Reduction of Galois Representations of slope 1

Sandra Rozensztajn (1) , Shalini Bhattacharya (2) , Eknath Ghate (3)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 Bar-Ilan University [Israël]
3 School of Mathematics [TIFR]

We compute the reductions of irreducible crystalline two-dimensional representations of GQp of slope 1, for primes p≥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.

Domaines: Théorie des nombres [math.NT]

Sandra Rozensztajn, Shalini Bhattacharya, Eknath Ghate. Reduction of Galois Representations of slope 1. Journal of Algebra, In press, 508, pp.98 -- 156. ⟨10.1016/j.jalgebra.2018.04.023⟩. ⟨hal-01402388⟩