you suppose Langlands functoriality known then the fact that your variety shows up in the Tannakian category generated by motives of Shimura varieties implies its L function is automorphic (tensor product functoriality).
If you suppose Langlands functoriality and your variety shows up potentially in the motive of a Shimura variety then its L-function is automorphic (existence of automorphic induction which implies for example Artin conjecture).
About the intersection cohomology of Shimura varieties: it is now pretty well understood and I think there is no reason why any variety would show up potentially in it. More precisely the Langlands parameters of automorphic representations showing up in the intersection cohomology of Shimura varieties factor through some representation $r_\mu:\;^L G_E\rightarrow GL_n$ where $G$ is the group attached to the Shimura variety (well to be more serious I woud have to invoke cohomological Arthur's parameter but it would take 5 hours to write this in details). Thus I clearly think the class of varieties that show up potentially in the cohomology of Shimura varieties has some serious restrictions...
Now there is another thing I did not speak about: the cohomology of non-compact Shimura varieties that may not be pure. For this little is known and it may be possible some interesting Galois representations that do not show up in the intersection cohomology of Shimura varieties show up in the cohomology...I know some people are looking at this (I won't give any name, even if I'm tortured) but as I said up to now little is known.
Well, I will stop here since this is an endless story and you can speak about this during hours...