2023 Fiscal Year Final Research Report
Number theory of real quadratic fields from the perspective of non-holomorphic modular forms
| Project/Area Number |
20K14292
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| Research Category |
Grant-in-Aid for Early-Career Scientists
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| Allocation Type | Multi-year Fund |
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| Review Section |
Basic Section 11010:Algebra-related
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| Research Institution | Kyushu University (2022-2023)
Nagoya University (2020-2021) |
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Principal Investigator |
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| Project Period (FY) |
2020-04-01 – 2024-03-31
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| Keywords | モックモジュラー形式 / 双曲型アイゼンシュタイン級数 / サイクル積分 / Rademacher記号 |
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| Outline of Final Research Achievements |
We investigated a mysterious analytic object introduced over 80 years ago, known as the “cycle integrals” of modular forms, using the recent advancements in the theory of mock modular forms. Our perspectives on number theory and topology differ from the original motivation behind the notion of the cycle integrals. As a result, for instance, although the cycle integrals of the number-theoretical object called the elliptic modular j-function remain mysterious, we discovered that considering further cycle integrals of their generating function provides a geometric information, such as the number of intersections of a certain pair of geodesics.
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| Free Research Field |
整数論
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| Academic Significance and Societal Importance of the Research Achievements |
極めて古典的な対象であるにも関わらず，その数論的な役割が謎に包まれていた「サイクル積分」について，その数論的な側面に留まらず，トポロジーの研究との深い繋がりを見出せたことは，意外性もあり，今後の研究に大きな展望を与えるものである．また，モックモジュラー形式という，比較的新しく，また未だ国内に専門家が多くなかった理論を積極的に導入して，その新たな応用を与えたことも，意義が大きいと考える．
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