Study on automorphic forms of several variables

我们研究的目的如下：(1)研究Serre的p-进Eisenstein级数。(2)研究模形式的mod-p性质。关于目标(1)，我们得到以下结果。Serre的p-进Eisenstein级数概念的推广在以前的研究(2001-2003)中成功地应用于Siegel模形式。我们将这一概念推广到厄米特模形式的情形，并证明了p-进厄米特Eisenstein级数与秩为2的厄米特形式的亏格theta级数之间的重合性。关于这项工作，我们写了两篇论文：[1]S.Nagaoka，on p-进Hermitian Eisenstein级数，Proc.amer.Math.Soc。134,2533-2540(2006)[2]T.Munemoto和S.Nagaoka，关于p-进Hermitian Eisenstein级数的注记，Abh.Math.Sem.Hamburg 76,247-260(2006)在[1]中，我们处理了高斯场的情况。我们将结果推广到类数为1的虚二次域的情形。关于对象(2)，我们成功地解决了这一领域中的一个未决问题。也就是说，证明了具有一定同余性质的模形式的存在性。满足模1模p同余的模形式在具有正特征的模形式理论中占有重要地位。在第二级情形中，在前一时期(2001-2003年)中证明了其存在性。我们的结果给出了这个问题的完整答案。这一结果的主要部分在[3]S.Boecherer和S.Nagaoka关于Siegel模形式的mod p性质上公布，这一结果被接受发表在《数学年鉴》上。这位研究人员作为特邀演讲者在日本数学学会年会上就这一结果发表了演讲。

The purposes of our investigation are stated as follows :(1) Study of Serre's p-adic Eisenstein series.(2) Study of mod p properties of modular forms.About the object (1), we got the following result. The generalization of the notion of Serre's p-adic Eisenstein series succeeded in the case of Siegel modular forms in the previous investigation (2001-2003). We generalized the notion to the case of Hermitian modular forms, and proved the coincidence between a p-adic Hermitian Eisenstein series and a genus theta series for Hermitian forms of rank 2. Concerning this work, we wrote two papers :[1] S.Nagaoka, On p-adic Hermitian Eisenstein series, Proc.Amer.Math.Soc. 134, 2533-2540 (2006)[2] T.Munemoto and S.Nagaoka, Note on p-adic Hermitian Eisenstein series, Abh.Math.Sem.Univ.Hamburg 76, 247-260 (2006)In [1], we treated the case of Gaussian field. We generalized the result to the case of imaginary quadratic fields with class number one..About the object (2), we succeeded to solve a pending problem in this field. That is, the existence of a modular form with certain congruence property is proved. A modular form satisfying congruence one modulo p plays an important role in the theory of modular forms with positive characteristic. In the degree 2 case, the existence was proved during in the previous period (2001-2003). Our result gives a complete answer to this problem. The main part of this result was announced in[3] S.Boecherer and S.Nagaoka, On mod p properties of Siegel modular forms,and this is accepted for publication in the "Mathematische Annalen". The researcher presented a lecture on this result at the annual meeting of the Mathematical Society of Japan as an invited speaker.