Explicit study on automorphic structure in algebra and zeta functions.

在这四年中，本课题主要研究员在oberwoll -fach组织了一次会议，三次春季会议，两次秋季研讨会（其中一次是JSPS支持的日德研讨会），数论小型研讨会，模形式和Zeta函数会议，共举办了8次会议，并与许多外国研究人员和在日本的合作研究人员进行了有效的研究。我们注重与模形式周围的广泛区域进行通信，采用模形式的梯度环，在第一和第二种情况下采用顶点算子代数，在第三种情况下采用混合区域。我们发表了5篇论文集，共970页。具体地说，主要研究者提出了积分的Siegel模形式与半积分权的Shimura型对应猜想，研究了标量值和向量值Siegel模形式与微分算子的结构、Borcherds积表达式、模形式上微分算子的完整系统、eta积的正性、提升得到的koecher - mass级数形式、有理权的模形式。从他们的论文列表中可以看出，我们项目的其他研究人员还积极研究了共形场论、球面和惠特克函数、预齐次向量空间、酉群上的模形式、椭圆根、p进模形式、l包、阿德尔几何、逆定理、顶点算子代数、组合设计、弱球面空间、逆伽罗瓦问题和CAP形式。

In these four years, the principal researcher of this subject organized a conference at Oberwol-fach, three Spring Confecences, two Autumn Workshops (one of which is Japan-Germany Seminar supported by JSPS), Mini-workshop on Number Theory, and Meeting on Modular Forms and Zeta Functions, hence 8 conferences in total, and studied effectively with many foreign researchers and cooperative researchers in Japan, In particular, in Spring Conference, we focused to communicate with wide areas arround modular forms, and took Graded rings of modular forms, Vertex operator algebras in first and second one, and more mixed areas in the third one. We published five Proceedings of 970 total pages. More concretely, the principal researcher proposed a conjecture on Shimura type correspondence between Siegel modular forms of integral and half integral weight, studied structures of scalar valued and vector valued Siegel modular forms and differential operators, Borcherds product expression, holonomic system coming from differential operators on modular forms, positivity of eta products, Koecher-Maass series of forms obtained by lifting, modular forms of rational weight. As it can be seen from the list of their papers, the other researchers of our project also actively studied on conformal field theory, spherical and Whittaker functions, prehomogenous vector spaces, modular forms on unitary groups, elliptic root systems, p-adic modular forms, L-packet, adele geometry, converse theorem, vertex operator algebras, combinatorial designs, weakly spherical spaces, inverse Galois problem and CAP forms.