Explicit study of number theory of automorphic forms of several variables related to trace formulas.

2017 Fiscal Year Final Research Report

• Project/Area Number: 25247001
• Research Category: Grant-in-Aid for Scientific Research (A)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Osaka University
• Principal Investigator: Ibukiyama Tomoyoshi 大阪大学, その他部局等, 名誉教授 (60011722)
• Co-Investigator(Kenkyū-buntansha): 若槻 聡 金沢大学, 数物科学系, 准教授 (10432121) ; 佐藤 文広 津田塾大学, 付置研究所, 研究員 (20120884) ; 北山 秀隆 和歌山大学, 教育学部, 准教授 (20622567) ; 桂田 英典 室蘭工業大学, 工学研究科, 教授 (80133792)
• Co-Investigator(Renkei-kenkyūsha): Hayashida Shuichi 上越教育大学, 大学院学校教育研究科, 准教授 (80597766) ; Watanabe Takao 大阪大学, 大学院理学研究科, 教授 (30201198) ; Moriyama Tomonori 大阪大学, 大学院理学研究科, 准教授 (80384171) ; Ochiai Tadashi 大阪大学, 大学院理学研究科, 准教授 (90372606) ; Ikeda Tamotsu 京都大学, 大学院理学研究科, 教授 (20211716)
• Project Period (FY): 2013-04-01 – 2018-03-31
• Keywords: ジーゲル保型形式 / ヤコービ保型形式 / テータ関数 / ゲーゲンバウア関数 / 微分作用素 / ホロノミー系 / ゼータ関数 / コンパクト実形

Outline of Final Research Achievements

In study of the number theory, functions called automorphic forms are important objects which describe various arithmetic properties. We gave many results on these in this project. A theory of automorphic differential operators includes the classical theory of special functions such as Gegenbauer functions, and in our study, we gave three different complete constructions for these operators, which can be regarded as final results. We also gave explicit solutions of associated system of differential equations. We gave conjectures on comparison of
automorphic forms belonging to different domains, and structure theorems of automorphic forms by using constructive methods including automorphic differential operators. Some dimension formulas of automorphic forms and trace formulas of arithmetically important Hecke operators were given, and we clarified their relations to arithmetic invariants in some algebraic geometry over finite fields.

Free Research Field

整数論と多変数保型形式論