A global study of the moduli spaces of abelian varieties over the ring of integers

2010 Fiscal Year Final Research Report

• Project/Area Number: 20340001
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Hokkaido University
• Principal Investigator: NAKAMUR Iku Hokkaido University, 大学院・理学研究院, 教授 (50022687)
• Co-Investigator(Renkei-kenkyūsha): KATSURADA Hidenori 室蘭工業大学, 大学院・工学研究科, 教授 (80133792) ; WENG Lin 九州大学, 大学院・数理学研究院, 教授 (60304002)
• Project Period (FY): 2008 – 2010
• Keywords: モジュライ / アーベル多様体 / コンパクト化 / 安定性 / マッカイ対応 / 単純特異点 / 既約表現 / ディンキン図形

Research Abstract

We proved the following in the theory of moduli of abelian varieties :
Theorem : There is another canonical compactification SQ^<toric>_<g,k> of the moduli of abelian varieties different from SQ_<g,k> constructed by us in 1999. Moreover there is a canonical bijectivebirationalmorphism from SQ^<toric>_<g,k> onto SQ_<g,k> which induces the isomorphism of their normalizations.
There was also a progress in sharpening the 2-dimensional McKay correspondence, which explains the connection with the extended Dynkin diagram.

Research Products

• [Journal Article] On Ikeda's conjecture on the period of the Ikeda lift and its application (2011)
  • Author(s): Katsurada Hidenori ; Kawamura Hisaaki
  • Journal Title: 京都大学数理解析研究所報告集別冊『保型形式とその周期の構成と応用』
  • Peer Reviewed
• [Journal Article] Symmetries and the Riemann Hypothesis (2010)
  • Author(s): Weng Lin
  • Journal Title: Advanced Studies in Pure Mathematics (Math.Soc.Japan) 58 Pages: 173-223
  • Peer Reviewed
• [Journal Article] Stability and arithmetic (2010)
  • Author(s): Weng Lin
  • Journal Title: Advanced Studies in Pure Mathematics (Math.Soc.Japan) 58 Pages: 225-359
  • Peer Reviewed
• [Journal Article] On the Andrianov type identity for power series attached to Jacobi forms and its application (2010)
  • Author(s): Katsurada Hidenori ; Kawamura Hisaaki
  • Journal Title: ActaArithmetica 145 Pages: 233-265
  • Peer Reviewed
• [Journal Article] Rankin-Selberg method and periods of modular forms (2010)
  • Author(s): Katsurada Hidenori
  • Journal Title: MSJ Memoirs 21 Pages: 86-114
  • Peer Reviewed
• [Journal Article] Another canonical compactification of the moduli space of abelian varieties (2010)
  • Author(s): Nakamura Iku
  • Journal Title: Algebraic and Arithmetic Structures of Moduli Spaces (Sapporo, 2007), Advanced Studies of Pure Mathematics Math.Soc.Japan 58 Pages: 69-135
  • Peer Reviewed
• [Journal Article] McKay correspondence (2009)
  • Author(s): Nakamura Iku
  • Journal Title: Groups and symmetries, CRM Proc.and Lec.Notes 47 Pages: 267-298
  • Peer Reviewed
• [Journal Article] Zeta functions for G2 and their zeros (2009)
  • Author(s): Suzuki Masatoshi ; Weng Lin
  • Journal Title: Int.Math.Res.Not.IMRN 2 Pages: 241-290
  • Peer Reviewed
• [Journal Article] Zeta functions for Sp(2n) (2009)
  • Author(s): Weng Lin
  • Journal Title: J.Number Theory 129 3 Pages: 567-579
  • Peer Reviewed
• [Journal Article] The Riemann hypothesis for Weng's zeta function of Sp(4) over Q (2009)
  • Author(s): Suzuki Masatoshi
  • Journal Title: J.Number Theory 129 3 Pages: 551-567
• [Journal Article] Ikeda's conjecture on the period of the Ikeda lift (2009)
  • Author(s): Katsurada Hidenori ; Kawamura Hisaaki
  • Journal Title: 数理解析研究所講究録 1659 Pages: 68-81
• [Journal Article] The asymptotic behavior of the Takhtajan-Zograf metric (2008)
  • Author(s): Weng Lin ; Obitsu Kunio ; To Wing-Keung
  • Journal Title: Comm.Math.Phys.284 1 Pages: 227-261
  • Peer Reviewed
• [Journal Article] Deligne products of line bundles over moduli spaces of curves (2008)
  • Author(s): Weng Lin ; Zagier Don
  • Journal Title: Comm.Math.Phys.281 3 Pages: 793-803
  • Peer Reviewed
• [Journal Article] A certain Dirichlet series of Rankin-Selberg type associated with the Ikeda lifting (2008)
  • Author(s): Katsurada Hidenori ; Kawamura Hisaaki
  • Journal Title: Journal of Number Theory 128 7 Pages: 2025-2052
  • Peer Reviewed
• [Presentation] Extended Dynkin diagram in McKay correspondence (2011)
  • Author(s): Nakamura Iku
  • Organizer: Arithmetic and Algebraic Geometry 2011
  • Place of Presentation: University of Tokyo
  • Year and Date: 20110118-20110122
• [Presentation] A canonical morphism from SQ^<tario>_<g,k> to (2009)
  • Author(s): Nakamura Iku
  • Organizer: Moduli and SQ_<g,k> Discrete Groups
  • Place of Presentation: Kyoto University
  • Year and Date: 20090608-20090612
• [Presentation] Two compactifications of moduli spaces of abelian varieties (2009)
  • Author(s): Nakamura Iku
  • Organizer: Arithmetic and algebraic geometry related to moduli spaces
  • Place of Presentation: University of Tokyo
  • Year and Date: 20090119-20090123
• [Remarks] 2010.5.31-6.4, Nakamura, Iku: "Moduli of elliptic curves", 集中講義,埼玉大学理工学部
• [Remarks] 2009.9.3, Nakamura, Iku : "McKay correspondence", Seminar on Calalbi-Yau, Queen's University, Canada
• [Remarks] 2008.11.21, Nakamura, Iku : "Acanonical morphism from SQ^<torio>_<g,k> to SQ_<g,k>", Kyoto University
• [Remarks] 2008.9.11 and 18, Nakamura, Iku : "Compactification of moduli of abelian varieties", Seminar on Calalbi-Yau, Queen's University, Canada
• [Remarks] 2008.9.5, Nakamura, Iku : "Stability and compactification of moduli of abelian varieties",the Centre de recherches mathematiques, Montereal University
• [Remarks] ホームページのURL
  • URL: http://www.math.sci.hokudai.ac.jp/~nakamura/indexJ.html