The search for a p-adic local Langlands correspondence
Prof. Peter Schneider
University of Muenster, Germany.
Local Langlands correspondences are about relating representations of absolute Galois groups of local fields to representations of reductive linear groups over the same field. I will begin by reviewing the hierarchy of local Galois representations (discrete, l-adic, p-adic) as well as the classical local Langlands correspondence which deals with l-adic Galois representations. Then I will describe the very rough form a p-adic correspondence between on the one side p-adic Galois representations and on the other side p-adic Banach space representations of GL_n should have. Finally I will describe the basic ideas underlying recent work of Breuil and of Colmez in the case of 2-dimensional Galois representations over the field Q_p.