Almost strong approximations for definite quadratic spaces
Almost strong approximations for definite quadratic spaces
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DOI：
10.1007/s002220050169
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发表时间：
1997-08
影响因子：
3.1
通讯作者：
J. Hsia;M. Jöchner
中科院分区：
文献类型：
--
作者：
J. Hsia;M. Jöchner
积分二次型算术的一个基本结果是三变量或更多变量的非简并不定二次型的自旋群 Spin 的强逼近定理。（在实践中，人们调用旋量范数函数的核 OH 的相应版本。）这一结果归功于 Eichler 和 Kneser [Ei],[Kn] 在 20 世纪 50 年代的工作，并且自此在证明不定和定二次形式的许多重要定理。另一方面，众所周知，对于 i 不定二次形式的群 SO 来说，这样的强近似不可能成立，因为它不是单连通的，而 ii 对于定二次形式的群 Spin 来说，因为 SpinI 是紧的。本文的目的是证明对于定二次型存在一种“几乎”强近似定理，一个用于 OH m，一个用于 O m m！ 3 并表明它们在算术理论中也有重要的应用；例如，在属中类的渐近分布和确定形式的表示论中。预计这种几乎强的近似也将扩展到其他经典群，并将在以后的文章中进行讨论。我们简要概述本文的内容。术语和符号通常来自[OM]。为了简单和方便，我们在这里只考虑正定积分 Z 格。证明 OH 强近似定理的关键步骤是具有近似性质的不定表示的以下结果：
A fundamental result in the arithmetic of integral quadratic forms is the strong approximation theorem for the spin group Spin of a non-degenerate indefinite quadratic form in three or more variables.(In practice, one invokes the corresponding version for the kernel OH of the spinor norm function.) This result is due to the works of Eichler and Kneser [Ei],[Kn] in the 1950s, and has ever since played a central role in the proofs of a number of important theorems for both indefinite as well as definite quadratic forms. On the other hand, it is well-known that such a strong approximation cannot possibly hold for i the group SO of a indefinite quadratic form as it is not simply connected, and ii for the group Spin of a definite quadratic form as SpinI is compact. The purpose of this paper is to show that there exists a version of an``almost''strong approximation theorem for definite quadratic forms, one for OH m and one for O m m! 3 and show that they too have serious applications in the arithmetic theory; eg, in the asymptotic distributions of classes in a genus and in the representation theory of definite forms. It is anticipated that such almost strong approximations would also extend to other classical groups and will be treated in a future article. We give a brief outline of the contents of this paper. Terminology and notations are generally those from [OM]. For simplicity as well as convenience we shall consider here only positive definite integral Z-lattices. A key step in the proof of the strong approximation theorem for OH is the following result on indefinite representations with approximation property: