Arithmetic of automorphic forms of many variable : reconstruction of the foundation

按照我们在申请中的计划，我们在东京和神户举办了一系列的月度研讨会。因为我们已经在年度报告中报道了它们，所以我们在这里不写了。我们还省略了关于RIMS、京都大学和暑期学校的研讨会的细节。这些会议的会议记录可供查阅。Oda首席研究员与山形大学的Takahiro Hayata合作和Mie大学的Harutaka Koseki得到了SU(2，2)的中间离散级数的矩阵系数的显式公式。此外，他还得到了SU(2，2)和Sp(2，R)的大型离散级数的矩阵系数的显式公式。此外，在与Masao Tsuzuki(索菲亚大学)的一篇联合论文中。他在某些有界对称域的算术商上构造了模除数的格林函数，作为自同构形式。我们还解释了参与这一计划的研究人员的研究成果概要。这一结果不仅有了应用…进一步证明了自同构型空间的维数，但也具有理论意义。Ibukiyama和Saito在评价准齐次向量空间的Zeta函数的特殊值方面取得了显着的结果。这一结果不仅适用于自同构型空间的显式维公式，而且在理论上具有重要的意义。Sato和Hironaka对p-进球面函数的研究取得了很大进展。新的一点是，它们还可以处理“分支”球面函数的情况。村上春树和杉野的联合工作也取得了进展。他们关于SU(2，1)的本原theta函数的工作是一个有趣的结果。而且这也是正交群上自同构L函数理论的一个进展。关于p-进球面函数是其中的一个成果。渡边和森下正则(金泽大学)推进了对代数群的Hermite常数的研究，这可以说是非阿贝尔版本的《数的几何》。Katsurada的结果也很有趣，因为它是对分支p-进球面函数的研究。较少

As planned in our application, at Tokyo and at Kobe we had a series of monthly seminars. Since we have already reported about them in the annual reports, here we do not write about them. We also omit the details about the workshops at RIMS, Kyoto University and the summer schools. The proceedings on these meetings are available. The head investigator Oda in a joint work with Takahiro HAYATA (Yamagata Univ.) and Harutaka KOSEKI (Mie University) obtained an explicit formula for matrix coefficients of the middle discrete series of SU(2,2). Added to this he also obtained explicit formulae of matrix coefficients for the large discrete series of SU(2,2) and Sp(2,R). Moreover in a joint paper with Masao TSUZUKI (Sophia Univ.) he constructed Green functions of modular divisors on arithmetic quotients of certain bounded symmetric domains, as automorphic forms. We also explain the outline of the fruits of the research by the investigators joined to this plan. This result has not only have applic … More ations for the dimension of spaces of automorphic forms, but also theoretically significance. Ibukiyama and Saito had remarkable results on the evaluation of the special values of the zeta functions of prehomogeneous vector spaces. This result has not only application to the explicit dimension formulae of the spaces of automorphic forms, but also theoretically very important meaning. The investigation of p-adic spherical functions by Sato and Hironaka made much progress. The new point among others is that they can handle also the case of "ramified" spherical functions. The joint work of Murase and Sugano also advanced. Their work of the primitive theta function of SU(2,1) is one interesting result. But also the is a progress in the theory of automorphic L-functions on orthogonal groups A joint work with Shin-ichi Kato (Kyoto Univ.) on p-adic spherical functions is one of fruits. Watanabe together with Masanori Morishita (Kanazawa Univ.) pushed forward the investigation of Hermite constants for algebraic groups, Which is so to speak a non-abelian version of "Geometry of Numbers". The result of Katsurada is also interesting by regarding it as an investigation of ramified p-adic spherical functions. Less