2005 Fiscal Year Final Research Report Summary
From Stability, Riemann-Roch to Non-Abelian Zeta Functions
| Project/Area Number |
14340008
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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, Faculty of Mathematics, Associate Professor, 大学院・数理学研究院, 助教授 (60304002)
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| Co-Investigator(Kenkyū-buntansha) |
SATO Eiichi Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (10112278)
KAJIWARA Kenji Kyushu University, Faculty of Mathematics, Associate Professor, 大学院・数理学研究院, 助教授 (40268115)
NAKAYASHIKI Atsushi Kyushu University, Faculty of Mathematics, Associate Professor, 大学院・数理学研究院, 助教授 (10237456)
KOBAYASHI Ryoichi Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (20162034)
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| Project Period (FY) |
2002 – 2005
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| Keywords | Stability / Truncation / Eisenstein series / Zeta functions / lattices / Rankin-Selberg Method / degeneration / Takhtajan-Zograf metric |
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| Research Abstract |
Over all, we, guided by our paper of A Program on Geometric Arithmetic, carry on our researches in the past a few years.
(1)Develop a new cohomology theory for lattices over number fields, motivated by Tate's thesis.
(2)Introduce New yet genuine defnition of rank n non-abelian zeta and L functions.
(3)Establish fundamental properties of these functions
(4)Show that in case of rank two for number fields, our zeta functions satisfies the Riemann Hypothesis.
(5)Expose the relation between Geometric Truncation and Analytic Truncations
(6)Introduce non-abelian L functions and establish their basic properties : Based on 4, we introduce general non-abelian L functions using Langlands' fundamental theory of Eisenstein series and hence also establish their basic properties such as meromorphic continuation, functional equations and singularities.
(7)Based on (5) and (6),with substantial additional hard work, we are able to relate our non-abelian L functions and what we call Eisenstein periods, a special kind of Arthur's period.
(8)Examples :
(i)Using an advanced version of Rankin-Selberg and Zagier method due to Jacquet-Lapid-Rogawski, we write down the precise expression for our L functions associated to the so-caleld cusp forms.
(ii) As a direct application, using (i),we together with Henry Kim (University of Toronto) obtain a general formula for volumes of Arthur's truncated domains of fundemantal domains associated to split semi-simple groups over number fields. This result when letting the parameter go to infinity then recovers the famous formula of (Siegel and) Langlands on the volumes of fundamental domains associated to (SLn and) general split semi-simple groups over number fields.
(9)(i)We find a general formula for the volume of moduli space of semi-stable lattices.
(ii)We exposes a fundamental relation between spaciel values of non-abelian zeta functions and special values of classical Dedekind zeta functions.
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