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Higher Chow groups and higher dimensional class field theory

Research Project

Project/Area Number 22684001
Research Category

Grant-in-Aid for Young Scientists (A)

Allocation TypeSingle-year Grants
Research Field Algebra
Research InstitutionTohoku University

Principal Investigator

TAKAO Yamazaki  東北大学, 理学(系)研究科(研究院), 教授 (00312794)

Project Period (FY) 2010-04-01 – 2014-03-31
Project Status Completed (Fiscal Year 2013)
Budget Amount *help
¥12,610,000 (Direct Cost: ¥9,700,000、Indirect Cost: ¥2,910,000)
Fiscal Year 2013: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Fiscal Year 2012: ¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
Fiscal Year 2011: ¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
Fiscal Year 2010: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000)
Keywords数論幾何学 / 高次元類体論 / 高次 Chow 群 / ブラウアー群 / K-群 / モチビックホモロジー / モチーフ / 代数的サイクル / 高次Chow群 / Milnor K群 / 染川K群 / モチビック・コホモロジー / Brauer群 / モチビックコホモロジー / Suslinホモロジー / Milnor K-群
Research Abstract

Higher Chow groups of varieties are important objects in arithmetic geometry in general. When the base field is of arithmetic nature, higher Chow groups are known to be related with arithmetic objects such as the fundamental group and Brauer group. Over a local field, such relation has been studied for smooth projective varieties. We extended it to non-projective varieties. Besides, in a joint work with Bruno Kahn, we described the tensor product in the triangulated category of motives in terms of Somekawa K-groups, which provides us with a new method to compute the higher Chow groups of a product of varieties.

Report

(5 results)
  • 2013 Annual Research Report   Final Research Report ( PDF )
  • 2012 Annual Research Report
  • 2011 Annual Research Report
  • 2010 Annual Research Report
  • Research Products

    (52 results)

All 2014 2013 2012 2011 2010 Other

All Journal Article (8 results) (of which Peer Reviewed: 6 results) Presentation (44 results) (of which Invited: 4 results)

  • [Journal Article] Torsion points on hyperelliptic Jacobians via Anderson's p-adic soliton theory2013

    • Author(s)
      Yuken Miyasaka and Takao Yamazaki
    • Journal Title

      Tokyo Journal of Mathematics

      Volume: 36 Pages: 387-403

    • Related Report
      2013 Annual Research Report 2013 Final Research Report 2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Voevodsky's motives and Weil reciprocity2013

    • Author(s)
      Bruno Kahn and Takao Yamazaki
    • Journal Title

      Duke Mathematical Journal

      Volume: 162 Pages: 2751-2796

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] Brauer-Manin pairing, class field theory and motivic homology2013

    • Author(s)
      Takao Yamazaki
    • Journal Title

      Nagoya Mathematical Journal

      Volume: 210 Pages: 29-58

    • Related Report
      2013 Annual Research Report 2013 Final Research Report 2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] ヴォドスキーのモチーフとヴェイユ相互律2013

    • Author(s)
      山崎隆雄
    • Journal Title

      数理解析研究所講究録別冊

      Volume: 44 Pages: 165-181

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] Bruno Kahn, Takao Yamazaki2013

    • Author(s)
      Voevodsky's motives and Weil reciprocity
    • Journal Title

      Duke Mathematical Journal

      Volume: 162 Pages: 2751-2796

    • Related Report
      2013 Annual Research Report
  • [Journal Article] ヴォエヴォドスキーのモチーフとヴェイユ相互律2013

    • Author(s)
      山崎隆雄
    • Journal Title

      数理解析研究所講究録別冊

      Volume: 44

    • Related Report
      2013 Annual Research Report 2012 Annual Research Report
  • [Journal Article] Singular fiber of the Mumford system and rational solutions to the KdV hierarchy2010

    • Author(s)
      Pol Vanhaecke, Rei Inoue and Takao Yamazaki
    • Journal Title

      Communications on Pure and Applied Mathematics

      Volume: 63 Pages: 508-532

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] Singular fiber of the Mumford system and rational solutions to the KdV hierarchy2010

    • Author(s)
      Pol Vanhaecke, Rei Inoue, Takao Yamazaki
    • Journal Title

      Comm.Pure and Applied Math.

