Prospects for Mordell-Weil Lattices andAlgebraic Surfaces

• Project/Area Number: 20540051
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Rikkyo University
• Principal Investigator: SHIODA Tetsuji 立教大学, 名誉教授 (00011627)
• Project Period (FY): 2008 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2012: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2011: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2010: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2009: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2008: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: 代数幾何学 / モーデル・ヴェイユ格子 / 代数曲面 / K3曲面 / ガロア表現 / 球のつめこみ / 整点 / グレブナ基底 / 楕円曲面 / K３曲面 / モーデルヴェイュ格子 / 有理楕円曲面 / 乗法的不変式論 / 代数方程式 / ワイル群 / E_6,E_7,E_8 / abc-定理 / 整数点・整点 / フェルマー曲面 / ネロン・セヴェリ群 / 代数的サイクル / 球の詰め込み / K3 曲面

Research Abstract

We have studied selected topics on Mordell-Weil lattices (MWL) of elliptic surfaces: (1) K3 surfaces and sphere packing problem, (2) Existence and finiteness theorem of semi-stable extremal elliptic surfaces of any arithmetic genus, (3) Galois representations and algebraic equations arising from MWL, (4) Construction of multiplicative excellent families of elliptic surfaces with MWL of type E_6, E_7 or E_8, (5) Integral sections and Groebner basis, (6) Lines as generators of Neron-Severi group of Fermat surfaces.

Research Products

• [Journal Article] Elliptic fibrations ofmaximal rank on a supersingular K3 surface (2013)
  • Author(s): T. Shioda(塩田)
  • Journal Title: Izvestia RAS, Ser. Math. Volume: 77(3) Issue: 3 Pages: 139-148
  • DOI: 10.4213/im8017
  • Peer Reviewed
• [Journal Article] Multiplicative excellent family of type E_6 (2012)
  • Author(s): T. Shioda
  • Journal Title: Proc. Japan Acad. 86A Pages: 21-26
  • Peer Reviewed
• [Journal Article] Multiplicative excellent family of type E_6 (2012)
  • Author(s): Tetsuji Shioda
  • Journal Title: Proc.Japan Acad. Volume: 88A Issue: 3 Pages: 46-51
  • DOI: 10.3792/pjaa.88.46
  • Peer Reviewed
• [Journal Article] Lines on Fermat surfaces (2010)
  • Author(s): M. Schuett, T. Shioda, R.van Luijk
  • Journal Title: Journal of Number Theory Volume: 130 Pages: 1939-1963
  • Peer Reviewed
• [Journal Article] Elliptic Surfaces (2010)
  • Author(s): M. Schuett, T. Shioda
  • Journal Title: Advanced Studies in Pure Mathematics Volume: 60 Pages: 51-160
  • Peer Reviewed
• [Journal Article] Groebner basis, Mordell-Weil lattices and deformation of singularities (2010)
  • Author(s): T. Shioda
  • Journal Title: I. Proc. Japan Acad. Volume: 86(A) Pages: 21-26
  • Peer Reviewed
• [Journal Article] Groebner basis, Mordell-Weil lattices and deformation of singularities (2010)
  • Author(s): T. Shioda
  • Journal Title: II. Proc. Japan Acad. Volume: 86(A) Pages: 27-32
  • Peer Reviewed
• [Journal Article] R.van Luijk : Lines on Fermat surfaces (2010)
  • Author(s): M.Schuett, T.Shioda
  • Journal Title: Journal of Number Theory 130 Pages: 1939-1963
  • Peer Reviewed
• [Journal Article] Elliptic Surfaces : Advanced Studies in Pure (2010)
  • Author(s): M.Schuett, T.Shioda
  • Journal Title: Mathematics 60 Pages: 51-160
  • Peer Reviewed
• [Journal Article] Correction and supplement : Generators of the Neron-Severi group of a Fermat surface (2010)
  • Author(s): N.Aoki, T.Shioda
  • Journal Title: Comment.Math.Univ.St,Pauli 59 Pages: 65-75
  • Peer Reviewed
• [Journal Article] Groebner basis, Mordell- Weil lattices and deformation of singularities (2010)
  • Author(s): T.Shioda
  • Journal Title: I.Proc.Japan Acad. 86(A) Pages: 21-26
  • Peer Reviewed
