On the λ-adic representations associated to some simple Shimura varieties
On the λ-adic representations associated to some simple Shimura varieties
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关于与一些简单 Shimura 品种相关的 λ-adic 表示
DOI：
10.1007/bf02100620
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发表时间：
1992
影响因子：
3.1
通讯作者：
R. Kottwitz
中科院分区：
文献类型：
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作者：
R. Kottwitz
许多作者研究了来自GL 2及其内部形式的Shimura簇的2-adic表示，首先是在II~上，最近是在任意全真实的域上。朗兰兹已经草拟了一个处理所有志村品种的程序（见[L1，L2]）和[L3]的介绍），但在这个程序可以实施之前还有很多工作要做。幸运的是，GL 2并不是唯一产生简单志村品种的群体。CM域上除代数上的第二类对合也得到了“酉”群。相应的志村品种，首先调查志村[SH]，特别是简单的几个方面。首先，它们是具有PEL结构（极化、自同态和水平结构）的阿贝尔簇的紧模空间。此外，内窥镜的各种现象，这是预期的复杂化的一般情况下，在这里只发挥次要的作用，因为第一次观察到的Rapoport-Zink [RZ]在9维除代数的情况下，在一个虚的二次扩张的II~。它断言存在的2进表示的通常排序的家庭志村品种上述。一旦更多地了解了这些“酉”群的自守表示，就有可能使定理1更加尖锐。建立定理1的一个关键要素是朗兰兹[L4]所称的”基本引理”的一个特例，Clozel [C]（以及Labesse [La]）已经给出了它的证明。
Many authors have studied the 2-adic representations coming from the Shimura varieties associated to GL 2 and its inner forms, first over II~ and more recently over arbitrary totally real fields. Langlands has sketched a program to handle all Shimura varieties (see [L1, L2]) and the introduction to [L3]), but much work remains to be done before this program can be carried out. Fortunately GL2 is not the only group to give rise to simple Shimura varieties. There are also the" unitary" groups obtained from involutions of the second kind on division algebras over CM fields. The corresponding Shimura varieties, first investigated by Shimura [Sh], are especially simple in several respects. First of all they are compact moduli spaces for abelian varieties with PEL structures (polarizations, endomorphisms, and level structures). Therefore it is relatively easy to study their reductions modulo p. Moreover the various phenomena of endoscopy, which are expected to complicate the general case considerably, play only a minor role here, as was first observed by Rapoport-Zink [RZ] in the case of division algebras of dimension 9 over an imaginary quadratic extension of II~.The main result of the paper is Theorem 1, which asserts the existence of 2-adic representations of the usual sort for the family of Shimura varieties mentioned above. Once more is known about the automorphic representations of these" unitary" groups, it should be possible to sharpen Theorem 1 substantially. A key ingredient in establishing Theorem 1 is a special case of what Langlands [L4] calls the" fundamental lemma," a proof of which has been given by Clozel [C](and also by Labesse [La]).