Purity Reigns Supreme
Purity Reigns Supreme
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纯洁至上
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发表时间：
2013
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通讯作者：
L. Clozel
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作者：
L. Clozel
本文证明了GL（n，AF）的尖点表示π的Ramanujan猜想，当F是全真实的或CM域，π是上同调表示，当F是全真实的时π是自对偶的，当F是CM时π是共轭自对偶的.我们将证明猜想只有在素数v的F，所有的数据是unramified。如果p是有理数素数除以v，这意味着F在p处是非分歧的，并且对于素数v′，π的所有因子πv′| p是不可分的。这样的结果是可访问的已经知道，因为工作的作者[7]依赖于Kottwitz的描述特定志村品种。哈里斯和泰勒[10]以及最近S.-W. [18][19]特别参见Harris、Labesse和作者的最后一章[9]。S.莫瑞尔在[17]中证明了一个结果对一个未指定的素数集为真。上同调表示与约化群G的有限维表示L相关联-这里考虑GL（n，F）-;在几何情况下，L也可以被看作是相关联的Shimura簇上的系数系统。至少对于偶数n，Ramanujan猜想的最近证明[9，18]需要L的最高权重的正则性：Shin在[18]中称L为“轻度正则”。我们将证明这个假设是不必要的，至少在好的约简的素数。关于上同调表示或代数表示的概念以及伴随的性质，我们参考[6]。如果F是复数，用c表示复共轭;它自然地作用于GL（n，AF）的表示。临时标题
The purpose of this note is to prove the Ramanujan conjecture for cuspidal representations π of GL(n,AF ) when F is either a totally real or a CM field, and π is a cohomological representation that is self–dual if F is totally real, or conjugate self–dual if F is CM . We will prove the conjecture only at primes v of F where all data are unramified. If p is the rational prime divided by v, this means that F is unramified at p, and that all factors πv′ of π for the primes v′|p are unramified. That such a result is accessible has been known since the work of the author [7] relying on Kottwitz’s description of particular Shimura varieties. Increasingly precise and general variants have been proved by Harris and Taylor [10] and then, recently, by S.–W. Shin [18] and as a result of the collective effort embodied in [S]. See in particular the final chapter [9] by Harris, Labesse and the author. S. Morel has proved [17], in particular cases, a result true for an unspecified set of primes. A cohomological representation is associated to a finite – dimensional representation L of the reductive group G – here GL(n, F ) – being considered ; in the geometric cases L can also be seen as a coefficient system on the associated Shimura variety. At least for even n, the recent proofs [9, 18] of the Ramanujan conjecture require a regularity property of the highest weight of L : Shin calls L “mildly regular” in [18]. We will show that this assumption is unnecessary, at least at the primes of good reduction. We refer to [6] for the notion of cohomological or algebraic representation, and for the attendant properties. If F is complex denote by c the complex conjugation ; it acts naturally on representations of GL(n,AF ). ∗Provisional title