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 $p\ge 5$, 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 très ramifiée, 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.

Refereed version. Some arguments have been simplified in Section 7

References in corpus (6)

• On the Breuil-Mézard conjecture
• Explicit reduction modulo p of certain 2-dimensional crystalline representations, II
• La correspondance de Langlands locale p-adique pour GL_2(Q_p)
• Sur la réduction des représentations cristallines de dimension 2 en poids moyens
• Reduction modulo p of two-dimensional crystalline representations of G_{Q_p} of slope less than three
• Black Box Galois Representations

Cited by in corpus (6)

• On certain finiteness questions in the arithmetic of modular forms
• Semi-stable representations as limits of crystalline representations
• An Iwahori theoretic mod $p$ Local Langlands Correspondence
• On the locus of $2$-dimensional crystalline representations with a given reduction modulo $p$
• On the existence of analytic families of G-stable lattices and their reductions
• A Refined Lifting Theorem for Supersingular Galois Representations