# Sur la compatibilité entre les correspondances de Langlands locale et globale pour U(3)

### Joël Bellaïche

Columbia University, New York, United States

## 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 all places.

## Cite this article

Joël Bellaïche, Sur la compatibilité entre les correspondances de Langlands locale et globale pour U(3). Comment. Math. Helv. 81 (2006), no. 2, pp. 449–470

DOI 10.4171/CMH/58