We compute the reductions of irreducible crystalline two-dimensional representations of G Qp 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 très ramifiée, in the relevant reducible nonsemisimple 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 nonstandard lattices in certain p-adic Banach spaces.• However, when v(a p ) = 1, the theorem shows that in each congruence class of weights mod (p − 1) there are further congruence classes of weights mod p whereV k,ap is irreducible.• A surprising trichotomy occurs for b = 2. This seems to be a more complicated manifestation of the dichotomy that occurs for weights r ≡ 1 mod (p − 1) when v(a p ) = 1 2 in [BG13].The proof of Theorem 1.1 uses the compatibility of the p-adic and mod p Local Langlands Correspondences, with respect to the process of reduction [Ber10]. This compatibility allows one to reduce the reduction problem to one on the 'automorphic side', namely, to computing the reduction of a lattice in a certain Banach space. Let G = GL 2 (Q p ) and let B(V k,ap ) be the unitary G-Banach space associated to V k,ap by the p-adic Local Langlands Correspondence. The reduction B(V k,ap ) ss of a lattice in this Banach space coincides with the image ofV ss k,ap under the (semisimple) mod p Local 1 Recent work by Arsovski [Ars15] includes a study of the reductionV ss k,ap when the slope is 1, for b = 3, p, though the reduction is not uniquely specified there. REDUCTIONS OF GALOIS REPRESENTATIONS OF SLOPE 1 3Langlands Correspondence defined in [Bre03b]. Since the mod p correspondence is by definition injective, it suffices to compute the reduction B(V k,ap ) ss .Let K = GL 2 (Z p ), a maximal compact open subgroup of G = GL 2 (Q p ), and let Z = Q × p be the center of G. Let X = KZ\G be the (vertices of the) Bruhat-Tits tree associated to G. The module Sym rQ2 p , for r = k − 2, carries a natural action of KZ, and the projectionconsists of all sections f : G → Sym k−2Q2 p of this local system which are compactly supported mod KZ. There is a G-equivariant Hecke operator T acting on this space of sections. Let Π k,ap be the locally algebraic representation of G defined by taking the cokernel of T − a p acting on the above space. Let Θ k,ap be the image of the integral sections ind G KZ Sym k−2Z2 p in Π k,ap . Then B(V k,ap ) is the completionΠ k,ap of Π k,ap with respect to the lattice Θ k,ap . The completionΘ k,ap , and sometimes by abuse of notation Θ k,ap itself, is called the standard lattice in B(V k,ap ). We have∼ =Θ ss k,ap . Thus, to computeV ss k,ap , it suffices to compute the reductionΘ ss k,ap of Θ k,ap . There are two main steps. First, we study a quotient P of Sym k−2F2 p whose Jordan-Hölder factors provide an upper bound on the possible Jordan-Hölder fac...

We describe an algorithm to compute the reduction modulo p of a crystalline Galois representation of dimension 2 of Gal(Q p /Qp) with distinct Hodge-Tate weights via the semi-simple modulo p Langlands correspondence. We give some examples computed with an implementation of this algorithm in SAGE. Proposition 2.3.1. Let Q be a finite length quotient of I(σ a (b)). Then the Jordan-Hoelder factors of Q are of the form π (a, λ, ω Here is a slightly more precise statement, which follows immediatly the results of [BL94, BL95, Bre03a] although it is not stated explicitely there: Proposition 2.3.2. Let Q be an irreducible representation of G, and π a nonzero map I(σ a (b)) → Q. Then there exists a unique λ such that π is zero on (T − λ)I(σ a (b)), and then Q is isomorphic toThe point is that any non-zero map I(σ a (b)) → Q where Q is irreducible is (a scalar multiple of) the obvious map, and not something else. Proof of Proposition 2.3.2. By Frobenius reciprocity we have:Hom G (I(σ a (b)), Q) = Hom K (σ a (b), Q| K ) = Hom K (σ a (b), soc K (Q)) But we know that for any irreducible Q, every representation that appears in soc K (Q) appears with multiplicity one (this is a standard computation for the principal series case, and [Bre03a, Théorème 3.2.4] for the supersingular case). So the dimension of Hom G (I(σ a (b)), Q) is at most one.

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We study the irreducible constituents of the reduction modulo p of irreducible algebraic representations V of the group Res K/Qp GL 2 for K a finite extension of Q p . We show that asymptotically, the multiplicity of each constituent depends only on the dimension of V and the central character of its reduction modulo p. As an application, we compute the asymptotic value of multiplicities that are the object of the Breuil-Mézard conjecture.

Abstract. We define deformation rings for potentially semi-stable deformations of fixed discrete series extended type in dimension 2. In the case of representations of the Galois group of Qp, we prove an analogue of the BreuilMézard conjecture for these rings. As an application, we give some results on the existence of congruences modulo p for newforms in S k (Γ 0 (p)).