Higher Chow groups of varieties over local fields

• Project/Area Number: 19740002
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: Tohoku University
• Principal Investigator: YAMAZAKI Takao Tohoku University, 大学院・理学研究科, 准教授 (00312794)
• Project Period (FY): 2007 – 2009
• Project Status: Completed (Fiscal Year 2009)
• Budget Amount: ¥3,960,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥660,000); Fiscal Year 2009: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2008: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2007: ¥1,100,000 (Direct Cost: ¥1,100,000)
• Keywords: 数論幾何学 / 高次元類体論 / 高次Chow群 / Brauer群 / K-群 / 代数学 / 数論幾何 / Milnor K群 / Artin L-関数 / 代数的完全可積分系

Research Abstract

Higher Chow groups of varieties is a wide generalization of the ideal class group in the algebraic number theory, and are regarded as a fundamental research theme. When varieties are defined over a local field, it naturally appears in arithmetic context, such as the higher dimensional class field theory and Brauer-Manin pairing. We developed the class field theory for a product of curves, and the Brauer-Manin pairing for open varieties (which was previously considered only for complete varieties).

Research Products

• [Journal Article] Singular fiber of the Mumford system and rational solutions to the KdV hierarchy. (2010)
  • Author(s): Pol Vanhaecke, Rei Inoue, Takao Yamazaki
  • Journal Title: Communications on Pure and Applied Mathematics 63 Pages: 508-532
  • Peer Reviewed
• [Journal Article] Singular fiber of the Mumford system and rational solutions to the KdV hierarchy (2010)
  • Author(s): Pol Vanhaecke, Rei Inoue, Takao Yamazaki
  • Journal Title: Communications on Pure and Applied Mathematics 63 Pages: 508-532
  • Peer Reviewed
• [Journal Article] A counterexample to generalizations of the Milnor-Bloch-Kato conjecture. (2009)
  • Author(s): Michael Spiess, Takao Yamazaki
  • Journal Title: Journal of K-theory 4,No.1 Pages: 77-90
  • Peer Reviewed
• [Journal Article] Milnor K-group attached to a torus and Birch-Tate conjecture. (2009)
  • Author(s): Takao Yamazaki
  • Journal Title: International Journal of Mathematics 20,No.7 Pages: 841-857
  • Peer Reviewed
• [Journal Article] Class field theory for a product of curves over a local field. (2009)
  • Author(s): Takao Yamazaki
  • Journal Title: Mathematische Zeitschrift 261,No.1 Pages: 109-121
  • Peer Reviewed
• [Journal Article] A counterexample to generalizations of the Milnor-Bloch-Kato conjecture (2009)
  • Author(s): Michael Spiess, Takao Yamazaki
  • Journal Title: Journal of K-theory 4 Pages: 77-90
  • Peer Reviewed
• [Journal Article] Milnor K-group attached to a torus and Birch-Tate conjecture (2009)
  • Author(s): Takao Yamazaki
  • Journal Title: International Journal of Mathematics 20 Pages: 841-857
  • Peer Reviewed
• [Journal Article] Class field theory for a product of curves over a local field (2009)
  • Author(s): Takao Yamazaki
  • Journal Title: Mathematische Zeitschrift 261 Pages: 109-121
  • Peer Reviewed
• [Journal Article] Class field theory for a product of curves over a local field (2009)
  • Author(s): Takao Yamazaki
  • Journal Title: Mathematische Zeitschrift 261, No. 1 Pages: 109-121
  • Peer Reviewed
• [Journal Article] Jacobian variety and integrable system - after Mumford, Beauville and Vanhaecke. (2007)
  • Author(s): Rei Inoue, Yukiko Konishi, Takao Yamazaki
  • Journal Title: Journal of Geometry and Physics 57,No.3 Pages: 815-831
  • Peer Reviewed
• [Journal Article] トーラス係数のMilnor K群とBirchi-Tate予想. (2007)
  • Author(s): 山崎隆雄
  • Journal Title: 第52回代数学シンポジウム報告集 Pages: 129-140
• [Journal Article] acobian variety and integrable system-after Mumford, Beauville and Vanhaecke (2007)
  • Author(s): Inoue, Konishi, Yamazaki
  • Journal Title: Journal of Geometry and Physics 57 Pages: 815-831
  • Peer Reviewed
• [Presentation] p-adic tau function and arithmetic geometry (2010)
  • Author(s): Takao Yamazaki, Sato theory
  • Organizer: Kyushu Algebraic Number Theory 2010
