Sur quelques représentations modulaires et <i>p</i> -adiques de GL ₂ ( <b>Q</b> _<i>p</i> ): I

Abstract

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 $\overline{\bf F}_{p}$ -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 $\overline{\bf F}_{p}$ -representations of ${\rm Gal}(\overline{\bf Q}_{p}/{\bf Q}_{p})$ .