A study on automorphic forms of several variables with symmetries of level structure

2020 Fiscal Year Final Research Report

• Project/Area Number: 17K05186
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Kyoto Sangyo University
• Principal Investigator: MURASE Atsushi 京都産業大学, 理学部, 教授 (40157772)
• Project Period (FY): 2017-04-01 – 2021-03-31
• Keywords: Borcherds積 / 対称性 / 階層構造 / ヤコビ保型形式 / ジーゲル保型形式 / モジュラー曲線 / 完全再生可能関数

Outline of Final Research Achievements

We investigated on relations between certain symmetries of automorphic forms of several variables and infinite product expansions of Borcherds type.　Our goal is to show that a family of automorphic forms with multiplicative symmetries of level structure has an infinite product expansion of Borcherds type. In this study, we showed that a family of Jacobi forms with multiplicative symmetries of level structure has an infinite product expansion. By using this result, we also showed that a family of Siegel modular forms of degree 2 with multiplicative symmetries of level structure has an infinite product expansion. We also proved that a plane curve with a single symmetry of Hecke type is in fact a modular curve, and that completely replicable functions are characterized by multiplicative symmetries of level structure.

Free Research Field

整数論

Academic Significance and Societal Importance of the Research Achievements

多変数保型形式の整数論は近年研究が進んでいるものの、いまだ未解決の問題が多く、研究テーマとして大変興味深い分野である。多変数保型形式の中でも、Borcherds型の無限積展開を持つものは、代数幾何学や数理物理学とも関係して、重要な研究対象である。本研究では、Borcherds型の無限積展開を持つことと、本研究で新しく導入された階層構造を持つHecke型の乗法対称性が同値であることを、ヤコビ保型形式と２次ジーゲル保型形式の場合に証明した。与えられた保型形式がBorcherds型の無限積展開を持つかどうかは、判定が難しい問題だったが、本研究によって、判定する新しい方法が得られた。