## Asian Journal of Mathematics

- Asian J. Math.
- Volume 8, Number 4 (2004), 653-684.

### Conditional Base Change for Unitary Groups

Michael Harris and Jean-Pierre Labesse

#### Abstract

It has been known for many years that the stabilization of the Arthur-Selberg trace formula would, or perhaps we should write "will," have important consequences for the Langlands functoriality program as well as for the study of the Galois representations on the l-adic cohomology of Shimura varieties. At present, full stabilization is still only known for SL(2) and U(3) and their inner forms [LL,R]. The automorphic and arithmetic consequences of stabilization for U(3) form the subject of the influential volume [LR]. Under somewhat restrictive hypotheses, one can sometimes derive the expected corollaries of the stable trace formula. Examples of such "pseudo-stabilization" include Kottwitz' analysis in [K2] of the zeta functions of certain "simple" Shimura varieties attached to twisted forms of unitary groups over totally real fields, and the proof in [L1] of stable cyclic base change of automorphic representations which are locally Steinberg at at least two places. These conditional results have been used successfully to provide non-trivial examples of compatible systems of l-adic representations attached to certain classes of automorphic representations of GL(n) [C3], and of non-trivial classes of cohomology of S-arithmetic groups [BLS, L1]. Conditional results also suffice for important local applications, such as the local Langlands conjecture for GL(n) [HT, He]. The present article develops a technique for obtaining conditional base change and functorial transfer. Let Un be a unitary group over a number field F attached to a quadratic extension E/F. The technique applies to quadratic base change from Un to GL(n)E, and to transfer between inner forms of unitary groups. Roughly speaking, if p is an automorphic representation of U which is locally supercuspidal at two places of F split in E, then the expected consequences of the stable trace formula hold for p; in particular p admits a base change to a cuspidal automorphic representation of GL(n)E (Theorem 2.2.2). Slightly more general results are available when F is totally real and E is totally imaginary, and when p is of cohomological type. Automorphic descent from GL(n)E to Un can be proved under analogous hypotheses (Theorem 2.4.1, Theorem 3.1.2). Finally, we prove transfer between distinct inner forms of unitary groups (Jacquet-Langlands transfer) under quite general local hypotheses (Theorem 2.1.2 and, in a more precise form, Theorem 3.1.6 and Proposition 3.1.7). As in [L1], all results are obtained from the simple version of the Arthur-Selberg trace formula, in which non-elliptic and non-cuspidal terms are absent.

#### Article information

**Source**

Asian J. Math., Volume 8, Number 4 (2004), 653-684.

**Dates**

First available in Project Euclid: 13 June 2005

**Permanent link to this document**

https://projecteuclid.org/euclid.ajm/1118669695

**Mathematical Reviews number (MathSciNet)**

MR2127943

**Zentralblatt MATH identifier**

1071.22025

#### Citation

Harris, Michael; Labesse, Jean-Pierre. Conditional Base Change for Unitary Groups. Asian J. Math. 8 (2004), no. 4, 653--684. https://projecteuclid.org/euclid.ajm/1118669695