Sur la compatibilite entre les correspondances de Langlands locale et globale pour U(3). (On the compatibility between local and global Langlands correspondances for U(3))

Abstract

Using a level-raising argument (and a result of Larsen on the image of Galois representations in compatible systems), we prove that for any automorphic representation $\pi$ for $\U(3)$, the $l$-adic Galois representation $\rho_l$ which is attached to $\pi$ by the work of Blasius and Rogawski, is the one expected by local Langlands correspondance at every finite place (at least up to semi-simplification and for a density one set of primes $l$). We rely on the work of Harris and Taylor, who have proved the same results (for $\U(n)$) assuming the base change of $\pi$ is square-integrable at one place. As a corollary, every automorphic representation which is tempered at an infinite number of places is tempered at every places.