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From Stability, Riemann-Roch to Non-Abelian Zeta Functions

Research Project

Project/Area Number 14340008
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionKYUSHU UNIVERSITY

Principal Investigator

WENG Lin  Kyushu University, Faculty of Mathematics, Associate Professor, 大学院・数理学研究院, 助教授 (60304002)

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)
高山 茂晴  東京大学, 大学院・数理科学研究科, 助教授 (20284333)
Project Period (FY) 2002 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥11,400,000 (Direct Cost: ¥11,400,000)
Fiscal Year 2005: ¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 2004: ¥2,600,000 (Direct Cost: ¥2,600,000)
Fiscal Year 2003: ¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 2002: ¥2,900,000 (Direct Cost: ¥2,900,000)
KeywordsStability / Truncation / Eisenstein series / Zeta functions / lattices / Rankin-Selberg Method / degeneration / Takhtajan-Zograf metric / stability / truncation / Eisenstein Series / zeta functions / Takhatajan-Zograf metric / Eisenstein senes / Rankin-selberg Method / Taketajan-zograf metne
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.

Report

(5 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • Research Products

    (56 results)

All 2006 2005 2004 2003 2002 Other

All Journal Article (53 results) Book (2 results) Publications (1 results)

  • [Journal Article] Deligne Pairings over Moduli Spaces of Punctured Riemann Surfaces2006

    • Author(s)
      K.Obitsu, W.-K.To, L.Weng
    • Journal Title

      Arithmetic Geometry and Number Theory(World Scientific) (to appear)

      Pages: 25-40

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Automorphic Forms, Eisenstein Series and Spectral Decompositions2006

    • Author(s)
      L.Weng
    • Journal Title

      Arithmetic Geometry and Number Theory(World Scientific) (to appear)

      Pages: 107-184

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Geometric Arithmetic : A Program.2006

    • Author(s)
      L.Weng
    • Journal Title

      Arithmetic Geometry and Number Theory(World Scientific) (to appear)

      Pages: 185-350

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Non-Abelian Zeta Functions for Function Fields.2005

    • Author(s)
      Weng Lin
    • Journal Title

      Amer.J. Math. 127・No.5

      Pages: 973-1017

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Generating functions related to the Okamoto polynomials for the Painleve' IV equation.2005

    • Author(s)
      H.Goto, K.Kajiwara
    • Journal Title

      Bull. Austral. Math. Soc. 71・No.3

      Pages: 517-526

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Soliton solutions for the non-autonomous discrete-time Toda lattice equation.2005

    • Author(s)
      K.Kajiwara, A.Mukaihira
    • Journal Title

      J. Phys. A 38・No.28

      Pages: 6363-6370

    • NAID

      120000981493

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Non-abelian zeta functions for function fields2005

    • Author(s)
      L.Weng
    • Journal Title

      Amer.J.Math. 127, no.5

      Pages: 973-1017

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Soliton solutions for the non-autonomous discrete-time Toda lattice equation.2005

    • Author(s)
      K.Kajiwara, A.Mukaihira
    • Journal Title

      Funkcial.Ekvac. 48, no.1

      Pages: 147-160

    • NAID

      120000981493

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Cubic pencils and Painleve Hamiltonians2005

    • Author(s)
      K.Kajiwara, T.Masuda, M.Noumi, Y.Ohta, Y.Yamada
    • Journal Title

      Funkcial.Ekvac. 48, no.1

      Pages: 147-160

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Construction of hypergeometric solutions to the $q$-Painleve equations2005

    • Author(s)
      K.Kajiwara, T.Masuda, M.Noumi, Y.Ohta, Y.Yamada
    • Journal Title

      Int.Math.Res.Not. no.24

      Pages: 1441-1463

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Generating functions related to the Okamoto polynomials for the Painleve IV equation2005

    • Author(s)
      H.Goto, K.Kajiwara
    • Journal Title

      Bull.Austral.Math.Soc. 71, no.3

      Pages: 517-526

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Non-Abelian Zeta Functions for Function Fields.2005

    • Author(s)
      Weng Lin
    • Journal Title

      Amer.J.Math. 127・No.5

      Pages: 973-1017

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Generating functions related to the Okamoto polynomials for the Painleve, IV equation.2005

    • Author(s)
      H.Goto, K.Kajiwara
    • Journal Title

      Bull.Austral.Math.Soc. 71・No.3

      Pages: 517-526

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Soliton solutions for the non-autonomous discrete-time Toda lattice equation.2005

    • Author(s)
      K.Kajiwara, A.Mukaihira
    • Journal Title

      J.Phys.A 38・No.28

      Pages: 6363-6370

    • NAID

      120000981493

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Cubic pencils and Painleve Hamiltonians2005

    • Author(s)
      K.Kajiwara, T.Masuda, M.Noumi, Y.Ohta, Y.Yamada
    • Journal Title

      Funkcialaj Ekvacioj 48・1(掲載予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] q-Painleve方程式の超幾何解2005

