2018 Fiscal Year Final Research Report
The geometry of Shimura varieties over positive characteristic and the development of Galois representations
| Project/Area Number |
15K04787
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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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| Research Field |
Algebra
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| Research Institution | Tohoku University (2016-2018)
Kagoshima University (2015) |
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Principal Investigator |
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| Research Collaborator |
Chida Masataka
Miyauchi Michitaka
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| Project Period (FY) |
2015-04-01 – 2019-03-31
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| Keywords | ガロア表現 / 法pガロア表現 / セール予想 / 保型性問題 / テータ作用素 / ジーゲル保型形式 / 楕円保型形式 |
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| Outline of Final Research Achievements |
The study conducted during the project was analysis of Serre weight. The Serre weight is a concept necessary to formulate modularity problem of the residual Galois representation, and specifies the possibility of the weights of the corresponding automorphic forms. In this research, we introduce the theta operators as one of the tools to analyze the Serre weight, and when the algebraic group is GSp4, we succeeded in giving a concrete expression of the theta operators. We use the modular lifting theorems to prove the Serre weight theorem in a fairly general case for GSp4 over totally real fields, and further a list of the possible Serre weight of corresponding automorphic forms was described completely in terms of the local properties of a given mod p Galois representation.
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| Free Research Field |
整数論
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| Academic Significance and Societal Importance of the Research Achievements |
保型性問題とは幾何的ガロア表現と保型表現という代数的対象と解析的対象の間にある種の関係を問う問題である。GL2の場合にはWiles が楕円曲線に付随するガロア表現に対して保型性問題を解決したが、その応用として350年も未解決であったフェルマー予想が解決された。またそれと同時に周辺分野の大きな発展をもたらした。今回の研究も目標は保型性問題の解決のための基礎を発展させることに主眼が置かれ、特に正標数体上での志村多様体上の幾何を用いて法p保型形式の性質(テータ作用や重さの還元)に関する当該研究者の成果はセール重さを調べることには必要不可欠な道具であると期待する。
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