Study of zeta functions related to Jacobi forms and Siegel modular forms

1.关于Siegel模形式：我们制定匡威定理的今井在这样一种方式，它可以适用于不一定尖点西格尔模形式的程度2。作为一个应用，我们利用Duke-Imamanti方法（与同事F.Sato和I.Makino的合作）重构了Siegel模形式的Saito-Kurokawa提升.桥本猜想的解决：我们利用Jacobi形式，Deligne-Serre关于权为1的模形式的定理（与S.Boecherer共同工作）解决了K.Hashimoto提出的关于Q上定四元数代数的that级数的猜想. Selberg zeta函数的研究：首席研究员通过Selberg迹公式和Selberg zeta函数研究了Maass波形的Shimura对应关系，并将对应关系的双射性问题简化为相关的两个Selberg zeta函数的简单关系。另一方面，他与S.Koyama和M.Nakasuji合作，得到了Q上不定四元数aogebras的算术Selberg zeta函数的显式形式和Brun-Titchmarsh型素测地线定理.关于多重L值：首席研究员和M.Kaneko制定了多重L值的（正则化）双重洗牌关系。我们还利用zeta正则化确定了相关多重L-函数在极点s=0处的主部.对预齐次向量空间的研究：同事F.Sato和K.Sugiyama证明了b-函数也有对应于向量空间的相关表示的分解。

1. On Siegel modular forms : We formulated the converse theorem of Imai in such a manner that it can be applica* to not necessarily cuspidal Siegel modular forms of degree two. As an application we reconstructed the Saito-Kurokawa lifting for Siegel modular forms of degree two by using Duke-Imamogle's method (joint work with the coworker F.Sato and I.Makino).2. Solution of the Hashimoto conjecture : We solved the conjecture presented by K.Hashimoto concerning the that series associated to definite quaternion algebras over Q by using Jacobi forms, the theorem of Deligne-Serre on modular forms of weight one (joint work with S.Boecherer).3. The study of Selberg zeta functions : The head investigator studied the Shimura correspondence for Maass wave forms via Selberg trace formulas and Selberg zeta functions and reduced the problem of bijectivity of the correspondence to a simple relation of the associated two Selberg zeta functions. On the other hand he with the co-operation of S.Koyama and M.Nakasuji obtained explicit forms of the arithmetic Selberg zeta functions attached to indefinite quaternion aogebras over Q and the prime geodesic theorem of Brun-Titchmarsh type.4. On multiple L-values : The head investigator and M.Kaneko formulated the (regularized) double shuffle relations for multiple L-values. We also determined the principal part at the pole s=0 of the associated multiple L-function by using the zeta regularization.5. The study of prehomogeneous vector spaces : The coworker F.Sato and K.Sugiyama showed that the b-function also has the decomposition corresponding to that of the associated representations of the vector space.