Explicit study of arithmetic structures of automorphic forms of several variables and trace formulas

在数论研究中，自同构形式函数及其L函数是描述各种算术性质的重要对象。我们在这个项目中给出了很多结果。其中，我们给出了一些自同构形式的结构定理和维数公式，并在此基础上提出了三个精确的新猜想，证明了定义和起源完全不同的几个自同构形式空间具有本质上相同的算术结构。我们还给出了三个新的提升猜想，这些猜想声称自同构形式的空间的某些部分来自于简单空间。利用与微分算子相关的特殊函数的理论，我们重新计算了一些L函数的显式值，并给出了1976年Zagier猜想的证明和一些数论应用。与其他新得到的结果一起，这些结果有助于自同构形式理论的发展。

In study of the number theory, functions called automorphic forms and their L functions are important objects which describe various arithmetic properties. We gave many results on these in this project. Among others, we gave structure theorems and dimension formulas of some automorphic forms, and based on these, we proposed three precise new conjectures which claim that several spaces of automorphic forms whose definitions and origins are completely different have essentially the same arithmetical structures. We also gave three new lifting conjectures which claim that some parts of such spaces of automorphic forms come from easier spaces. Using the theory we exploited on special functions associated with differential operators, we newly calculated explicit values of some L functions, and gave a proof of Zagier's conjecture in 1976 and some number-theoretic applications. Together with other newly obtained results, these contribute to the development of the theory of automorphic forms.