    • Author(s)
      梶原健司, 増田哲, 野海正俊, 太田泰広, 山田泰彦
    • Journal Title

      京都大学数理解析研究所講究録 (掲載予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Generating function associated with the determinant formula for the solutions of the Painleve II equation2005

    • Author(s)
      N.Joshi, K.Kajiwara, M.Mazzocco
    • Journal Title

      Asterisque (掲載予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] L^2-metrics, projective flatness and families of polarized abelian varieties2004

    • Author(s)
      To Wing-Keung, Weng Lin
    • Journal Title

      Trans. Amer. Math. Soc. 356・7

      Pages: 2685-2707

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Hypergeometric solutions to the q-Painleve equati ons2004

    • Author(s)
      K.Kajiwara, T.Masuda, M.Noumi, Y.Ohta, Y.Yamada
    • Journal Title

      International Mathematical Research Notices 2004・47

      Pages: 2497-2521

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Generating function associated with the determinant formula for the solutions of the Painleve II equation2004

    • Author(s)
      N.Joshi, K.Kajiwara, M.Mazzocco
    • Journal Title

      Asterisque 297

      Pages: 67-78

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The space of local fields as a module over the ring of local integrals of motion2004

    • Author(s)
      Atsushi Nakayashiki
    • Journal Title

      International Mathematics Research Notices 52

      Pages: 2797-2822

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary 2004 Annual Research Report
  • [Journal Article] L^2-metrics, projective flatness and families of polarized abelian varieties2004

    • Author(s)
      To Wing-Keung, Weng Lin
    • Journal Title

      Trans.Amer.Math.Soc. 356・7

      Pages: 2685-2707

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Generating function associated with the determinant formula for the solutions of the Painleve II equation.2004

    • Author(s)
      N.Joshi, K.Kajiwara, M.Mazzocco
    • Journal Title

      Asterisque No.297

      Pages: 67-78

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Hypergeometric solutions to the $q$-Painleve equations2004

    • Author(s)
      K.Kajiwara, T.Masuda, M.Noumi, Y.Ohta, Y.Yamada
    • Journal Title

      Int.Math.Res.Not. no.47

      Pages: 2497-2521

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The space of local fields as a module over the ring of local integrals of motion2004

    • Author(s)
      A.Nakayashiki
    • Journal Title

      Int.Math.Res.Not. no.52

      Pages: 2797-2822

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The chiral space of local operators in ${rm SU}(2)$-invariant Thirring mode2004

    • Author(s)
      A.Nakayashiki
    • Journal Title

      Comm.Math.Phys. 245, no.2

      Pages: 279-296

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On the existence of pluricanonical forms on varieties with infinite fundamental group2004

    • Author(s)
      S.Takayama
    • Journal Title

      Amer.J.Math. 126, no.6

      Pages: 1221-1235

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Seshadri constants and a criterion for bigness of pseudo-effective line bundles2004

    • Author(s)
      S.Takayama
    • Journal Title

      Complex analysis in several variables---Memorial Conference of Kiyoshi Oka's Centennial Birthday, Adv.Stud.Pure Math., Math.Soc., Japan, Tokyo 42

      Pages: 307-312

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] L^2-metrics, projective flatness and families of polariz ed abelian varieties2004

    • Author(s)
      To Wing-Keung, Weng Lin
    • Journal Title

      Trans.Amer.Math.Soc. 356・7

      Pages: 2685-2707

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Hypergeometric solutions to the q-Painleve equations2004

    • Author(s)
      K.Kajiwara, T.Masuda, M.Noumi, Y.Ohta, Y.Yamada
    • Journal Title

      International Mathematical Research Notices 2004・47

      Pages: 2497-2521

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Okamoto多項式のHankel行列式表示に付随する母函数2004

    • Author(s)
      後藤弘道, 梶原健司
    • Journal Title

      九州大学応用力学研究所研究集会報告 15ME-S3

      Pages: 220-226

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Cremona変数と楕円差分Painleve方程式2004

    • Author(s)
      梶原健司, 増田哲, 野海正俊, 太田泰広, 山田泰彦
    • Journal Title

      京都大学数理解析研究所講究録 1400

      Pages: 197-263

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On the existence of pluricanonical forms on varieties with infinite fundamental group2004

    • Author(s)
      S, takayama
    • Journal Title

      Amer J Math. 126・6

      Pages: 1221-1235

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Analytic truncation and Rankin Selberg versus algebraic truncation and non-abelian zeta2003

    • Author(s)
      L.Weng
    • Journal Title

      Algebraic number theory and related topics (Japanese) (Kyoto, 2002). Surikaisekikenkyusho Kokyuroku No.1324

      Pages: 7-21

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On a $q$-difference Painlev・III equation. II.Rational solutions.2003

    • Author(s)
      K.Kajiwara
    • Journal Title

      J.Nonlinear Math.Phys. 10, no.3

      Pages: 282-303

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] $〜_{10}E_9$ solution to the elliptic Painleve equation.2003

