CONDITIONAL BASE CHANGE FOR UNITARY GROUPS
CONDITIONAL BASE CHANGE FOR UNITARY GROUPS
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单一组的有条件基础变化
DOI：
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
M. J.
中科院分区：
文献类型：
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作者：
M. J.
导论.多年来，人们已经知道阿瑟-塞尔伯格迹公式的稳定性将，或者我们应该写“将”，对朗兰兹函子性程序以及对志村簇的l-adic上同调的伽罗瓦表示的研究产生重要影响。目前，完全稳定性仍然只知道SL（2）和U（3）及其内部形式[LL，R]。U（3）稳定化的自守和算术结果形成了有影响力的卷[LR]的主题。在有些限制性的假设下，有时可以导出稳定迹公式的预期推论。这种“伪稳定化”的例子包括Kottwitz在[K2]中对附在全真实的域上的酉群的扭曲形式上的某些“简单”志村变种的zeta函数的分析，以及在[L1]中对自守表示的稳定循环基变化的证明，这些自守表示至少在两个地方是局部Steinberg的。这些条件结果已被成功地用于提供非平凡的例子，兼容系统的l-adic表示附加到某些类的自同构表示GL（n）[C3]，和非平凡类的上同调的S-算术群[BLS，L1]。条件的结果也足以为重要的本地应用，如本地Langlands猜想GL（n）[HT，他]。本文发展了一种获得条件碱基变化和函子转移的技术。设Un是数域F上的酉群，它与二次扩张E/F相连。该技术适用于从Un到GL（n）E的二次基变换，以及酉群的内部形式之间的转换。粗略地说，如果π是U的一个自守表示，它在E中F分裂的两个地方是局部超尖点的，那么稳定迹公式的预期结果对π成立;特别是π允许一个基变换到GL（n）E的尖点自守表示（定理2.2.2）。当F是全真实的而E是全虚的，并且π是上同调型时，可以得到稍微更一般的结果。从GL（n）E到Un的自守下降可以在类似的假设下被证明（定理2.4.1，定理3.1.2）。最后，我们在相当一般的局部假设下证明了酉群的不同内部形式之间的转移（Jacquet-Langlands转移）（定理2.1.2，以及更精确的形式，定理3.1.6和命题3.1.7）。与文献[L1]一样，所有结果都是从Arthur-Selberg迹公式的简单形式得到的，其中不存在非椭圆项和非尖点项。作为一个主要的应用，我们得到了类似于[C3]和[HT]的结果，对于附在Hermitian形式的酉群上的Shimura簇的上同调，或者更确切地说，对于满足超尖点性假设的部分上同调（定理3.1.4）。这个项目的最初动机是在厄米特形式的酉群上构造几乎等价的上同调自守形式族的非平凡例子，其中对L-
Introduction. 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 π 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 π; in particular π 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 π 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. As a principal application, we obtain results similar to those of [C3] and [HT] for the cohomology of Shimura varieties attached to unitary groups of hermitian forms, or rather for the part of the cohomology satisfying the supercuspidality hypotheses (Theorem 3.1.4). An initial motivation for this project was the construction of nontrivial examples of families of nearly equivalent cohomological automorphic forms on unitary groups of hermitian forms, to which the analysis of special values of L-