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Local-global compatibility for l = p , I

Thomas Barnet-Lamb, Toby Gee, David Geraghty, Richard Taylor (2012)

Annales de la faculté des sciences de Toulouse Mathématiques

We prove the compatibility of the local and global Langlands correspondences at places dividing l for the l -adic Galois representations associated to regular algebraic conjugate self-dual cuspidal automorphic representations of GL n over an imaginary CM field, under the assumption that the automorphic representations have Iwahori-fixed vectors at places dividing l and have Shin-regular weight.

Locally analytic vectors of unitary principal series of  GL 2 ( p )

Ruochuan Liu, Bingyong Xie, Yuancao Zhang (2012)

Annales scientifiques de l'École Normale Supérieure

The p -adic local Langlands correspondence for  GL 2 ( p ) attaches to any 2 -dimensional irreducible p -adic representation V of  G p an admissible unitary representation Π ( V ) of  GL 2 ( p ) . The unitary principal series of  GL 2 ( p ) are those Π ( V ) corresponding to trianguline representations. In this article, for  p > 2 , using the machinery of Colmez, we determine the space of locally analytic vectors Π ( V ) an for all non-exceptional unitary principal series Π ( V ) of  GL 2 ( p ) by proving a conjecture of Emerton.

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