Study on automorphic forms, automorphic L-functions and Shintani functions.

1. Sintani函数与L-函数的构造研究了四元数斜厄米特型酉群上的局部Sintani函数。2.三次酉群上的自守形式（与A.Murase合作）（1）Kudla提升的Fourier-Jacobi展开式：设f是U（1，1）上的全纯尖点形式。S.Kudla构造了U（2，1）上的一个全纯尖形L（f）.利用本原theta函数得到了L（f）的一个精化的Fourier-Jacobi展开式。作为应用，我们给出了L（f）不为零的一个判据，它是用f的U（1）-周期表示的。（2）Siegel-Weil公式：证明了（U（2，1），U（2，2））的Siegel-Weil公式。我们不需要正则化过程。（3）内积公式：计算了L（f）与f的Petersson内积之比<L（f），L（f）>/<f，f>。它由特殊值L（f;1）和分歧素数处的局部数据写成。3. Jacobi形式的Kudla提升我们认为Kudla提升是Jacobi形式到U（2，1）的提升。当二次数域的类数为1时，它与Jacobi Hecke作用相容。作为Jacobi形式的应用，我们给出了Resnikoff-Tai关于SU（2，1）上自守型分次代数结构的结果在高斯场情形下的另一个简单证明。

1.Sintani functions and construction of L-functionsWe investigated local Shintani functions on the unitary groups of quaternion skew hermitian forms. Moreover, to construct automorphic L-functions, we calculated a functional equation of Eisenstein series explicitly.2.Automorphic forms on unitary groups of degree three (joint work with A.Murase)(1)Fourier-Jacobi expansion of Kudla lift : Let f be a holomorphic cusp form on U(1,1). S.Kudla constructed a holomorphic cusp form L(f) on U(2,1). We obtained a refined Fourier-Jacobi expansion of L(f) by primitive theta functions. As an application, we gave a criterion for non-vanishing of L(f) in terms of U(1)-period of f.(2)Siegel-Weil formula : We proved Siegel-Weil formula for the pair ((U(2,1),U(2,2)). We do not need regularization process.(3)Inner product formula : The ratio <L(f),L(f)>/<f,f> of Petersson inner prodects of L(f) and f is calculated. It is written by the special value L(f;1) and local data at ramified primes. Consequently we gave another criterion for the non-vanising of L(f).3.Kudla lift from Jacobi formsWe considered Kudla lift as a lifting from Jacobi forms to U(2,1). When the calss number of the quadratic number fields is one, it is compatible with Jacobi Hecke action. As an application of Jacobi form version, we gave another simple proof of Resnikoff-Tai's result on the structure of thegraded algebra of automorphic forms on SU(2,1) in Gaussian field case.