  • Place of Presentation: Kyushu University
  • Year and Date: 2010-03-19
• [Presentation] Sato theory, p-adic tau function and arithmetic geometry (2010)
  • Author(s): Takao Yamazaki
  • Organizer: Kyushu Algebraic Number Theory 2010
  • Place of Presentation: Kyushu University
  • Year and Date: 2010-03-19
• [Presentation] Higher dimensional class field theory for a product of curves (2010)
  • Author(s): Takao Yamazaki
  • Organizer: Japan-Korea Number Theory 2010 at Seoul
  • Place of Presentation: Korea
  • Year and Date: 2010-02-20
• [Presentation] Zero-cycles on a variety over a (two dimensional) local field. (2010)
  • Author(s): Takao Yamazaki
  • Organizer: Workshop on period integral and motif
  • Place of Presentation: University of Tokyo
  • Year and Date: 2010-02-17
• [Presentation] Zero-cycles on a variety over a(two dimensional)local field (2010)
  • Author(s): Takao Yamazaki
  • Organizer: Workshop on period integral and motif
  • Place of Presentation: University of Tokyo
  • Year and Date: 2010-02-17
• [Presentation] Higher dimensional class field theory for a product of curves (2010)
  • Author(s): Takao Yamazaki
  • Organizer: Japan-Korea Number Theory 2010 at Seoul
  • Place of Presentation: Seoul National University, Korea
  • Year and Date: 2010-01-20
• [Presentation] Milnor K-theory attached to semi-abelian varieties and class field theory. (2009)
  • Author(s): Takao Yamazaki
  • Organizer: Workshop on arithmetic geometry in Tambara
  • Place of Presentation: Numata
  • Year and Date: 2009-05-26
• [Presentation] Milnor K-theory attached to semi-abelian varieties and class field theory (2009)
  • Author(s): Takao Yamazaki
  • Organizer: Workshop on arithmetic geometry in Tambara
  • Place of Presentation: Numata
  • Year and Date: 2009-05-26
• [Presentation] Counter examples to variants of the Milnor-Bloch-Kato conjecture. (2009)
  • Author(s): Takao Yamazaki
  • Organizer: Industrious Number Theory
  • Place of Presentation: Korea Institute for Advanced Study
  • Year and Date: 2009-03-12
• [Presentation] Counter examples to variants of the Milnor-Bloch-Kato conjecture. (2008)
  • Author(s): Takao Yamazaki
  • Organizer: P-adic method and its applications in arithmetic geometry at Sendai
  • Place of Presentation: Tohoku University
  • Year and Date: 2008-11-06
• [Presentation] Degenerate fibers of the Mumford system and rational solutions to the KdV hierarchy (2008)
  • Author(s): Takao Yamazaki
  • Organizer: Number Theory and Physics at the Crossroads
  • Place of Presentation: Banff, Canada
  • Year and Date: 2008-09-23
• [Presentation] Degenerate fibers of the Mumford system and rational solutions to the KdV hierarchy (2008)
  • Author(s): Takao Yamazaki
  • Organizer: Number Theory and Physics at the Crossroads
  • Place of Presentation: バンプ、カナダ
  • Year and Date: 2008-09-23
• [Presentation] Generalized Jacobians and rational solutions for the Mumford system. (2008)
  • Author(s): Takao Yamazaki
  • Organizer: Journees integrables, Universite de Poitiers
  • Place of Presentation: France
  • Year and Date: 2008-06-06
• [Presentation] Generalized Jacobians and rational solutions for the Mumford system (2008)
  • Author(s): Takao Yamazaki
  • Organizer: Journees integrables
  • Place of Presentation: ボアティエ、フランス
  • Year and Date: 2008-06-06
• [Presentation] Recent development on the even Mumford system. (2007)
  • Author(s): Takao Yamazaki
  • Organizer: Seminaire groupes, algabres, et geometrie
  • Place of Presentation: Universite de Poitiers, France
  • Year and Date: 2007-11-21
• [Presentation] Milnor K-group attached to a torus and Birch-Tate conjecture. (2007)
  • Author(s): Takao Yamazaki
  • Organizer: Symposium on Algebra
  • Place of Presentation: Kobe University
  • Year and Date: 2007-08-07
• [Presentation] Milnor K-group attached to a torus and Birch-Tate conjecture (2007)
  • Author(s): 山崎隆雄
  • Organizer: 代数学シンポジウム
  • Place of Presentation: 神戸大学
  • Year and Date: 2007-08-07