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

Abstract

We describe the semisimplifications of the mod p reductions of certain crystalline two-demensional local Galois representations of slopes in (1, 2) and all weights.The proof uses the compatibility between the p-aadic and mod p Local Langlands Correspondences for GL₂ (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 GL₂ (F_p).