We use the patching method of Taylor-Wiles and Kisin to construct a candidate for the p-adic local Langlands correspondence for GL n (F ), F a finite extension of Q p . We use our construction to prove many new cases of the Breuil-Schneider conjecture.

Our aim is to establish some new cases of the global Langlands correspondence for GLm. Along the way we obtain a new result on the description of the cohomology of some compact Shimura varieties. Let F be a CM field with complex conjugation c and Π be a cuspidal automorphic representation of GLm(AF ). Suppose that Π ∨ Π • c and that Π∞ is cohomological. A very mild condition on Π∞ is imposed if m is even. We prove that for each prime l there exists a continuous semisimple representation R l (Π) : Gal(F /F ) → GLm(Q l ) such that Π and R l (Π) correspond via the local Langlands correspondence (established by Harris-Taylor and Henniart) at every finite place w l of F ("local-global compatibility"). We also obtain several additional properties of R l (Π) and prove the RamanujanPetersson conjecture for Π. This improves the previous results obtained by Clozel, Kottwitz, Harris-Taylor and Taylor-Yoshida, where it was assumed in addition that Π is square integrable at a finite place. It is worth noting that the mild condition on Π∞ in our theorem is removed by a p-adic deformation argument, thanks to Chenevier-Harris.Our approach generalizes that of Harris-Taylor, which constructs Galois representations by studying the l-adic cohomology and bad reduction of certain compact Shimura varieties attached to unitary similitude groups. The central part of our work is the computation of the cohomology of the so-called Igusa varieties. Some of the main tools are the stabilized counting point formula for Igusa varieties and techniques in the stable and twisted trace formulas.Recently there have been results by Morel and Clozel-Harris-Labesse in a similar direction as ours. Our result is stronger in a few aspects. Most notably, we obtain information about R l (Π) at ramified places.

We consider certain families of automorphic representations over number fields arising from the principle of functoriality of Langlands. Let G be a reductive group over a number field F which admits discrete series representations at infinity. Let L G = G Gal(F/F) be the associated L-group and r : L G → GL(d, C) a continuous homomorphism which is irreducible and does not factor through Gal(F/F). The families under consideration consist of discrete automorphic representations of G(A F ) of given weight and level and we let either the weight or the level grow to infinity. We establish a quantitative Plancherel and a quantitative Sato-Tate equidistribution theorem for the Satake parameters of these families. This generalizes earlier results in the subject, notably of Sarnak (Prog Math 70:321-331, 1987) and Serre (J Am Math Soc 10(1):75-102, 1997). As an application we study the distribution of the lowlying zeros of the associated family of L-functions L(s, π, r ), assuming from the principle of functoriality that these L-functions are automorphic. We find that the distribution of the 1-level densities coincides with the distribution of the 1-level densities of eigenvalues of one of the unitary, symplectic and orthogonal ensembles, in accordance with the Katz-Sarnak heuristics. We provide a criterion based on the Frobenius-Schur indicator to determine this symmetry type. If r is not isomorphic to its dual r ∨ then the symmetry type is unitary. Otherwise there is a bilinear form on C d which realizes the isomorphism between r and r ∨ . If the bilinear form is symmetric (resp. alternating) then r is real (resp. quaternionic) and the symmetry type is symplectic (resp. orthogonal).

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We present a new construction of the p-adic local Langlands correspondence for GL 2 (Q p ) via the patching method of Taylor-Wiles and Kisin. This construction sheds light on the relationship between the various other approaches to both the local and the global aspects of the p-adic Langlands program; in particular, it gives a new proof of many cases of the second author's local-global compatibility theorem and relaxes a hypothesis on the local mod p representation in that theorem.