ON CERTAIN AUTOMORPHIC FORMS OF Sp(1, q) (ARAKAWA'S RESULTS AND RECENT PROGRESS)
ON CERTAIN AUTOMORPHIC FORMS OF Sp(1, q) (ARAKAWA'S RESULTS AND RECENT PROGRESS)
复制标题
关于 Sp(1, q) 的某些自同构形式（ARAKAWA 的结果和最新进展）
DOI：
10.1142/9789812774415_0014
复制
发表时间：
2006
期刊：
影响因子：
--
通讯作者：
Hiroaki Narita
中科院分区：
文献类型：
--
作者：
Hiroaki Narita
大约在20世纪80年代初，荒川恒夫教授开始了关于签名为（1+，q-）的真实的辛群Sp（1，q）上的某些非全纯自守形式的研究。在[14]中，我们将它们理解为生成四元数离散级数的Sp（1，q）上的自守形式（关于这种离散级数的定义，请参见Gross-Wallach [8]）。荒川为这项研究提供了两部已发表的著作[2]和[3]。他们处理一个明确的尺寸公式的空间，这样的自守形式就整洁的非均匀格子群。另一方面，荒川给我们留下了几个未发表的笔记，对这种形式的研究。在其中一个中，他制定了从椭圆尖点形式到Sp（1，q）上的自守形式的θ提升，这是受到Kudla关于从椭圆模形式到SU（1，q）上的全纯自守形式的θ提升的工作的启发。除此之外，还有他的未发表的工作旋量L-功能重视自守形式的Sp（1，1）以上。他认为这是遵循安德里亚诺夫的方法[1]（也见[4]）。本说明有两个目的。一个目的是调查荒川的结果的维数公式和其他解释我们最近的结果荒川的theta提升连同他的结果。对于后一项工作，我们证明了提升的象是上的有界自守形式，
Around the beginning of the 1980s, Professor Tsuneo Arakawa initiated the research of certain non-holomorphic automorphic forms on the real symplectic group Sp (1, q) of signature (1+, q-). In [14] we understood them as automorphic forms on Sp (1, q) generating quaternionic discrete series (for the definition of this discrete series, see Gross-Wallach [8]). Arakawa gave two published works [2] and [3] for the research. They deal with an explicit dimension formula for the spaces of such automorphic forms with respect to neat non-uniform lattice subgroups. On the other hand, Arakawa left us several unpublished notes on the study of such forms. In one of them he formulated a theta lifting from elliptic cusp forms to automorphic forms on Sp (1, q), which is inspired by Kudla's work [13] on a theta lifting from elliptic modular forms to holomorphic automorphic forms on SU (1, q). In addition to this, there is his unpublished work on the spinor L-function attached to the automorphic forms on Sp (1, 1) above. He considered it by following the method of Andrianov [1](see also [4]). The aim of this note is two-fold. One aim is to survey Arakawa's result on the dimension formula and the other to explain our recent result on Arakawa's theta lifting together with his result. As for the latter work, we proved that the images of the lifting are bounded automorphic forms on