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2004 Fiscal Year Final Research Report Summary

Research on special values of the standard L-functions of Siegel modular forms and related fields

Research Project

Project/Area Number 15540003
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionMuroran Institute of Technology

Principal Investigator

KATSURADA Hidenori  Muroran Institute of Technology, Professor, 工学部, 教授 (80133792)

Co-Investigator(Kenkyū-buntansha) TAKEGAHARA Yugen  Muroran Institute of Technology, Associate Professor, 工学部, 教授 (10211351)
HASEGAWA Yuji  Muroran Institute of Technology, Associate Professor, 工学部, 助教授 (30287982)
KOHSAKA Yoshihito  Muroran Institute of Technology, Associate Professor, 工学部, 助教授 (00360967)
Project Period (FY) 2003 – 2004
KeywordsKoecher-Maass series / Saito-Kurokawa lift / Special values / Congruence of modular forms / Standard zeta function / Basis Problem / Doi-Hida-Ishi conjecture / Block-Kato conjecture
Research Abstract

(1)I considered the relation between special values of the standard zeta functions and the congruence a Siegel cusp form of arbitrary degree. In particular, I proposed a conjecture concerning the congruence of the Saito-Kurokawa lift of a cuspidal Hecke eigenform with respect to SL(2,Z) and the special value of its L-function, and proved it under certain condition. Furthermore we gave some numerical evidence for our conjecture. I gave talks on these topics at the workshop held at Hakuba on October 2004, and at the conference held at Karatsu on March 2005.
(2)Let M and N be square free positive integers, and φ and χ be Dirichlet characters mod M and N, respectively. For a cuspidal Hecke eigenform f of neben type φ I gave an explicit formula for the special values of the standard zeta function of f twisted by χ. I gave numerical examples for these values and checked some results derived from the Block-Kato conjecture.
(3)I proved that the space of Eisenstein series of square free level of quadratic nebentype is spanned by the genus theta series jointly with R. Schulze-Pillot. By using this result, I proved that the space of cusp forms of square free level of quadratic nebentype is spanned by the theta series jointly with S.Boecherer and R.Schulze-Pillot. They gave talks on these topics at the symposium held at Rikkyo University on September 2004.
(4)I gave an explicit form of the Koecher-Maass series for the real analytic Eisenstein series jointly with T.Ibukiyama. I gave a talk on this topic at the symposium held at Rikkyo University on September 2004.

  • Research Products

    (23 results)

All 2005 2004 2003

All Journal Article (23 results)

  • [Journal Article] Special values of the standard zeta functions for elliptic modular forms2005

    • Author(s)
      H.Katsurada
    • Journal Title

      Experimental Mathematics 14(発刊予定)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Linearized stability analysis of stationary solutions for surface diffusion with boundary conditions2005

    • Author(s)
      Harald Garcke
    • Journal Title

      SIAM Journal of Mathematical Analysis 36

      Pages: 1031-1056

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Special values of the standard zeta functions for Elliptic modular forms2005

    • Author(s)
      H.Katsurada.
    • Journal Title

      Experimental Mathematics 14 (To appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Linearized stability analysis of stationary solutions for surface diffusion with boundary conditions2005

    • Author(s)
      H.Garcke, K.Ito, Y.Kohsaka
    • Journal Title

      SIAM Journal of Mathematical Analysis 36

      Pages: 1031-1056

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Crossed homomorphisms and the Schur-Zassenhaus theorem2004

    • Author(s)
      淺井恒信
    • Journal Title

      数理解析研究所講究録

      Pages: 23-31

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Generating functions for permutation representations2004

    • Author(s)
      Y.Takegahara
    • Journal Title

      J.Algebra 281

      Pages: 68-82

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Hall's relations in finite groups2004

    • Author(s)
      Y.Takegahara
    • Journal Title

      J.Algebra 271

      Pages: 312-326

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] An explicit formula for the Koecher-Maass Dirichiet series for the Ikeda lifting2004

    • Author(s)
      T.Ibukiyama
    • Journal Title

      Abhandlungen aus dem Mathematichen Seminar der Universitaet Hamburg 74

      Pages: 101-121

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] An explicit formula for the Koecher-Maass Dirichlet series for the Ikeda lifting2004

    • Author(s)
      T.Ibukiyama, H.Katsurada
    • Journal Title

      Abhand.Math.Sem.der Univ.Hamburg 74

      Pages: 101-121

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On P.Hall's relations in finite groups, II2004

    • Author(s)
      Y.Takegahara
    • Journal Title

      RIMS Kokyuroku 1407

      Pages: 63-70

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Crossed homomorphisms and the Schur-Zassenhaus theorem2004

    • Author(s)
      T.Asai, Y.Takegahara, N.Chigira, T.Niwasaki
    • Journal Title

      RIMS Kokyuroku

      Pages: 23-31

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Hall's relations infinite groups2004

    • Author(s)
      Y.Takegahara
    • Journal Title

      J.Algebra 271

      Pages: 312-326

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Stability analysis of stationarysolutions for surface diffusion flow equation2004

    • Author(s)
      H.Garcke., K.Ito, Y.Kohsaka
    • Journal Title

      RIMS Kokyuroku 1356

      Pages: 27-43

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] An explicit form of the Koecher-Maass Dirichlet series for Klingen's Eisenstein series2003

    • Author(s)
      T.Ibukiyama
    • Journal Title

      J.Number Theory 71

      Pages: 223-256

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Special values of the standard zeta functions2003

    • Author(s)
      H.Katsurada
    • Journal Title

      Developments in Math. 11

      Pages: 337-356

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A remark on the coincidence of Hecke eigenforms2003

    • Author(s)
      H.Katsurada
    • Journal Title

      Comment.Math.Univ.St.Pauli 52

      Pages: 47-54

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On some p-adic properties of Siegel Eisenstein series2003

    • Author(s)
      H.Katsurada
    • Journal Title

      J.Number Theory 104

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Crossed homomorphisms from rank-2 abelian to exceptional p-groups2003

    • Author(s)
      T.Asai
    • Journal Title

      J.Algebra

      Pages: 213-237

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Automorphisms of X_0^*(p)2003

    • Author(s)
      Matthew Baker
    • Journal Title

      Journal of Number Theory

      Pages: 72-87

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] An explicit form of the Koecher-Maass Dirichlet series for Klingen's Eisenstein series2003

    • Author(s)
      T.Ibukiyama, H.Katsurada
    • Journal Title

      J.Number Theory 71

      Pages: 223-256

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On some p-adic properties of Siegel Eisenstein series2003

    • Author(s)
      H.Katsurada, S.Nagaoka
    • Journal Title

      J.Number Theory

      Pages: 100-117

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Crossed homomorphisms from rank-2 abelian to exceptional p-groups2003

    • Author(s)
      T.Asai, T.Niwasaki, Y.Takegahara
    • Journal Title

      J.Algebra 281

      Pages: 212-237

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Automorphisms of X_0^*(p)2003

    • Author(s)
      M.Baker, Y.Hasegawa
    • Journal Title

      J.Number Theory

      Pages: 72-87

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2006-07-11  

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