• [Journal Article] Groebner basis, Mordell- Weil lattices and deformation of singularities (2010)
  • Author(s): T.Shioda
  • Journal Title: II.Proc.Japan Acad. 86(A) Pages: 27-32
  • Peer Reviewed
• [Journal Article] Elliptic surfaces (2010)
  • Author(s): Matthias Schuett, Tetsuji Shioda
  • Journal Title: Advanced Studies in Pure Mathematics Volume: 60 Pages: 51-160
  • Peer Reviewed
• [Journal Article] Correction and supplement : Generators of the Neron-Severi group of a Fermat surface (2010)
  • Author(s): Noboru Aoki, Tetsuji Shioda
  • Journal Title: Comment.Math.Univ.St, Pauli Volume: 59 Pages: 65-75
  • Peer Reviewed
• [Journal Article] Groebner basis, Mordell-Weil lattices and deformation of singularities, I. (2010)
  • Author(s): SHIODA, Tetsuji
  • Journal Title: Proc.Japan Acad. 86(A) Pages: 21-26
  • Peer Reviewed
• [Journal Article] Groebner basis, Mordell-Weil lattices and deformation of singularities, II. (2010)
  • Author(s): SHIODA, Tetsuji
  • Journal Title: Proc.Japan Acad. 86(A) Pages: 27-32
  • Peer Reviewed
• [Journal Article] Some explicit integral polynomials with Galois group W(E_8) (2009)
  • Author(s): T. Shioda
  • Journal Title: Proc.Japan Acad. Volume: 85(A) Pages: 118-121
  • Peer Reviewed
• [Journal Article] Some explicit integral polynomials with Galois group W(E_8) (2009)
  • Author(s): T.Shioda
  • Journal Title: Proc.Japan Acad. 85A Pages: 118-121
  • Peer Reviewed
• [Journal Article] Some explicit integral polynomials with Galois group W(E_8) (2009)
  • Author(s): SHIODA, Tetsuji
  • Journal Title: Proc.Japan Acad. 85(A) Pages: 118-121
  • Peer Reviewed
• [Journal Article] K3 surfaces and sphere packings (2008)
  • Author(s): T. Shioda
  • Journal Title: J. Math. Soc. Japan Volume: 60 Pages: 1083-1105
  • NAID: 10024905303
  • Peer Reviewed
• [Journal Article] The abc-theorem, Davenport's inequality and elliptic surfaces, Proc. (2008)
  • Author(s): T. Shioda
  • Journal Title: Japan Acad. Volume: 84A Pages: 51-56
• [Journal Article] K3 surfaces and sphere packings (2008)
  • Author(s): T.Shioda
  • Journal Title: J.Math.Soc.Japan 60 Pages: 1083-1105
  • NAID: 10024905303
  • Peer Reviewed
• [Journal Article] The abc-theorem, Davenport's inequality and elliptic surfaces (2008)
  • Author(s): T.Shioda
  • Journal Title: Proc.Japan Acad. 84A Pages: 51-56
  • NAID: 40015988007
  • Peer Reviewed
• [Journal Article] K3 surfaces and sphere packings (2008)
  • Author(s): SHIODA, Tetsuji
  • Journal Title: J. Math. Soc. Japan 60 Pages: 1083-1105
  • NAID: 10024905303
  • Peer Reviewed
• [Journal Article] The abc-theorem, Davenport's inequality and elliptic surfaces (2008)
  • Author(s): SHIODA, Tetsuji
  • Journal Title: Proc. Japan Acad. 84A
  • NAID: 40015988007
  • Peer Reviewed
• [Journal Article] Multiplicative excellent families of elliptic surfaces of type E_7 and E_8
  • Author(s): A. Kumar, T. Shioda
  • Journal Title: Algebra and Number Theory
  • URL: http://arXiv.org/pdf/1204.1531
  • Peer Reviewed
• [Journal Article] Lines on Fermat surfaces
  • Author(s): M.Schuett, T.Shioda, R.van Luijk
  • Journal Title: Journal of Number Theory (受理済み)
  • Peer Reviewed
• [Presentation] Manin's map and Mordell-Weil lattices (2013)
  • Author(s): 塩田徹治
  • Organizer: Workshopon Arithmetic and Algebraic Geometry
  • Place of Presentation: 東京大学大学院数理科学科
  • Year and Date: 2013-01-29
• [Presentation] Excellent families of rational elliptic surfaces (2012)
  • Author(s): 塩田徹治
  • Organizer: Workshop on Arithmetic and Algebraic Geometry
  • Place of Presentation: 東京大学大学院数理科学科
