### Description

Let F/Q be a number field where p is unramified and r : Gal(F /F ) → GL_3(Q_p ) a

continuous Galois representation. We assume that r is automorphic for U(3) and the

p-adic local parameters of r at p are tamely potentially crystalline, with Hodge-Tate

weights (0,1,2).

The local/global compatibility conjecture in the p-adic local Langlands correspon-

dence predicts that the r-eigenspace in the integral ́etale cohomology on the adelic

points of U(3) with infinite level at p, should only depend on the p-adic local para-

meter associated to r, in some explicit way.

In this talk we prove the local/global compatibility conjecture when considering a

tame level at p, under mild technical hypotheses on the mod p-reduction of r. More

precisely, we show that the integral structure cut out by the global ́etale cohomology

on the tame ́etale local system giving rise to r depends only on the p-adic local

parameter.

The proof relies on the explicit construction of local Galois deformation rings in

dimension three, the description of their special fiber in automorphic terms via the

Breuil-M ́ezard conjecture, a new technique (which is a mixture of both global and

local methods) to compute the mod p reduction of Z_p -lattices in tame K-types.

This is a joint work with Dan Le, Viet-Bao Le Hung and Brandon Levin.