Reductions of Galois representations for slopes in $(1,2)$

Abstract

We describe the semisimplifications of the mod $p$ reductions of certain crystalline two-dimensional local Galois representations of slopes in $(1,2)$ and all weights. The proof uses the compatibility between the $p$-adic and mod $p$ Local Langlands Correspondences for ${\mathrm{GL}}_2({\mathbb{Q}}_p)$. We also give a complete description of the submodules generated by the second highest monomial in the mod $p$ symmetric power representations of ${\mathrm{GL}}_2({\mathbb{F}}_p)$.