Reduction of Galois Representations of slope 1

Abstract

We compute the reductions of irreducible crystalline two-dimensional representations of $G_{\mathbf{Q}_p}$ of slope 1, for primes $p \geq 5$, and all weights. We describe the semisimplification of the reductions completely. In particular, we show that the reduction is often reducible. We also investigate whether the extension obtained is peu or tr\`es ramifi\'ee, in the relevant reducible non-semisimple cases. The proof uses the compatibility between the $p$-adic and mod $p$ Local Langlands Correspondences, and involves a detailed study of the reductions of both the standard and non-standard lattices in certain $p$-adic Banach spaces.