On the Saito-Kurokawa lifting
On the Saito-Kurokawa lifting
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关于斋藤黑川吊装
DOI：
10.1007/bf01389101
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发表时间：
1983
影响因子：
3.1
通讯作者：
I. Piatetski
中科院分区：
文献类型：
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作者：
I. Piatetski;I. Piatetski
通过算例计算，H. Saito和N. Kurokawa证明，对于任意偶数k，存在从权为2k-2的经典全纯模形式空间到权为k的全纯Siegel模形式空间的映射[11]。在Maass [12]、Andrianov [13]和Zagier的一系列论文中，Saito-Kurokawa猜想得到了证明。一个美丽的阐述证明和历史的问题可以发现在谈话Zagier伊莱。N.黑川计算他的数值例子，以反驳拉马努金猜想。同时，R.豪和我Piatetski-Shapiro [-2].从文[3]的工作中，我们可以推出，如果我们有一个从小秩群到大秩群的Weil提升，那么这个像不满足Ramanujan猜想。因此，所有尖点Weil提升都是反例。(For韦尔升力的精确定义见第1节。）以R.豪和我。Piatetski-Shapiro，证明Weil提升是尖点自守表示是纯局部的。换句话说，某些局部分量是超尖点的。另一方面，正如我们将看到的，在斋藤-黑川类型的例子中，原因基本上是全局性的。在本文中，我们提出了一种新的方式来看待Saito-Kurokawa的例子。让我们对我们的问题作一个概括的描述。设G是整体域k上的任意约化群，P= MS是真抛物子群，其中M是Levi子群，S是幂单根.设z是MA的不可约自守表示。根据RP Lang的一个定理，
From calculations of numerical examples, H. Saito and N. Kurokawa conjectured that there exists a map from the space of classical holomorphic modular forms of weight 2k-2 into the space of holomorphic Siegel modular forms of weight k, for any even number k [11]. In a series of papers by Maass [12], Andrianov [13] and Zagier the Saito-Kurokawa conjecture was proved. A beautiful exposition of the proof and the history of the question can be found in the talk of Zagier Eli. N. Kurokawa computed his numerical examples in order to disprove the Ramanujan conjecture. At the same time, counter-examples of a different type were given by R. Howe & I. Piatetski-Shapiro [-2]. From the work of [3], we can deduce that if we have a Weil lifting from a group of small rank to a group of a bigger rank then the image does not satisfy the Ramanujan conjecture. Thus all cuspidal Weil liftings are counter examples.(For the precise definition of the Weil lifting see Sect. 1.) In the examples of R. Howe and I. Piatetski-Shapiro, the proof that the Weil lifting is a cuspidal automorphic representation is purely local. In other words some local component is supercuspidal. On the other hand, as we shall see, in the examples of Saito-Kurokawa type the reasons are essentially global. In this paper we present a new way of viewing the Saito-Kurokawa examples. Let us give a general description of our problem. Let G be any reductive group over a global field k, and let P= MS be a proper parabolic subgroup, where M is a Levi subgroup and S is the unipotent radical. Let z be an irreducible automorphic representation of M A. From a theorem of RP Lang-