Sur quelques représentations modulaires et p -adiques de GL ₂ ( Q p ): I

Let p be a prime number and F a complete local field with residue field of characteristic p . In 1993, Barthel and Livné proved the existence of a new kind of F̄ₚ -representations of GL 2 ( F ) that they called ‘supersingular’ and on which one knows almost nothing. In this article, we determine all the supersingular representations of GL 2 ( Q p ) with their intertwinings. This classification shows a natural bijection between the set of isomorphism classes of supersingular representations of GL 2 ( Q p ) and the set of isomorphism classes of two-dimensional irreducible F̄ₚ -representations of Gal( Q̄ₚ/ Qₚ) .