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My understanding is this: one can attach 2-d Galois representations to classical modular eigenforms because one can look in the etale cohomology of modular curves. For Hilbert modular forms the naive analogy breaks down, because the middle cohomology of the Hilbert modular varieties will (conjecturally at least, and possibly this is known in this case) be built up from tensor products of the 2-dimensional Galois representations attached to the automorphic representations contributing to the cohomology, and one can't unravel the prodands from the product.

One way of resolving this problem is to instead use Shimura curves. By Jacquet-Langlands, cuspidal automorphic forms on GL_2 over a totally real field (of odd degree over Q say) biject with cuspidal automorphic forms on a quaternion algebra ramified at all but one infinite place. And we have the happy coincidence that the associated algebraic group satisfies Deligne's axioms for a Shimura variety, and we can again look in the cohomology of a curve to construct the Galois representation.

This trick relies on two things, one local and one global: the local thing is that GL_2(R) has an inner form which is compact mod centre, and the global thing is that the quaternion algebra satisfies Deligne's axioms for a Shimura variety.

Now let's try and generalise all of this to the symplectic case, so G=GSp_{2g} with g>1. If the base field is Q then Weissauer and others constructed the Galois representations attached to a Siegel modular form in the case g=2 by looking in the etale cohomology of a Siegel modular 3-fold. Now what about if the base is bigger? Can one pull off the same trick?

Local question: does GSp_4(R) have an inner form which is compact mod centre?

Global question: if so, does GSp_4(F) (F totally real) sometimes have an inner form which is compact mod centre at all but one infinite place, and for which Deligne's axioms hold? If so, might one hope to see the Galois representations attached to Hilbert-Siegel modular forms over F here?

[Edit: the local question is solved below by Hansen. I thought that the papers he linked to would deal with the global question too, but now I suspect they don't. I've put a 150-point bounty on for the global question.]

[Edit: because of bounty daftness I can now no longer accept any answer for this question.]

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1 Answer 1

Local question: Yes, $GU2(\mathbb{H})$, where $\mathbb{H}$ is the Hamilton quaternions.

Global question: Not an answer, but perhaps useful to you - There are two quite relevant papers of Claus Sorenson which can be found here and here. The first paper constructs the Galois representations attached to Hilbert-Siegel modular forms over totally real fields. The second paper concerns level-lowering for GSp4 - along the way, he proves a Jacquet-Langlands transfer to an inner form of GSp4 compact at all infinite places and split at all finite places (at least when F has even degree and pi satisfies the usual conditions).

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These papers of Sorenson probably answer everything I am asking; I need to find the time to look at them before I am sure though. –  Kevin Buzzard Apr 30 '10 at 0:04
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OK so I finally got around to looking at these papers---and they don't answer the global question :-( When Sorenson constructs Galois representations he does not look in the etale cohomology of some Shimura variety, he uses base change to move first to GL(4) and then to a rank 4 unitary group, and then uses Harris-Taylor. In particular I still don't know if the Shimura variety I was wondering about exists. –  Kevin Buzzard May 13 '10 at 12:05
    
So I guess the natural candidate for the inner form in the global question would be a unitary group in two variables over a quaternion algebra D/F split at just one infinite place - a brief look at chapter 8 in Milne's notes on Shimura varieties made me believe it should be possible to do this and get a PEL Shimura variety, but I'm not sure. –  jnewton May 13 '10 at 14:00
    
Have you tried asking Tilouine? I feel like he would know about this. –  Rob Harron May 13 '10 at 17:21
    
Yes I'm sure he'd know. But nowadays, for questions like this---when it's just "an idle question" rather than "something I need right now for my research" I am more minded to ask here than to ask an expert. –  Kevin Buzzard May 13 '10 at 17:55

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