2009 Fiscal Year Final Research Report
Stability, Global Galois Representations and Non-Abelian Zeta Functions
| Project/Area Number |
18340012
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Kyushu University |
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Principal Investigator |
WENG Lin Kyushu University, 大学院・数理学研究院, 教授 (60304002)
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| Co-Investigator(Renkei-kenkyūsha) |
KOBAYASHI Ryoichi 名古屋大学, 多元数理科学研究科, 教授 (20162034)
FUTAKI Akito 東京工業大学, 理工学研究科, 教授 (90143247)
NAKAMURA Iku 北海道大学, 理学研究科, 教授 (50022687)
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| Project Period (FY) |
2006 – 2009
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| Keywords | 非可換L-関数 / Eisenstein級数 / Galois表現 / 安定性 / リーマン予想 |
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| Research Abstract |
In terms of mathematics, there are some remarkable advances : (1) We introduce genuine non-abelian zeta functions and study their basic properties ; (2) We intensively study semi-stable lattices and expose their relation with Arthur truncation of trace formula ; (3) We find a natural relation between our theory of zetas and Langlands' theory on Eisenstein systems ; (4) This then with an advanced Rankin-Selberg and Zagier method leads to the discovery of abelian zetas associated to (reductive group, maximal parabolic subgroup)s and hence an exposition of a hidden role played by symmetry in the Riemann Hypothesis ; (5) We initiate a program on using stability to establish a general (non-abelian) Class Field Theory for p-adic number fields and function fields over finite fields, under the framework of Tannakian category theory.
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