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.