Reductions of Galois representations of slope $2$

Abstract

Let $p$ be a prime $\ge 5$. We determine the semisimplifications of the mod $p$ reductions of irreducible two-dimensional crystalline representations of the Galois group of ${\mathbb Q}_p$ of slope $2$ and weights $k\ge3p+4$ with $k \not \equiv 6 \bmod (p-1)$. The proof relies on the compatibility between the $p$-adic and the mod $p$ Local Langlands Correspondences for $\mathrm{GL}_2(\mathbb{Q}_p)$.