Study of congruences and p-adic properties for modular forms with several variables

2021 Fiscal Year Final Research Report

• Project/Area Number: 18K03229
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Fukuoka Institute of Technology
• Principal Investigator: Kikuta Toshiyuki 福岡工業大学, 情報工学部, 助教 (60569953)
• Project Period (FY): 2018-04-01 – 2022-03-31
• Keywords: p進モジュラー形式 / Siegelモジュラー形式 / 法p特異モジュラー形式 / テータ級数 / テータ作用素

Outline of Final Research Achievements

1. An evaluation formula for the weight (filtration) of an element of the image ImΘ of the mod p power Θ-operator was studied and results were obtained in some special cases. 2. In a fairly general case, we showed that all mod p singular modular forms are represented by linear combinations of theta series. In the cases of some levels and some singular ranks, the levels of the corresponding theta series were specified. 3. In the case where such as the base field and weights are special, the concrete structure of the graded algebra over the ring of rational integers formed by the Hermite modular forms was determined. 4. There are two papers that were in the process of submission to journals before this research period, but were published in journals during this research period.

Free Research Field

整数論

Academic Significance and Societal Importance of the Research Achievements

本研究の成果は、Serreによって展開された(１変数の)p進モジュラー形式の理論が、どの程度平行して多変数化されるか、1変数と多変数の場合の違いを一部明らかにする。特に、多変数の場合にのみ成り立つ特有の事象の追究により、新たな理論の形成を担う。これにより、多変数のp進モジュラー形式の理論の発展に寄与する。