    • Author(s)
      K.Kajiwara, T.Masuda, M.Noumi, Y.Ohta, Y.Yamada
    • Journal Title

      J.Phys.A 36, no.17

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Residues of $q$-hypergeometric integrals and characters of affine Lie algebras2003

    • Author(s)
      A.Nakayashiki
    • Journal Title

      Comm.Math.Phys. 240, no.1-2

      Pages: 197-241

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Local simple connectedness of resolutions of log-terminal singularities2003

    • Author(s)
      S.Takayama
    • Journal Title

      Internat.J.Math. 14, no.8

      Pages: 825-836

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Seshadri constants and a criterion for bigness of pseudo-effective line bundles2003

    • Author(s)
      S.Takayama
    • Journal Title

      Math.Z. 244, no.1

      Pages: 179-199

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Iitaka's fibrations via multiplier ideals2003

    • Author(s)
      S.Takayama
    • Journal Title

      Trans.Amer.Math.Soc. 355, no.1

      Pages: 37-47

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Segre's Reflexivity and an Inductive Characterization of Hyperquadrics2002

    • Author(s)
      E.Sato, Y.Kachi
    • Journal Title

      Memoirs of Amer. Math. 160・763

      Pages: 1-116

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Refined Brill-Noether locus non-abelian zeta functions for elliptic curves2002

    • Author(s)
      L.Weng
    • Journal Title

      Algebraic geometry in East Asia (Kyoto, 2001)(World Scientific)

      Pages: 245-262

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Stability and new non-abelian zeta functions.2002

    • Author(s)
      L.Weng
    • Journal Title

      Number theoretic methods, Dev.Math.(Kluwer Acad.Publ., Dordrecht) 8

      Pages: 405-419

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Zeta-functions defined by two polynomials.2002

    • Author(s)
      K.Matsumoto, L.Weng
    • Journal Title

      Number theoretic methods, Dev Math.(Kluwer Acad.Publ., Dordrecht) 8

      Pages: 233-262

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Generating function associated with the rational solutions of the Painleve II equation.2002

    • Author(s)
      K.Iwasaki, K.Kajiwara, T.Nakamura
    • Journal Title

      J.Phys.A 35, no.16

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] $q$-Painleve systems arising from $q$-KP hierarchy2002

    • Author(s)
      K.Kajiwara, M.Noumi, Y.Yamada
    • Journal Title

      Lett.Math.Phys. 62, no.3

      Pages: 259-268

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A determinant formula for a class of rational solutions of PainleveV equation.2002

    • Author(s)
      T.Masuda, Y.Ohta, K.Kajiwara
    • Journal Title

      Nagoya Math.J. 168

      Pages: 1-25

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Discrete dynamical systems with $W(A_{m-1}^{(1)}timesA_{n-1}^{(1)})$ symmetry2002

    • Author(s)
      K.Kajiwara, M.Noumi, Y.Yamada
    • Journal Title

      Lett.Math.Phys. 60, no.3

      Pages: 211-219

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On the cohomology of theta divisors of hyperelliptic Jacobians. Integrable systems,2002

    • Author(s)
      A.Nakayashiki
    • Journal Title

      topology, and physics (Tokyo, 2000), Contemp.Math., Amer.Math.Soc., Providence, RI, 2002. 309

      Pages: 177-183

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Euler characteristics of theta divisors of Jacobians for spectral curves.2002

    • Author(s)
      A.Nakayashiki, F.Smirnov
    • Journal Title

      The Kowalevski property (Leeds, 2000), CRM Proc.Lecture Notes, Amer.Math.Soc., Providence, RI 32

      Pages: 239-246

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On form factors of the ${rm SU}(2)$ invariant Thirring model2002

    • Author(s)
      A.Nakayashiki, Y.Takeyama
    • Journal Title

      MathPhys odyssey, 2001, Prog.Math.Phys., Birkhueser Boston, Boston, MA 23

      Pages: 357-390

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A survey on varieties with large fundamental groups.2002

    • Author(s)
      S.Takayama
    • Journal Title

      Communications in arithmetic fundamental groups (Kyoto, 1999/2001). Surikaisekikenkyusho Kokyuroku No.1267

      Pages: 14-25

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Non-Abelian Zeta Functions for function Fields

    • Author(s)
      Weng Lin
    • Journal Title

      Amer.J.Math. (to appear)

    • Related Report
      2004 Annual Research Report
  • [Book] Arithmetic Geometry and Number Theory2006

    • Author(s)
      L WENG, I NAKAMURA(eds)
    • Total Pages
      358
    • Publisher
      World Scientific
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Book] Proceedings of Conference on L Functions2006

    • Author(s)
      L WENG, M KANEKO(eds)
    • Total Pages
      350
    • Publisher
      World Scientific
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Publications] To, W.K, Weng.L.: "L^2-metrics, projective flatness and families of polarized abeliam varieties"Transactions of American Math. Soc. (発表予定). (2004)

    • Related Report
      2003 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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