Motive theory and algebraic cycles based on Weil reciprocity

• Project/Area Number: 15K04773
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Tohoku University
• Principal Investigator: Yamazaki Takao 東北大学, 理学研究科, 教授 (00312794)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥5,850,000 (Direct Cost: ¥4,500,000、Indirect Cost: ¥1,350,000); Fiscal Year 2017: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2016: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000); Fiscal Year 2015: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
• Keywords: 代数学 / 数論幾何 / 代数幾何学 / 整数論 / モチーフ / 代数幾何

Outline of Final Research Achievements

A triangulated category of motives with modulus is constructed in such a way that it enables us to handle non-homotopy invariant phenomena. This is a generalization of Voevodsky's triangulated category of motives. Our category has intimate relationship with reciprocity sheaves that was introduced in our previous work. We also studied the mixed Hodge structures with modulus, which is a counter-part of motives with modulus in the Hodge theory. We proved that the Nori category arising from curves with modulus is equivalent to Laumon 1-motives. This is an extension of a result of Ayoub and Barbieri-Viale that considers the case without modulus.

Research Products

• [Int'l Joint Research] l'Universite de Rennes 1(フランス)
• [Int'l Joint Research] 國立中央大學(台湾)
• [Int'l Joint Research] Institut de Mathematiques de Jussieu(France)
• [Int'l Joint Research] Universitat de Rennes 1(France)
• [Int'l Joint Research] Freie Universiat Berlin(Germany)
• [Int'l Joint Research] National Chiao Tung University(Taiwan)
• [Journal Article] Nori motives of curves with modulus and Laumon $1$-motives (2018)
  • Author(s): Florian Ivorra, Takao Yamazaki
  • Journal Title: Canadian Journal of Mathematics Volume: 印刷中 Issue: 4 Pages: 868-897
  • DOI: 10.4153/cjm-2017-037-x
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Rational torsion on the generalized Jacobian of a modular curve with cuspidal modulus (2016)
  • Author(s): Takao Yamazaki, Yifan Yang
  • Journal Title: Documenta Mathematica Volume: 21 Pages: 1669-1690
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Reciprocity sheaves (2016)
  • Author(s): Bruno Kahn, Shuji Saito, Takao Yamazaki, Kay Ruelling
  • Journal Title: Compositio Mathematica Volume: 152 Issue: 9 Pages: 1851-1898
  • DOI: 10.1112/s0010437x16007466
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Torsion points on Jacobian varieties via Anderson's p-adic soliton theory (2016)
  • Author(s): Shinichi Kobayashi,Takao Yamazaki
  • Journal Title: Asian Journal of Mathematics Volume: 20 Issue: 2 Pages: 323-352
  • DOI: 10.4310/ajm.2016.v20.n2.a5
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Suslin homology of relative curves with modulus (2016)
  • Author(s): Kay Ruelling, Takao Yamazaki
  • Journal Title: Journal of the London Mathematical Society Volume: 93 Issue: 3 Pages: 567-589
  • DOI: 10.1112/jlms/jdw006
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Suslin homology of relative curves with modulus (2016)
  • Author(s): Kay Ruelling and Takao Yamazaki
  • Journal Title: Journal of the London Mathematical Society Volume: 未定
  • Int'l Joint Research
• [Journal Article] Algebraic integrable systems related to spectral curves with automorphisms (2015)
  • Author(s): Rei Inoue, Pol Vanhaecke and Takao Yamazaki
  • Journal Title: Journal of Geometry and Physics Volume: 87 Pages: 198-216
  • DOI: 10.1016/j.geomphys.2014.07.008
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] Mixed Hodge structures with modulus (2018)
  • Author(s): Takao Yamazaki
  • Organizer: Motives in Tokyo
  • Int'l Joint Research / Invited
• [Presentation] Nori motives of curves with modulus and Laumon 1-motives (2017)
  • Author(s): Takao Yamazaki
  • Organizer: Regulators in Niseko
  • Invited
• [Presentation] Generalized Jacobians of modular and Drinfeld modular curves (2017)
  • Author(s): Takao Yamazaki
  • Organizer: 北陸数論研究会
  • Invited
• [Presentation] Nori motives of curves with modulus and Laumon 1-motives (2016)
  • Author(s): Takao Yamazaki
  • Organizer: Differential forms in algebraic geometry
  • Place of Presentation: University of Freiburg, ドイツ
  • Year and Date: 2016-09-20
  • Int'l Joint Research / Invited
• [Presentation] Nori motives of curves with modulus and Laumon 1-motives (2016)
  • Author(s): Takao Yamazaki
  • Organizer: Japan-Taiwan Joint conference on Number theory 2016
  • Place of Presentation: Taipei、台湾
  • Year and Date: 2016-09-10
  • Int'l Joint Research / Invited
• [Presentation] Motives with modulus (2016)
  • Author(s): Takao Yamazaki
  • Organizer: Algebraic K-theory and Motivic Cohomology
  • Place of Presentation: Mathematisches Forschungsinstitut Oberwolfach, ドイツ
  • Year and Date: 2016-06-29
  • Int'l Joint Research / Invited
• [Presentation] Rational torsion on the generalized Jacobian of a modular curve with cuspidal modulus (2016)
  • Author(s): Takao Yamazaki
  • Organizer: Hakodate workshop on arithmetic geometry 2016
  • Place of Presentation: Hakodate
  • Year and Date: 2016-05-30
  • Int'l Joint Research / Invited
• [Presentation] Motives with modulus (2016)
  • Author(s): Takao Yamazaki
  • Organizer: Generalizations of A1-Homotopy Invariance in Algebraic Geometry and Homotopy Theory
  • Place of Presentation: Usedom, ドイツ
  • Year and Date: 2016-04-05
  • Int'l Joint Research / Invited
• [Presentation] Motives with modulus (2016)
  • Author(s): Takao Yamazaki
  • Organizer: Motives in Tokyo, 2016
  • Place of Presentation: Tokyo University
  • Year and Date: 2016-02-15
  • Int'l Joint Research / Invited
• [Book] 初等整数論―数論幾何への誘い― (2015)
  • Author(s): 山崎隆雄
  • Total Pages: 237
  • Publisher: 共立出版
• [Funded Workshop] Hakodate workshop on arithmetic geometry 2017 (2017)
• [Funded Workshop] Hakodate workshop on arithmetic geometry 2016 (2016)
  • Place of Presentation: Hakodate
  • Year and Date: 2016-05-30