  • Year and Date: 2012-02-17
• [Presentation] Excellent families of rational elliptic surfaces (2012)
  • Author(s): Tetsuji Shioda
  • Organizer: Arithmetic and Algebraic Geometry
  • Place of Presentation: 東京大学大学院数理科学科(招待講演)
  • Year and Date: 2012-02-17
• [Presentation] Cubic surfaces via Mordell-Weil Lattices--Multiplicative excellent family of type E_6 (2011)
  • Author(s): 塩田徹治
  • Organizer: インターシティー数論セミナー
  • Place of Presentation: Groningen Univ.オランダ
  • Year and Date: 2011-12-15
• [Presentation] Cubic surfaces via Mordell-Weil Lattices ---Multiplicative excellent family of type E_6 (2011)
  • Author(s): Tetsuji Shioda
  • Organizer: Dutch Intercity Number Theory Seminar
  • Place of Presentation: Groningen Univ.(Netherlands)(招待講演)
  • Year and Date: 2011-12-15
• [Presentation] Cubic surfaces via Mordell-Weil Lattices re-revisited (2011)
  • Author(s): 塩田徹治
  • Organizer: Workshop on Arithmetic and Algebraic Geometry
  • Place of Presentation: 法政大学.
  • Year and Date: 2011-09-07
• [Presentation] Cubic surfaces via Mordell-Weil Lattices re-revisited (2011)
  • Author(s): Tetsuji Shioda
  • Organizer: Arithmetic and Algebraic Geometry
  • Place of Presentation: 法政大学理工学部(招待講演)
  • Year and Date: 2011-09-07
• [Presentation] Remarkable configurations of curves on K3 by Dolgachev- Kondo- Katsura (2011)
  • Author(s): T.Shioda
  • Organizer: Workshop on Arithmetic and Algebraic Geometry
  • Place of Presentation: 東京大学(招待講演)
  • Year and Date: 2011-01-19
• [Presentation] On the remarkable configurations of curves on K3 by Dolgachev-Kondo-Katsura (2011)
  • Author(s): Totsuji Shioda
  • Organizer: Arithmetic and Algebraic Geometry
  • Place of Presentation: 東京大学
  • Year and Date: 2011-01-19
• [Presentation] Mordell-Weil lattices and Galois representations: old and new (2010)
  • Author(s): 塩田徹治
  • Organizer: Workshop on Arithmetic on Surfaces, Lorentz Center
  • Place of Presentation: Leiden Univオランダ
  • Year and Date: 2010-11-25
• [Presentation] Excellent families of rational elliptic surfaces (2010)
  • Author(s): 塩田徹治
  • Organizer: Workshop on Reflection Groups,Lattices and Algebraic Geometry
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-11-24
• [Presentation] Excellent families of rational elliptic surfaces (2010)
  • Author(s): T.Shioda
  • Organizer: Workshop on Reflection Groups,Lattices and Algebraic Geometry
  • Place of Presentation: 名古屋大学(招待講演)
  • Year and Date: 2010-11-24
• [Presentation] Excellent families of rational elliptic surfaces (2010)
  • Author(s): Tetsuji Shioda
  • Organizer: Reflection Groups, Lattices and Algebraic Geometry
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-11-24
• [Presentation] Mordell-Weil lattices and Galois representations : old and new (2010)
  • Author(s): T.Shioda
  • Organizer: Workshop on Arithmetic on Surfaces
  • Place of Presentation: Lorentz Center, Leiden Univ(招待講演)
  • Year and Date: 2010-10-25
• [Presentation] Mordell-Weil lattices and Galois representations-old and new. (2010)
  • Author(s): Tetsuji Shioda
  • Organizer: Arithmetic on Surfaces
  • Place of Presentation: Lorentz Center, Leiden Univ
  • Year and Date: 2010-10-25
• [Presentation] Galois representations and Mordell-Weil lattices (2010)
  • Author(s): Tetsuji Shioda
  • Organizer: Algebraic Geometry Seminar
  • Place of Presentation: Hannover Univ
  • Year and Date: 2010-07-19
• [Presentation] Cubic surfaces via Mordell-Weil lattice revisited (2010)
  • Author(s): 塩田徹治
  • Organizer: Workshop on Elliptic fibrations and K3 surfaces
  • Place of Presentation: Humboldt Univ Berlin ドイツ.
