2023 Fiscal Year Final Research Report
Arithmetic, geometric and p-adic applications of multivariable modular forms
| Project/Area Number |
18K03210
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 11010:Algebra-related
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| Research Institution | Osaka Metropolitan University (2022-2023)
Osaka City University (2019-2021)
Kyoto University (2018) |
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Principal Investigator |
Yamana Shunsuke 大阪公立大学, 大学院理学研究科, 教授 (50633301)
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| Project Period (FY) |
2018-04-01 – 2024-03-31
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| Keywords | モジュラー形式 / L関数 / p進L関数 / 志村多様体 / 肥田族 / 市野-池田公式 / 積分表示 / 周期積分 |
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| Outline of Final Research Achievements |
(1)We construct the four-variable p-adic triple product L-functions associated to three Hida families of elliptic modular forms. As an application we prove the exceptional zero conjecture for the triple product of p-ordinary elliptic curves. (2)When the product of the central character is trivial, Atsushi Ichino proved a formula for the central value of the triple product L-series in terms of a period integral. We extend this formula to the case when the product of the central character is a quadratic character. (3)We compute the restriction of Hilbert-Eisenstein series and prove a relation between the first derivative of a certain p-adic L-function and the p-adic logarithm of a Stark-Heegner point. (4)We construct a five-variable p-adic L-function attached to Hida families on the denite unitary groups U(3) and U(2) by using the Ichino-Ikeda formula. (5)We also construct a five-variable p-adic L-function attached to Hida families on the quasi-split unitary groups U(2,1) and U(1,1).
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| Free Research Field |
数論
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| Academic Significance and Societal Importance of the Research Achievements |
L関数とモジュラー形式には、複素数体をp進体に取り替えた類似物も存在し、p進L関数やp進モジュラー形式と呼ばれます。p進L関数を用いて、L関数の数論的性質をp進的視点から考えることが岩澤理論です。p進数の新しい側面として複素数で離散的なパラメータがp進数では連続的となり、Eisenstein級数以外にも肥田族などp進モジュラー形式の連続族が存在し、複素変数に類似した円分変数の他に、反円分変数や連続族のパラメータを加えた多変数p進L関数など様々なタイプのp進L関数を考えられることがあります。本研究では、3重積L関数などの多変数p進L関数を構成し、その例外零点や微分値を考察しました。
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