The image of Colmez’s Montreal functor

Abstract

Abstract. We prove a conjecture of Colmez concerning the reduction modulo p of invariant lattices in irreducible admissible unitary p-adic Banach space representations of GL 2 (Qp) with p ≥ 5. This enables us to restate nicely the p-adic local Langlands correspondence for GL 2 (Qp) and deduce a conjecture of Breuil on irreducible admissible unitary completions of locally algebraic representations.

Speculations on the mod p representation theory of p-adic groups

“…In this connection, Corollary 8.11 of [23], which treats the case of GL(2, Q p ), is extremely suggestive. I note that Helm has announced in [17] a conjectural answer to the analogous question in the case of -adic representations of GL(n, F ), with F a p-adic field, p = .…”

Patching and the $p$-adic local Langlands correspondence

“…At present, the existence of such a correspondence is only known for GL 1 (F ) (where it is given by local class field theory), and for GL 2 (Q p ) (cf. [Col10;Paš13]). We do not prove that our construction gives a purely local correspondence (and it would perhaps be premature to conjecture that it should), but we are able to say enough about our construction to prove many new cases of the Breuil-Schneider conjecture, and to reduce the general case of the Breuil-Schneider conjecture (under some mild technical hypotheses) to standard conjectures related to automorphy lifting theorems.…”

Multivariable $$(\varphi ,\Gamma )$$ ( φ , Γ ) -modules and smooth o-torsion representations

“…The so-called "Montréal-functor" associates to a smooth mod p h representation π of GL 2 (Q p ) (first restricting π to a Borel subgroup B 2 (Q p )) an étale (ϕ, Γ)-module over Z/p h ((X)). By Paškūnas's work [24] this induces a bijection for certain p-adic Banach space representations of GL 2 (Q p ).…”

“…By now the p-adic Langlands correspondence for GL 2 (Q p ) is very well understood through the work of Berger [1], Breuil [4,5,6], Colmez [12], [13], Emerton [15], Kisin [21], Paškūnas [24] (see [7] for an overview). The starting point of Colmez's work is Fontaine's [18] theorem that the category of modulo p h Galois representations of Q p is equivalent to the category of étale (ϕ, Γ)-modules over Z/p h ((X)).…”