      Volume: 63 Pages: 508-532

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Presentation] Voevodsky's motives and Weil reciprocity2014

    • Author(s)
      Takao Yamazaki
    • Organizer
      2014 NCTS Mini-Workshop on Number Theory, NCTS
    • Place of Presentation
      台湾国立理論科学研究所
    • Year and Date
      2014-03-12
    • Related Report
      2013 Final Research Report
  • [Presentation] p-adic soliton theory and arithmetic geometry2014

    • Author(s)
      Takao Yamazaki
    • Organizer
      Arithmetic and Algebraic Geoemtry 2014
    • Place of Presentation
      東京大学数理科学研究科
    • Year and Date
      2014-01-30
    • Related Report
      2013 Final Research Report
  • [Presentation] p-adic soliton theory and torsion points on Jacobian varieties2014

    • Author(s)
      Takao Yamazaki
    • Organizer
      East Asia Number Theory Conference 2014
    • Place of Presentation
      九州大学西新プラザ
    • Year and Date
      2014-01-24
    • Related Report
      2013 Final Research Report
  • [Presentation] Towards non-homotopy invariant motive theory2014

    • Author(s)
      Takao Yamazaki
    • Organizer
      談話会
    • Place of Presentation
      京都大学
    • Year and Date
      2014-01-08
    • Related Report
      2013 Final Research Report
  • [Presentation] Reciprocity sheaves : towards non-homotopy invariant motive theory2013

    • Author(s)
      Takao Yamazaki
    • Organizer
      Motives in Tokyo, 2013
    • Place of Presentation
      東京大学数理科学研究科
    • Year and Date
      2013-11-27
    • Related Report
      2013 Final Research Report
  • [Presentation] Reciprocity sheaves : towards non-homotopy invariant motive theory2013

    • Author(s)
      Takao Yamazaki
    • Organizer
      Arithmetic, differential and geometry
    • Place of Presentation
      東北大学
    • Year and Date
      2013-10-31
    • Related Report
      2013 Final Research Report
  • [Presentation] Torsion points on Jacobian varieties via Anderson's p-adic soliton theory2013

    • Author(s)
      Takao Yamazaki
    • Organizer
      Seminaire Groupes, Algebre et Geometrie
    • Place of Presentation
      フランス・ポアティエ大学
    • Year and Date
      2013-09-19
    • Related Report
      2013 Final Research Report
  • [Presentation] Reciprocity sheaves2013

    • Author(s)
      Takao Yamazaki
    • Organizer
      Workshop on reciprocity sheaves
    • Place of Presentation
      八ヶ岳自然文化園
    • Year and Date
      2013-07-31
    • Related Report
      2013 Annual Research Report 2013 Final Research Report
  • [Presentation] p-adic soliton theory and arithmetic geometry2013

    • Author(s)
      Takao Yamazaki
    • Organizer
      談話会
    • Place of Presentation
      北海道大学
    • Year and Date
      2013-05-30
    • Related Report
      2013 Final Research Report
  • [Presentation] p-adic soliton theory and torsion points on Jacobian varieties2012

    • Author(s)
      Takao Yamazaki
    • Organizer
      代数的整数論とその周辺
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2012-12-11
    • Related Report
      2013 Annual Research Report 2013 Final Research Report
  • [Presentation] Algebraic cycles on a product of curves over a p-adic field2012

    • Author(s)
      Takao Yamazaki
    • Organizer
      Seminaire autour des cycles algebriques
    • Place of Presentation
      フランス・ジュシュー数学研究所
    • Year and Date
      2012-11-21
    • Related Report
      2013 Final Research Report
  • [Presentation] Torsion points on Jacobian varieties via Anderson's p-adic soliton theory2012

    • Author(s)
      Takao Yamazaki
    • Organizer
      Seminaire de theorie des nombres
    • Place of Presentation
      フランス・ジュシュー数学研究所
    • Year and Date
      2012-11-19
    • Related Report
      2013 Final Research Report
  • [Presentation] Voevodsky's category of motives and Milnor K-groups attached to semi-abelian varieties2012

    • Author(s)
      Takao Yamazaki
    • Organizer
      Number Theory Day in Keio
    • Place of Presentation
      慶應大学
    • Year and Date
      2012-09-13
    • Related Report
      2013 Final Research Report
  • [Presentation] Torsion points on Jacobian varieties via Anderson's p-adic soliton theory2012

    • Author(s)
      Takao Yamazaki
    • Organizer
      Tohoku-Fudan Workshop
    • Place of Presentation
      東北大学
    • Year and Date
      2012-05-17
    • Related Report
      2013 Final Research Report 2012 Annual Research Report
  • [Presentation] On Theta divisor, Jacobian, KdV and KP equations2012

