Construction of some families of 2-dimensional crystalline representations

Abstract

We construct explicitly some analytic families ofétale (ϕ, Γ)-modules, which give rise to analytic families of 2-dimensional crystalline representations. As an application of our constructions, we verify some conjectures of Breuil on the reduction modulo p of those representations, and extend some results (of Deligne, Edixhoven, Fontaine and Serre) on the representations arising from modular forms.Résumé. -Nous construisons explicitement des familles analytiques de (ϕ, Γ)-moduleś etales, qui donnent lieuà des familles analytiques de représentations cristallines de dimension 2. Comme application de nos constructions, nous vérifions des conjectures de Breuil quant a la réduction modulo p de ces représentations, et nousétendons des résultats (de Deligne, Edixhoven, Fontaine et Serre) sur les représentations associées aux formes modulaires.

A Local-Global Compatibility Conjecture in the p-adic Langlands Programme for GL_2/Q

“…Condition (5) has been verified by Berger [5] (improving on earlier results of various authors [7,9,10,15]). We now turn to condition (7).…”

Explicit Reduction Modulo p of Certain Two-Dimensional Crystalline Representations

“…A few years ago this question seemed to be regarded as almost intractible for weights k > p: as far as we know, the only results for high weights were those of Berger, Li and Zhu ( [BLZ04]) who showed that for v(a p ) sufficiently large (an explicit bound depending on k) the answer was the same as for the case a p = 0 (which was already known). Not only that, computational evidence collected by one of us (KB) seemed to indicate that the answer to the question was in general rather subtle.…”

Serre’s modularity conjecture (II)

“…Theorem 4.1 (2)(i) for weights k ≤ p − 1 is proven in [18]. The weight k = p + 1 case is proved in [38] when the lift is non-ordinary at p: note that then residually the representation is irreducible at p of Serre weight 2 by results of Berger-Li-Zhu [8]. Alsoρ does arise from a newform of level prime to p and weight p + 1 by the weight part of Serre's conjecture together with multiplication by the Hasse invariant (see 12.4 of [26]), or Corollary 1 of Section 2 of [21], using Lemma 2 of [13] to avoid the hypothesis N > 4 in [21].…”

“…We have the following lemma which uses easy extensions of results in [22] and [7] (for (i)) and [8] (see prop. 6.1.1. for (iii)).…”

“…The case k(ρ) = p + 1, p = 2; crystalline lifts of weight p + 1. By [8], we know that such lifts are ordinary (Lemma 3.5, recall that we are supposing F v = Q p ). Ordinary lifts of weight p + 1 are the lifts that are extensions of an unramified free rank one representation by a free rank one representation with action of I v by χ p p .…”