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

点击翻译按钮获取中文摘要

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.