We prove the weight part of Serre's conjecture in generic situations for forms of U (3) which are compact at infinity and split at places dividing p as conjectured by [Her09]. We also prove automorphy lifting theorems in dimension three. The key input is an explicit description of tamely potentially crystalline deformation rings with Hodge-Tate weights (2, 1, 0) for K/Qp unramified combined with patching techniques. Our results show that the (geometric) Breuil-Mézard conjectures hold for these deformation rings. M denote the O-flat and reduced quotient of R τ,β M such that Spec R τ,β,∇ M

Abstract. Suppose that F/F + is a CM extension of number fields in which the prime p splits completely and every other prime is unramified. Fix a place w|p of F . Suppose that r : Gal(F /F ) → GL 3 (Fp) is a continuous irreducible Galois representation such that r| Gal(F w /Fw) is upper-triangular, maximally non-split, and generic. If r is automorphic, and some suitable technical conditions hold, we show that r| Gal(F w /Fw) can be recovered from the GL 3 (Fw)-action on a space of mod p automorphic forms on a compact unitary group. On the way we prove results about weights in Serre's conjecture for r, show the existence of an ordinary lifting of r, and prove the freeness of certain Taylor-Wiles patched modules in this context. We also show the existence of many Galois representations r to which our main theorem applies.

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Let p be an odd rational prime and F a p-adic field. We give a realization of the universal p-modular representations of GL 2 (F) in terms of an explicit Iwasawa module. We specialize our constructions to the case F = Q p , giving a detailed description of the invariants under principal congruence subgroups of irreducible admissible p-modular representations of GL 2 (Q p), generalizing previous work of Breuil and Paskunas. We apply these results to the local-global compatibility of Emerton, giving a generalization of the classical multiplicity one results for the Jacobians of modular curves with arbitrary level at p. Contents 1. Introduction 6625 1.1. Notation 6630 2. Reminders on the universal representations for GL 2 6633 2.1. Construction of the universal representation 6633 3. Structure theorems for universal representations 6637 3.1. Refinement of the Iwahori structure 6637 3.2. The case F = Q p 6647 4. Study of K t and I t invariants 6649 4.1. Invariants for the Iwasawa modules R − ∞,• 6649 4.2. Invariants for supersingular representations 6656 5. The case of principal and special series 6660 6. Global applications 6663 Acknowledgements 6665 References 6666

Let F/Q be a CM field where p splits completely and let r¯:Galfalse(Q¯/Ffalse)→ GL 3false(boldF¯pfalse) be a Galois representation whose restriction to Gal(Q¯p/Fw) is ordinary and strongly generic for all places w above p. In this paper, we specify the set of Serre weights in which r¯ can be modular. To this aim, we develop a technique in integral p‐adic Hodge theory to describe extensions of rank‐one Breuil modules.