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 \operatorname{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 {\operatorname{GL}}_2(\mathbb F_p) .