    • Author(s)
      Takao Yamazaki
    • Organizer
      TIMS Seminar on Arithmetic Geometry
    • Place of Presentation
      国立台湾大学
    • Year and Date
      2012-03-14
    • Related Report
      2013 Final Research Report
  • [Presentation] Torsion points on the quotient of a Fermat Jacobian via Anderson's p-adic soliton theory2012

    • Author(s)
      Takao Yamazaki
    • Organizer
      I. Mini-workshop on number theory
    • Place of Presentation
      台湾国立理論科学研究所
    • Year and Date
      2012-03-07
    • Related Report
      2013 Final Research Report
  • [Presentation] p-adic Sato theory and arithmetic geometry2012

    • Author(s)
      Takao Yamazaki
    • Place of Presentation
      香川セミナー, 香川大学
    • Year and Date
      2012-02-18
    • Related Report
      2013 Final Research Report
  • [Presentation] Somekawa's K-groups and Voevodsky's Hom groups2012

    • Author(s)
      Takao Yamazaki
    • Organizer
      Arithmetic geometry seminar
    • Place of Presentation
      北海道大学
    • Related Report
      2013 Final Research Report
  • [Presentation] Somekawa's K-groups and Voevodsky's Hom groups2012

    • Author(s)
      Takao Yamazaki
    • Organizer
      Arithmetic geometry seminar, Hokkaido University
    • Place of Presentation
      北海道大学
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] p-adic Sato theory and arithmetic geometry2011

    • Author(s)
      Takao Yamazaki
    • Organizer
      Algebra Seminar
    • Place of Presentation
      広島大学
    • Year and Date
      2011-12-16
    • Related Report
      2013 Final Research Report
  • [Presentation] Voevodsky's motif and Weil reciprocity2011

    • Author(s)
      Takao Yamazaki
    • Organizer
      International Workshop on motives in Tokyo 2011
    • Place of Presentation
      東京大学数理科学研究科
    • Year and Date
      2011-12-12
    • Related Report
      2013 Final Research Report
  • [Presentation] Voevodsky's motif and Weil reciprocity2011

    • Author(s)
      Takao Yamazaki
    • Organizer
      International Workshop on motives in Tokyo
    • Place of Presentation
      東京
    • Year and Date
      2011-12-12
    • Related Report
      2011 Annual Research Report
  • [Presentation] Voevodsky's motif and Weil reciprocity2011

    • Author(s)
      Takao Yamazaki
    • Organizer
      代数的整数論とその周辺
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2011-11-29
    • Related Report
      2013 Final Research Report
  • [Presentation] Voevodsky's motif and Weil reciprocity2011

    • Author(s)
      Takao Yamazaki
    • Organizer
      Algebraic Number Theory and Related Topics, Kyoto
    • Place of Presentation
      京都
    • Year and Date
      2011-11-29
    • Related Report
      2011 Annual Research Report
  • [Presentation] Torsion on theta divisors of hyperelliptic Jacobians and p-adic tau functions2011

    • Author(s)
      Takao Yamazaki
    • Organizer
      Algebra seminar
    • Place of Presentation
      慶應大学
    • Year and Date
      2011-11-14
    • Related Report
      2013 Final Research Report
  • [Presentation] Torsion on theta divisors of hyperelliptic Jacobians and p-adic tau functions. Geometrie arithmetique et motivique2011

    • Author(s)
      Takao Yamazaki
    • Organizer
      Centre International de Rencontres Mathematique
    • Place of Presentation
      フランス
    • Year and Date
      2011-09-22
    • Related Report
      2013 Final Research Report
  • [Presentation] Class field theory for open varieties over local fields2011

    • Author(s)
      Takao Yamazaki
    • Organizer
      談話会
    • Place of Presentation
      千葉大学
    • Year and Date
      2011-06-14
    • Related Report
      2013 Final Research Report
  • [Presentation] Class field theory for open varieties over local fields. NCTS number theory seminar2011

    • Author(s)
      Takao Yamazaki
    • Organizer
      NCTS number theory seminar
    • Place of Presentation
      台湾国立理論科学研究所
    • Year and Date
      2011-04-19
    • Related Report
      2013 Final Research Report
  • [Presentation] Class field theory for open varieties over local fields2011

    • Author(s)
      Takao Yamazaki
    • Organizer
      NCTS Number theory seminar
    • Place of Presentation
      新竹、台湾
    • Year and Date
      2011-04-19
    • Related Report
      2011 Annual Research Report
  • [Presentation] Class field theory for open varieties over local fields2010

