General study of modular forms and arithmetic varieties

首席研究员的目的是研究希尔伯特模尖点的性质。我们参加了一些工作坊和研讨会，以获得有关模块化形式的信息和讨论。我们组织了一个关于希尔伯特模形式的研讨会。通过这次研讨会，我们了解了一些基本的和最近的结果，我们的结果如下：1.我们建立了模形式环和模形式无限积环的结构的一些结果; 2.我们建立了完全数的存在性及其类比的一些结果; 3.我们用图的形式描述了椭圆曲线的塞尔默群。

The purpose of head investigator was to study properties of Hilbert modular cusps. We joined some workshops and seminars to get information and discuss about modular forms. We organized a workshop on Hilbert modular forms. Thanks to this workshop, we understood fundamental and recent results.Our results are as follows :1.We established some results on the structure of rings of modular forms and infinite products of modular forms.2.We established some results on the existence of perfect numbers and their analogue.3.We described Selmer groups of elliptic curves in terms of graphs.