Reductions of Galois Representations for Slopes in (1,2)

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 $GL_2(\Q_p)$. We also give a complete description of the submodules generated by the second highest monomial in the mod $p$ symmetric power representations of ${GL}_2(\F_p)$.

2010 Mathematics Subject Classification: Primary: 11F80

Keywords and Phrases: Reductions of Galois representations, Local Langlands Correspondence, Hecke operators.

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