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SUMMARY:p-adically completed cohomology and the p-adic Langlands program -
  Emerton\, M (Northwestern)
DTSTART:20090730T090000Z
DTEND:20090730T100000Z
UID:TALK19266@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:Speaking at a general level\, a major goal of the p-adic Langl
 ands program (from a global\, rather than local\, perspective) is to find 
 a p-adic generalization of the notion of automorphic eigenform\, the hope 
 being that every p-adic global Galois representation will correspond to su
 ch an object. (Recall that only those Galois representations that are moti
 vic\, i.e. that come from geometry\, are expected to correspond to classic
 al automorphic eigenforms). \n\nIn certain contexts (namely\, when one has
  Shimura varieties at hand)\, one can begin with a geometric definition of
  automorphic forms\, and generalize it to obtain a geometric definition of
  p-adic automorphic forms. However\, in the non-Shimura variety context\, 
 such an approach is not available. Furthermore\, this approach is somewhat
  remote from the representation-theoretic point of view on automorphic for
 ms\, which plays such an important role in the classical Langlands program
 . \n\nIn this talk I will explain a different\, and very general\, approac
 h to the problem of p-adic interpolation\, via the theory of p-adically co
 mpleted cohomology. This approach has close ties to the p-adic and mod p r
 epresentation theory of p-adic groups\, and to non-commutative= Iwasawa th
 eory. \n\nAfter introducing the basic objects (namely\, the p-adically com
 pleted cohomology spaces attached to a given reductive group)\, I will exp
 lain several key conjectures that we expect to hold\, including the conjec
 tural relationship to Galois deformation spaces. Although these conjecture
 s seem out of reach at present in general\, some progress has been made to
 wards them in particular cases. I will describe some of this progress\, an
 d along the way will introduce some of the tools that we have developed fo
 r studying p-adically completed cohomology\, the most important of these b
 eing the Poincare duality spectral sequence. \n\nThis is joint work with F
 rank Calegari.
LOCATION:Seminar Room 1\, Newton Institute
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