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2001 Fiscal Year Final Research Report Summary

Galois representations arising from etale cohomology

Research Project

Project/Area Number 11640013
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionThe University of Tokyo

Principal Investigator

SAITO Takeshi  School of Mathematical Sciences, The University of Tokyo, Professor, 大学院・数理科学研究科, 教授 (70201506)

Co-Investigator(Kenkyū-buntansha) KURIHARA Masato  Tokyo Metropolitan University, Faculty of Sciences, Assistant Professor, 理学部, 助教授 (40211221)
TERASOMA Tomohide  School of Mathematical Sciences, The University of Tokyo, Assistant Professor, 大学院・数理科学研究科, 助教授 (50192654)
KATO Kazuya  Kyoto University, School of Sciences, Professor, 大学院・理学系研究科, 教授 (90111450)
FUJIWARA Kazuhiro  Nagoya University, School of Polymathematics, Professor, 多元数理研究科, 教授 (00229064)
SAITO Shuji  Nagoya University, School of Polymathematics, Professor, 多元数理研究科, 教授 (50153804)
Project Period (FY) 1999 – 2001
Keywordsetale cohomology / Galois representation / weight spectral sequence / semi-stable reduction / log scheme / Hasse-Weil L function / resolution of singularities / conductor formula
Research Abstract

1999 and 2000. The main result in this period is a proof of the conductor formula. This is a joint work with one of the investigators, K. Kato. Bad reduction of a variety over a local field causes ramification of the inertia action. The conductor formula tells us how to measure it in terms of differential forms on a regular model. We proved it under the assumption that the reduced closed fiber is a divisor with normal crossings. I also studied ramification of local fields with imperfect residue field and defined a filtration by ramification groups on the absolute Galois group.
2001. 1. For a variety over a local field with semi-stable reduction. Rapoport and Zink defined the weight spectral sequence. I gave an elementary construction of the spectral sequence and proved its functoriality. From this, I deduced the independence of 1 of the alternating sum of the traces of the composition of the actions of an element of the Weil group and an algebraic correspondence on 1-adic cohomology. As an application, I proved that the classical conjecture that the bad Euler factor of the Hasse-Weil L function of an algebraic surface is independent of 1 is reduced to a part of Tate conjecture that the Kunneth projectors are defined by algebraic cycles and the weight-monodromy conjecture.
2. I proved that a smooth family of curves defined on the interior of log regular scheme is extended to the boundary if it is extended at each generic point of the boundary.

  • Research Products

    (14 results)

All Other

All Publications (14 results)

  • [Publications] Saito, Takeshi: "Log smooth extension of family of curves and semi-stable reduction"Journal of Algebraic Geometry. (掲載予定).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Saito, Takeshi, Abbes, Ahmed: "Ramification of local fields with imperfect residue fields"American Journal of Mathematics. (掲載予定).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Saito, Takeshi: "Parity in Bloch's conductor formula"B.Erez (ed.), The Proceedings of the conference, Ramification of Arithmetic schemes. (掲載予定).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Nakamura, Iku, Terasoma, Tomohide: "Moduli space of elliptic curves with Heisenberg level structure"Progr. Math., Moduli of abelian varieties. 195. 299-324 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kato, Kazuya: "Bloch's conductor formula"Proceedings of the Jangjeon Mathematical Society. 1. 91-95 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 斎藤毅: "数論幾何におけるガロワ表現"数学. 53-2. 337-348 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 斎藤毅: "Fermat予想 1"岩波書店. 180 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Saito, Takeshi: "Log smooth extension of family of curves and semi-stable reduction"Journal of Algebraic Geometry. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Abbes, Ahmed, Saito, Takeshi: "Ramification of local fields with imperfect residue fields"American Journal of Mathematics. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Saito, Takeshi: "Parity in Bloch's conductor formula"B. Erez (ed.), The Proceedings of the conference, Ramification of Arithmetic schemes. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Nakamura, Iku, Terasoma, Tomohide: "Moduli space of elliptic curves with Heisenberg level structure"Progr. Math., Moduli of abelian varieties. 195. 299-324 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kato, Kazuya: "Bloch's conductor formula"Proceedings of the Jangjeon Mathematical Society. 1. 91-95 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Saito, Takeshi: "Galois representations in arithmetic geometry II"Sugaku. 53-2. 337-348 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Saito, Takeshi: "Iwanami-Shoten"Fermat's Last Theorem 1. 180 (2000)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2003-09-17  

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