Analogue of Shimura curves in the symplectic case? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T07:38:27Z http://mathoverflow.net/feeds/question/23002 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23002/analogue-of-shimura-curves-in-the-symplectic-case Analogue of Shimura curves in the symplectic case? Kevin Buzzard 2010-04-29T16:13:19Z 2010-05-20T23:07:03Z <p>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.</p> <p>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.</p> <p>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.</p> <p>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?</p> <p>Local question: does GSp_4(R) have an inner form which is compact mod centre?</p> <p>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?</p> <p>[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.]</p> <p>[Edit: because of bounty daftness I can now no longer accept any answer for this question.]</p> http://mathoverflow.net/questions/23002/analogue-of-shimura-curves-in-the-symplectic-case/23008#23008 Answer by David Hansen for Analogue of Shimura curves in the symplectic case? David Hansen 2010-04-29T16:58:38Z 2010-04-29T16:58:38Z <p>Local question: Yes, \$GU2(\mathbb{H})\$, where \$\mathbb{H}\$ is the Hamilton quaternions.</p> <p>Global question: Not an answer, but perhaps useful to you - There are two quite relevant papers of Claus Sorenson which can be found <a href="http://www.math.princeton.edu/~csorense/gal_final.pdf" rel="nofollow">here</a> and <a href="http://www.math.princeton.edu/~csorense/lower.pdf" rel="nofollow">here</a>. 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 <em>all</em> infinite places and split at all finite places (at least when F has even degree and pi satisfies the usual conditions).</p>