canonical heights on varieties defined over a number field and related topics

• Project/Area Number: 18740008
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: Osaka University (2007-2008); Kyoto University (2006)
• Principal Investigator: KAWAGUCHI Shu Osaka University, 大学院・理学研究科, 准教授 (20324600)
• Project Period (FY): 2006 – 2008
• Project Status: Completed (Fiscal Year 2008)
• Budget Amount: ¥2,380,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥180,000); Fiscal Year 2008: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2007: ¥700,000 (Direct Cost: ¥700,000); Fiscal Year 2006: ¥900,000 (Direct Cost: ¥900,000)
• Keywords: 標準的高さ / Arakelov幾何 / アラケロフ幾何

Research Abstract

高さとは、代数体上に定義された代数多様体の点(より一般には部分代数多様体)の算術的な「大きさ,複雑さ」を測る量と考えられる.代数多様体が「良い」自己写像を持つときには,その写像に関して良く振る舞う高さが存在することがある.このような高さを標準的高さという.例えば,代数体上に定義されたアーベル多様体の算術的性質を調べるときに基本的なNeron-Tateの高さは,アーベル多様体のn倍射に関してよく振る舞う標準的高さである.当該研究では,標準的高さ関数の構成,および,代数多様体の有理点の自己写像に関する算術的性質を以下の場合に調べた.代数多様体が複数の「良い」射を持つ場合,アフィン平面の多項式自己同型,代数多様体の射の「良い」列,正の位相エントロピーをもつ代数曲面の自己同型.さらに,Joseph H. Silverman氏と共同で,射影空間の二つの射に付随する標準的高さ関数が一致するときに,二つの射が深く関係していることを示した

Research Products

• [Journal Article] Dynamics of projective morphisms having identical canonical heights (2008)
  • Author(s): Shu Kawaguchi and Joseph H. Silverman, Addendum
  • Journal Title: Proc. Lond. Math. Soc. (3)97, no.1 Pages: 272-272
  • Peer Reviewed
• [Journal Article] Projective surface automorphisms with positive topological entropy from an arithmetic viewpoint (2008)
  • Author(s): Shu Kawaguchi
  • Journal Title: Amer. J. Math. 130 Pages: 159-186
  • Peer Reviewed
• [Journal Article] Projective surface automorphisms with positive topological entropy from an arithmetic viewpoint (2008)
  • Author(s): Shu kawaguchi
  • Journal Title: American Journal of Mathematics 130 Pages: 159-186
  • Peer Reviewed
• [Journal Article] Dynamics of projective morphisms having identical canonical heights (2007)
  • Author(s): Shu Kawaguchi and Joseph H. Silverman, Addendum
  • Journal Title: Proc. Lond. Math. Soc. (3)95, No. 2 Pages: 519-544
  • Peer Reviewed
• [Journal Article] Silverman, Dynamics of projective morphisms having identical canonical heights (2007)
  • Author(s): Shu Kawaguchi and Joseph H
  • Journal Title: Proc. Lond. Math. Soc. (3)95 Pages: 519-544
  • Peer Reviewed
• [Journal Article] Canonical heights for random iterations in certain varieties (2007)
  • Author(s): Shu Kawaguchi
  • Journal Title: Int. Math. Res. Not. IMRN 2007 Art. ID rnm 023 Pages: 33-33
  • Peer Reviewed
• [Journal Article] Canonical heights for random iterations in certain varieties (2007)
  • Author(s): Shu Kawaguchi
  • Journal Title: International Mathematics Research Notices 2007 Pages: 33-33
  • Peer Reviewed
• [Journal Article] Dynamics of projective morphisms having identical canonical heights (2007)
  • Author(s): Shu Kawaguchi and Joseph H. Silverman
  • Journal Title: Proceedings of the London Mathematics Society 95 Pages: 519-544
  • Peer Reviewed
• [Journal Article] Canonical height functions for affine plane polynomial automorphisms (2006)
  • Author(s): Shu Kawaguchi
  • Journal Title: Math. Ann 335 Pages: 285-301
  • Peer Reviewed
• [Journal Article] Canonical heights, invariant currents and dynamical eigensystems for line bundles (2006)
  • Author(s): Shu Kawaguchi
  • Journal Title: J. Reine Angew. Math. 597 Pages: 135-173
  • Peer Reviewed
• [Journal Article] Canonical height functions for affine plane automorphisms (2006)
  • Author(s): Shu Kawaguchi
  • Journal Title: Mathematische Annalen 335 Pages: 285-310
• [Journal Article] Canonical heights, invariant currents, and dynamical eigensystems of morphisms for line bundles (2006)
  • Author(s): Shu Kawaguchi
  • Journal Title: Journal fur die reine und angewandte Mathematik 597 Pages: 135-174
• [Journal Article] Canonical heights for random iterations in certain varieties
  • Author(s): Shu Kawaguchi
  • Journal Title: International Mathematics Research Notices (accepted)
• [Journal Article] Dynamics of projective morphisms having identical canonical heights
  • Author(s): Shu Kawaguchi, Joseph H.Silverman
  • Journal Title: Proceedings of the London Mathematics Society (accepted)
• [Presentation] (2009)
  • Organizer: seminar on Arakelov geometry at the Institut de Mathematiques de Jussieu
• [Presentation] 代数多様体の射に関する標準的高さ関数 (2008)
  • Author(s): 川口周
  • Organizer: 代数学シンポジウム
  • Place of Presentation: 盛岡駅前アイーナ
  • Year and Date: 2008-08-06
• [Presentation] (2008)
  • Organizer: 代数学シンポジウム
  • Place of Presentation: 盛岡
• [Presentation] (2007)
  • Organizer: International conference on Birational automorphisms of compact complex manifold and Dynamical systems
  • Place of Presentation: 名古屋大学
• [Presentation] (2006)
  • Organizer: Seminars on Arakelov geometry and Shimura varieties
  • Place of Presentation: Centre de Recerca Matematica
• [Presentation] (2006)
  • Organizer: Seminaire Autour de la geometrie d'Arakelov, Institut Mathematique de Jussieu