  • Year and Date: 2010-07-14
• [Presentation] Cubic surfaces via Mordell-Weil lattice revisited (2010)
  • Author(s): T.Shioda
  • Organizer: Workshop on Elliptic fibrations and K3 surfaces
  • Place of Presentation: Humboldt Univ Berlin(招待講演)
  • Year and Date: 2010-07-14
• [Presentation] Cubic surfaces via Mordell-Weil lattice revisited (2010)
  • Author(s): Tetsuji Shioda
  • Organizer: Workshop on Elliptic fibrations and K3 surfaces
  • Place of Presentation: Humboldt Univ Berlin
  • Year and Date: 2010-07-14
• [Presentation] Some new features of K3 surfaces in characteristic p (2010)
  • Author(s): 塩田徹治
  • Organizer: Algebraic Geometry in characteristic p and related topics
  • Place of Presentation: 東京大学
  • Year and Date: 2010-02-17
• [Presentation] Some new features on K3 surfaces in characterisitic p (2010)
  • Author(s): SHIODA, Tetsuji
  • Organizer: Conference "Algebraic Geometry in Cha-racterisitic p and related topics"
  • Place of Presentation: 東京大学・数理科学研究科
  • Year and Date: 2010-02-17
• [Presentation] Mordell-Weil lattices and elliptic K3 surfaces (2010)
  • Author(s): 塩田徹治
  • Organizer: Workshop on Elliptic Fibrations andF-theory, IPMU
  • Place of Presentation: 東京大学
  • Year and Date: 2010-01-06
• [Presentation] Mordell-Weil lattices and elliptic K3 surfaces (2010)
  • Author(s): SHIODA, Tetsuji
  • Organizer: IPMU Conference "Elliptic Fibrations and F-theory"
  • Place of Presentation: IPMU, 東京大学(柏)数物宇宙連携機構
  • Year and Date: 2010-01-06
• [Presentation] 有理楕円 曲面の整切断(整点)について-グレブナ基 底と E8 格子 (2009)
  • Author(s): 塩田徹治
  • Organizer: 数学談話会
  • Place of Presentation: 京都大学数理解 析研究所
  • Year and Date: 2009-11-04
• [Presentation] モーデル・ ヴェイユ格子とグレブナ基底 (2009)
  • Author(s): 塩田徹治
  • Organizer: 数学談話会
  • Place of Presentation: 神戸大学
  • Year and Date: 2009-10-21
• [Presentation] モーデル・ヴェイユ格子とグレブナ基底 (2009)
  • Author(s): SHIODA, Tetsuji
  • Organizer: 数学談話会
  • Place of Presentation: 神戸大学
  • Year and Date: 2009-10-21
• [Presentation] Some Topics on Mordell-Weil lattices (2009)
  • Author(s): 塩田徹治
  • Organizer: 代数学セミナー
  • Place of Presentation: 埼玉大学
  • Year and Date: 2009-09-15
• [Presentation] Some Topics on Mordell-Weil lattices (2009)
  • Author(s): T.Shioda
  • Organizer: 代数学セミナー
  • Place of Presentation: 埼玉大学(招待講演)
  • Year and Date: 2009-09-15
• [Presentation] Some topics on Mordell-Weil lattices. (モーデル・ヴェイユ格子のトピックス) (2009)
  • Author(s): SHIODA, Tetsuji
  • Organizer: 代数学セミナー
  • Place of Presentation: 埼玉大学
  • Year and Date: 2009-09-15
• [Presentation] Mordell-Weil lattices and Galois representations Old and new (2009)
  • Author(s): 塩田徹治
