On modular forms associated with indefinite quadratic forms of signature (2,n−2)
On modular forms associated with indefinite quadratic forms of signature (2,n−2)
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DOI：
10.1007/bf01361138
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发表时间：
1977-06
影响因子：
1.4
通讯作者：
T. Oda
中科院分区：
文献类型：
--
作者：
T. Oda
本文的目的是推广Shintani [1]，Niwa [2]和Zagier [3]的一些结果.本文包括五个主要定理，并给出了它们的概要。设Q是定义在R中的格L上的一个符号为（2，n：-2）的固定不定整数二次型，F1是Q的单位群.用G和K分别表示Q的真实的正交群和G的极大紧群的单位分支。则商X= G/K是Hermitian有界对称整环，F1在其上适当地不连续作用.对于偶数k，设SdX; F_1）表示X上权为k的全纯尖形空间，~ k_（n-4）/2（F_o（N），Z）表示权为k-（n-4）/2的椭圆尖形空间，其中F_o（N）的整数N和乘子Z由Q决定。本文综述了Shintani [1]关于二次型Q上的θ级数的结果，这些结果在本文中起着基础作用。我们在§ 3中陈述定理t，并在§ 4中证明它。
The aim of this paper is a generalization of some results of Shintani [1], Niwa [2], and Zagier [3]. This paper contains five main theorems, of which we give an outline. Let Q be a fixed indefinite integral quadratic form of signature (2, n:-2), which is defined on a lattice L in R, and let F 1 be the unit group of Q. We denotes by G and K the identity component of the real orthogonal group for Q and the maximal compact group of G respectively. Then the quotient X= G/K is a Hermitian bounded symmetric domain, on which F 1 acts properly discontinuously. For even integer k, let SdX; F1) stand for the space ofholomorphic cusp forms of weight k on X for F1, and let~ k_~, _4~/2 (Fo (N), Z) stand for the space of elliptic cusp forms of weight k-(n-4)/2, where the integer N and the multiplicator Z of Fo (N) are determined by Q.In § 2, we review the results of Shintani [1] on theta series attached to the quadratic form Q, which play a basic role in this paper. We state Theorem t in § 3, and spend § 4 in proving it. In § 4, first we consider an integral