A note on Fourier coefficients of cusp forms on GL n
A note on Fourier coefficients of cusp forms on GL n
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DOI：
10.1515/forum.2006.007
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发表时间：
2006-01
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通讯作者：
Henry H. Kim
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文献类型：
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作者：
Henry H. Kim
在这篇简短的笔记中，我们证明了[R-Sa]中GL4的假设H是GL4的尖形表示的外平方的泛函性[Ki]的应用，并对GL2的尖形表示的Sato-Tate猜想作了一些改进。假设H的结果通过Rudnick和Sarnak [R-Sa]的工作表明，GL4的倒自同构形式的零的非水平相关遵循GUE（高斯酉系综）预测。（精确的表述见[R-Sa]的定理1.1。）我们感谢Peter Sarnak教授的建议，即GL4的假说H可能遵循外部方形的功能。设p 1 / 4 npp是GLmðAQÞ的反转自同构表示。对于pp未分叉的p a素数，设为diag - a1；p；……;点;pÞ为对应的Satake参数。对于正整数k，设apðpÞ 1 / 4 a 1；P þ þ a m；然后Rudnick和Sarnak [R-Sa]做了如下陈述
In this short note we prove Hypothesis H for GL4 in [R-Sa] as an application of the functoriality of the exterior square of cuspidal representations of GL4 [Ki], and some improvements toward Sato-Tate conjecture on cuspidal representations of GL2. The result on Hypothesis H implies via the work of Rudnick and Sarnak [R-Sa] that the nlevel correlations of the zeros of cuspidal automorphic forms of GL4 follow the GUE (Gaussian unitary ensemble) predictions. (See Theorem 1.1 of [R-Sa] for the precise statement.) We thank Prof. Peter Sarnak for his suggestion that Hypothesis H for GL4 may follow from the functoriality of the exterior square. Let p 1⁄4 N pp be a cuspidal automorphic representation for GLmðAQÞ. For p a prime at which pp is unramified, let diagða1;p; . . . ; am;pÞ be the corresponding Satake parameter. For a positive integer k, let apðpÞ 1⁄4 a 1;p þ þ a m;p. Then Rudnick and Sarnak [R-Sa] made the following