  • Organizer: 日本数学会・代数学シンポジウム
  • Place of Presentation: 明治大学
  • Year and Date: 2009-08-04
• [Presentation] Mordell-Weil lattices and Galois representations (2009)
  • Author(s): T.Shioda
  • Organizer: 日本数学会・代数学シンポジウム
  • Place of Presentation: 明治大学(招待講演)
  • Year and Date: 2009-08-04
• [Presentation] Mordell-Weil lattices and Galois representations―Old and new. (モーデル・ヴェイユ格子とガロア表現20年) (2009)
  • Author(s): SHIODA, Tetsuji
  • Organizer: 日本数学会・代数学シンポジウム
  • Place of Presentation: 明治大学
  • Year and Date: 2009-08-04
• [Presentation] Mordell-Weil lattices and integral sections (2009)
  • Author(s): 塩田徹治
  • Organizer: Conference "Diophantine Geometry: into the Millenniumm
  • Place of Presentation: ETH-Zurich.スイス
  • Year and Date: 2009-06-05
• [Presentation] Mordell--Weil lattices and integral sections (2009)
  • Author(s): T.Shioda
  • Organizer: Conference "Diophantine Geometry : into the Millenniumm"
  • Place of Presentation: ETH-Zurich.(招待講演)
  • Year and Date: 2009-06-05
• [Presentation] Mordell-Weil lattices and integral sections (2009)
  • Author(s): SHIODA, Tetsuji
  • Organizer: Conference "Diophantine Geometry : in tothe Millenniumm"
  • Place of Presentation: FIM-Mathematical Research Institute ETH-Zurich
  • Year and Date: 2009-06-05
• [Presentation] Curves on Fermat surfaces revisited (2009)
  • Author(s): SHIODA, Tetsuji
  • Organizer: Conference“Arithmetic and Algebraic Geometry related to Moduli Spaces"
  • Place of Presentation: Univ Tokyo
  • Year and Date: 2009-01-20
• [Presentation] Mordell-Weil lattices of certain elliptic K3 surfaces, Conference on Arithmetic of K3 surfaces (2008)
  • Author(s): 塩田徹治
  • Organizer: BIRS (Banff)
  • Place of Presentation: カナダ
  • Year and Date: 2008-12-02
• [Presentation] Profile of some beautiful K3 surfaces in char p
  • Author(s): 塩田徹治
  • Organizer: 楕円曲面に関するミニワークショップ
  • Place of Presentation: 立教大学数学科
  • Invited
• [Presentation] Manin's map and Mordell-Weil lattices
  • Author(s): 塩田徹治
  • Organizer: Workshop on Arithmetic and Algebraic Geometry
  • Place of Presentation: 東京大学大学院数理科学科
  • Invited
• [Presentation] ３次曲面と２７直線を決定するアルゴリズム
  • Author(s): 塩田徹治
  • Organizer: 代数幾何ミニ研究集会
  • Place of Presentation: 埼玉大学理学部数学科
  • Invited
• [Book] abc-定理,楕円曲面,モーデル・ヴェイユ格子東京大学数理科学レクチャーノート (2008)
  • Author(s): 塩田 徹治
  • Total Pages: 78
  • Publisher: 東京大学大学院数理科学研究科
• [Book] abc-定理,楕円曲面,モーデル・ヴェイユ格子(東京大学数理科学レクチャーノート4) (2008)
  • Author(s): 塩田徹治
  • Total Pages: 78
  • Publisher: 東京大学大学院数理科学研究科
• [Book] abc-定理, 楕円曲面, モーデル・ヴェイュ格子(東京大学数理科学レクチャーノート 4) (2008)
  • Author(s): 塩田徹治
  • Total Pages: 78
  • Publisher: 東京大学大学院 数理科学研究科