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 $k\ge 3p+4$ with $k\not\equiv 6\mathrm{mod}(p-1)$. The proof relies on the compatibility between the $p$-adic and the mod $p$ Local Langlands Correspondences for ${\mathrm{GL}}_2({\mathbb{Q}}_p)$.

arXiv comment74 pages. Comments welcome