Open Access
December, 2004 Conditional Base Change for Unitary Groups
Michael Harris, Jean-Pierre Labesse
Asian J. Math. 8(4): 653-684 (December, 2004).


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.


Download Citation

Michael Harris. Jean-Pierre Labesse. "Conditional Base Change for Unitary Groups." Asian J. Math. 8 (4) 653 - 684, December, 2004.


Published: December, 2004
First available in Project Euclid: 13 June 2005

zbMATH: 1071.22025
MathSciNet: MR2127943

Rights: Copyright © 2004 International Press of Boston

Vol.8 • No. 4 • December, 2004
Back to Top