Automorphic forms of several variables and spherical functions of homoge-neous spaces

齐性空间的球函数或Shintani函数对于研究自守形式的周期的Fourier展开以及与自守形式相关的L-函数是很重要的，本文主要研究了经典群的Shintani函数。特别地，我们证明了Shintani函数的存在性和唯一性，并在某些情况下得到了它们的显式公式。我们应用它们来研究自守L-函数的分析和算术性质。（Muarase Sugano）这将是我们的下一个目标，定义适当的几何对象相关联的球面子群和解释的Shintani函数在其上同调群。

The spherical function or the Shintani function of homogeneous spaces are important in order to study the Fourier expansion of period of automorphic forms and L-functions associated to automorphic forms.We studied mainly the Shintani function of the classical groups. in particular we proved uniqueness and existence of the Shintani function and in some cases we obtained explicit formulas for them. We applied them to examine analytic and arithmetic properties of automorphic L-functions. (Muarase-Sugano)It would be our next objective to define suitable geometrical objects associated to spherical subgroups and to interpret the Shintani function in terms of their cohomology groups.