    • Author(s)
      Takao Yamazaki
    • Organizer
      International workshop on motives in Tokyo, part 6
    • Place of Presentation
      東京大学数理科学研究科
    • Year and Date
      2010-12-16
    • Related Report
      2013 Final Research Report
  • [Presentation] Class field theory for open varieties over local fields2010

    • Author(s)
      山崎降雄
    • Organizer
      International workshop on motives in Tokyo, part 6
    • Place of Presentation
      東京大学
    • Year and Date
      2010-12-16
    • Related Report
      2010 Annual Research Report
  • [Presentation] Krichever correpondence / p-adic tau function and its application to arithmetic geometry2010

    • Author(s)
      Takao Yamazaki
    • Organizer
      Workshop on p-adic Sato theory and arithmetic geometry
    • Place of Presentation
      東北大学
    • Year and Date
      2010-09-30
    • Related Report
      2013 Final Research Report
  • [Presentation] Brauer groups and 0-cycles on open varieties over a local field2010

    • Author(s)
      Takao Yamazaki
    • Organizer
      日本数学会代数学分科会特別講演
    • Place of Presentation
      名古屋大学多元数理研究科
    • Year and Date
      2010-09-24
    • Related Report
      2013 Final Research Report
  • [Presentation] Brauer groups and O-cycles on open varieties over a local field2010

    • Author(s)
      山崎隆雄
    • Organizer
      日本数学会代数学分科会特別講演
    • Place of Presentation
      名古屋大学
    • Year and Date
      2010-09-24
    • Related Report
      2010 Annual Research Report
  • [Presentation] Brauer-Manin pairing for open varieties over local fields2010

    • Author(s)
      Takao Yamazaki
    • Organizer
      Academia Sinica algebra seminar
    • Place of Presentation
      台湾
    • Year and Date
      2010-07-06
    • Related Report
      2013 Final Research Report
  • [Presentation] Brauer groups and 0-cycles on varieties over a local field (survey)2010

    • Author(s)
      Takao Yamazaki
    • Organizer
      代数セミナー
    • Place of Presentation
      東北大学
    • Year and Date
      2010-05-20
    • Related Report
      2013 Final Research Report
  • [Presentation] Reciprocity sheaves: towards non-homotopy invariant motive theory

    • Author(s)
      Takao Yamazaki
    • Organizer
      Arithmetic, differential and geometry
    • Place of Presentation
      東北大学
    • Related Report
      2013 Annual Research Report
  • [Presentation] Reciprocity sheaves: towards non-homotopy invariant motive theory

    • Author(s)
      Takao Yamazaki
    • Organizer
      Motives in Tokyo, 2013
    • Place of Presentation
      東京大学
    • Related Report
      2013 Annual Research Report
  • [Presentation] p-adic soliton theory and torsion points on Jacobian varieties

    • Author(s)
      Takao Yamazaki
    • Organizer
      East Asia Number Theory Conference 2014
    • Place of Presentation
      九州大学
    • Related Report
      2013 Annual Research Report
  • [Presentation] p-adic soliton theory and arithmetic geometry

    • Author(s)
      Takao Yamazaki
    • Organizer
      Arithmetic and Algebraic Geoemtry 2014
    • Place of Presentation
      東京大学
    • Related Report
      2013 Annual Research Report
  • [Presentation] Voevodsky's motives and Weil reciprocity

    • Author(s)
      Takao Yamazaki
    • Organizer
      2014 NCTS Mini-Workshop on Number Theory
    • Place of Presentation
      新竹市,台湾
    • Related Report
      2013 Annual Research Report
  • [Presentation] Voevodsky's category of motives and Milnor K-groups attached to semi-abelian varieties

    • Author(s)
      Takao Yamazaki
    • Organizer
      Number Theory Day in Keio
    • Place of Presentation
      慶應義塾大学
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Torsion points on Jacobian varieties via Anderson's p-adic soliton theory

    • Author(s)
      Takao Yamazaki
    • Organizer
      Seminaire de theorie des nombres
    • Place of Presentation
      フランス,パリ
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Algebraic cycles on a product of curves over a p-adic field

    • Author(s)
      Takao Yamazaki
    • Organizer
      Seminaire autour des cycles algebriques
    • Place of Presentation
      フランス,パリ
    • Related Report
      2012 Annual Research Report
    • Invited

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Published: 2010-08-23   Modified: 2019-07-29  

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