2022 Fiscal Year Final Research Report
Applications of p-adic Langlands correspondence to Iwasawa main conjecture
| Project/Area Number |
19K03404
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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 | Saga University |
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Principal Investigator |
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| Project Period (FY) |
2019-04-01 – 2023-03-31
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| Keywords | 岩澤主予想 / p進ラングランズ対応 / 局所イプシロン予想 / p進ガロア表現 |
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| Outline of Final Research Achievements |
I studied the Iwasawa main conjecture, in particular, Kato'c conjecture (the generalized Iwasawa main conjecture and the local epsilon conjecture). For the generalized Iwasawa main conjecture, I constructed zeta morphisms for rank two universal deformations using several theories in p-adic Langlands correspondence. As an application of it, I could prove a theorem which states that Iwasawa main conjecture for a modular form holds if the conjecture for a congruent modular holds. For the local epsilon conjecture, I proved a theorem which states that the local epsilon conjecture for a de Rham (phi, Gamma)-module holds if the conjecture for its associated p-adic differential equation holds (this is a joint work with Tetsuya Ishida (Saga Univ)).
As another application of zeta morphism, I constructed the p-adic L-function over the Coleman-Mazur eigen curve (this is a joint work with Chan-Ho Kim (KIAS)).
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| Free Research Field |
整数論
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| Academic Significance and Societal Importance of the Research Achievements |
現在発展中のp進ラングランズ対応の理論が岩澤主予想と深く関わっていることを発見した本研究は、p進ラングランズ対応と岩澤理論双方の分野において価値のあることであると認識している。また、その応用として得られた、合同な保型形式に対する岩澤主予想の同値性に関する定理は、pで超カスプ表現となる保型形式に対する岩澤主予想についてのおそらく初めての結果であり、岩澤主予想の研究分野において大変価値の結果であると思っている。
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