Reductions of Galois representations for slopes in \((1,2)\) (Q273538)

Summary: 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 \({\mathrm {GL}}_2(\mathbb F_p)\).