On Relations of Dimensions of Automorphic Forms of $Sp(2, R)$ and Its Compact Twist $Sp(2)$ (II)
On Relations of Dimensions of Automorphic Forms of $Sp(2, R)$ and Its Compact Twist $Sp(2)$ (II)
复制标题
论$Sp(2, R)$的自守形式的维数关系及其紧扭曲$Sp(2)$ (II)
DOI：
10.2969/aspm/00710007
复制
发表时间：
1985
期刊：
影响因子：
--
通讯作者：
T. Ibukiyama
中科院分区：
文献类型：
--
作者：
K. Hashimoto;T. Ibukiyama
本文给出了Sp(2， R)（矩阵大小为4）的自同构形式与其紧捻Sp(2)之间的一些良好的全局维关系。当所讨论的离散子群的p进补全（对于一个固定素数p）是极大紧的（见[24]）时，一位作者已经证明了这种关系。在本文中，我们处理离散子群，它们的p进补全是最小仿似的。我们的目的是将SLz和SU(2)之间的Eichler-Jacquet-Langlands对应推广到更高次的辛情况。这种对应关系应该通过比较所有赫克算符的轨迹来证明。我们的结果表明，对于Sp（2， R）和Sp(2)的一些明确定义的离散子群，至少在T(I)上存在迹关系（§2主定理I）。此外，它们还为朗兰兹关于稳定共轭类的哲学提供了有意义的例证（§2主要定理II）。粗略地说，这种比较分为无限处的字符关系（或多或少为人所知）和有限处的算术。我们的重点是明确地执行算术部分的比较。我们的定理似乎是除GLn（参见[24]）外关于这类关系的第一个全局结果。在第1节中，经过简短的介绍，我们给出了关于一般n的Sp（n， R）和Sp(n)之间的问题的精确公式，例如如何明确地选择离散子群。对于这些明确选择的离散子群的自同构形式，我们提出了两个猜想（首先在[21]，[23]中给出）：作为Hecke代数模块的新形式之间的维数重合和同构存在。对于n= 1，这些只不过是Eichler的定理[10]，[11]，以上猜想是他的结果的自然推广。Langlands b[34]已经给出了一个关于任何约化代数群的自同构形式对应的相当一般的哲学，但是我们知道他的哲学目前还没有给出非常详细的表述，例如
In this paper, we show some good global dimensional relations between automorphic forms of Sp(2, R) (matrix size four) and its compact twist Sp(2). One of the authors has already shown such relations when the p-adic completions (for a fixed prime p) of the discrete subgroups in question are maximal compact (See [24]). In this paper, we treat discrete subgroups whose p-adic completions are minimal parahoric. Our aim is a generalization of Eichler-Jacquet-Langlands correspondence between SLz and SU(2) to the symplectic case of higher degree. Such correspondence should be proved by comparison of the traces of all the Hecke operators. Our results mean that there exist relations of traces at least for T(I) for some explicitly defined discrete subgroups of Sp(2, R) and Sp(2) (§ 2 Main Theorem I). Besides, they give meaningful examples for Langlands philosophy on stable conjugacy classes (§ 2 Main Theorem II). Roughly speaking, such comparison is divided into character relations at infinite places (which are more or less known) and arithmetics at finite places. Our point is to execute the comparison of the arithmetical part explicitly. It seems that our Theorems are the first global results on such relations except for GLn (cf. also [24]). In Section 1, after a brief introduction, we give a precise formulation on our problems between Sp(n, R) and Sp(n) for general n, e.g. on how to choose discrete subgroups explicitly. For automorphic forms with respect to these explicitly chosen discrete subgroups, we propose there two conjectures (which were first given in [21], [23]): coincidence of dimensions and existence of an isomorphism between new forms as Hecke algebra modules. For n= 1, these are nothing but the theorems by Eichler [10], [11], and the above conjectures are a natural generalization of his results. Langlands [34] has given a quite general philosophy on correspondence of automorphic forms of any reductive algebraic groups, but we understand that his philosophy does not give very detailed formulation at present for such typical and explicit cases as