Arithmetic of cohomological automorphic representations of orthogonal groups and theta series for indefinite quadratic forms

构造了一类与Sp（2n，R）的非调和导函子模相关的真实的解析Siegel-Eisenstein级数.进一步研究了它的Fourier展开式，证明了非平凡的Fourier系数只对Siegel幂幺子群的“不定”特征成立.我们可以把这一事实理解为与导出函子模的一个实质不变量，即其分布特征的波前集有关，并给出了每个Fourier系数的显式公式，使我们能够计算其各种扭曲的Mellin变换，从而把Maass的工作推广到一种特殊的真实的解析自守表示上.结果得到了一个由Rankin-Selberg型Dirichlet级数表示的公式，其中系数包含Maass波形的测地积分。这些研究将用于构造Sp（2，A）的非回火上同调Saito-Kurokawa表示.

We construct a kind of real analytic Siegel-Eisenstein series which associate to certain nontempered derived functor module of Sp (2n, R). A close study of its Fourier expansion gives us that the nontrivial Fourier coefficients are supported only for “indefinite" character of the Siegel unipotent subgroup. One can understand this fact to be related to a substancial invariant of the derived functor module, namely, the wave front set of its distribution character.We also give explicit formula of each Fourier coefficient, which enable us to compute its variously twisted Mellin transform, which give ar generalization of the works of Maass to a special real analytic automorphic representation. As a result we obtain a formula of it written by a Rankin-Selberg type Dirichlet series, where the coefficients include geodesic integrals of Maass wave forms. These study will be applied to construct a nontempered cohomological Saito-Kurokawa representation of Sp (2,A).