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Study of cohomology in arithmetic geometry

Research Project

Project/Area Number 20K14284
Research Category

Grant-in-Aid for Early-Career Scientists

Allocation TypeMulti-year Fund
Review Section Basic Section 11010:Algebra-related
Research InstitutionKyoto University

Principal Investigator

Koshikawa Teruhisa  京都大学, 数理解析研究所, 助教 (10791452)

Project Period (FY) 2020-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2023: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2022: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2021: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywords数論幾何学 / コホモロジー / 志村多様体 / Langlands対応 / p進Hodge理論 / 対数的幾何学 / プリズマティックコホモロジー / K3曲面 / 局所対称空間 / 保型表現 / 被覆群 / 対数的幾何 / 超ケーラー多様体 / 局所Langlands対応 / Hodge標準予想 / p進Hodge理論
Outline of Research at the Start

代数幾何学の手法を駆使することで, 整数論的問題の解決を目指すのが数論幾何学である. 本研究では特にコホモロジーと呼ばれるタイプの不変量について研究を行う. 近年Scholzeを中心に, p進Hodge理論と呼ばれるコホモロジーについての研究分野や志村多様体と呼ばれる特別な幾何的対象についての研究が大きく進展している. 本研究はこのような研究をさらに推し進めるものである. 最新の理論の適用できる範囲をさらに広げ, 志村多様体などのコホモロジーについてより詳しい分析を行う.

Outline of Final Research Achievements

Arithmetic geometry is a reaserch area where geometric perspective is used to study number theory. In this reaserch, I mainly studied invariants called cohomology. For example, we prove a result that certain parts of the cohomology of Shimura varieties, which are important geometric objects in number theory, vanish. It has application to number theory such as Langlands correspondence. I also contributed to classcailly known problems liek the Tate conjecture, the standard conjecture in the case of self-products of so-called K3 surfaces, an interesting class of geometric objects. Moreoever, I worked on the p-adic Hodge theory, which is a thoery specialized for a fixed prime number p. I introduced a logarithmic version of prismatic cohomology, which was found rather recently, and developed the foundation.

Academic Significance and Societal Importance of the Research Achievements

志村多様体では予期されていなかった成果を挙げるとともに、FarguesとScholzeとの局所Langlands対応の幾何化プログラムとの関係性を指摘することとなり、国際的にも大変な反響を得た。p進Hodge理論では対数的プリズマティックコホモロジーの基礎理論を構築し、国内外の研究者からもすでに用いられる理論となった。また、K3曲面に関係する特別な場合でのみであるが、古くから重要視されている代数幾何の予想について貢献することができた。これらの成果は学術的意義も十分にあると考えられるだけでなく、現在あるいは今後の国内外での研究を促進するような成果であったといえる。

Report

(5 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • Research Products

    (23 results)

All 2024 2023 2022 2021 2020 Other

All Int'l Joint Research (5 results) Journal Article (3 results) (of which Peer Reviewed: 3 results,  Open Access: 3 results) Presentation (15 results) (of which Int'l Joint Research: 11 results,  Invited: 14 results)

  • [Int'l Joint Research] University of California, Berkeley(米国)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] Universitat Munster/University of Bonn(ドイツ)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] University of Chicago/University of California, Berkeley(米国)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] Universitat Munster(ドイツ)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] University of Chicago(米国)

    • Related Report
      2021 Research-status Report
  • [Journal Article] The numerical Hodge standard conjecture for the square of a simple abelian variety of prime dimension2023

    • Author(s)
      Koshikawa Teruhisa
    • Journal Title

      manuscripta mathematica

      Volume: 173 Issue: 3-4 Pages: 1161-1169

    • DOI

      10.1007/s00229-023-01482-7

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] CM liftings of surfaces over finite fields and their applications to the Tate conjecture2021

    • Author(s)
      Ito Kazuhiro、Ito Tetsushi、Koshikawa Teruhisa
    • Journal Title

      Forum of Mathematics, Sigma

      Volume: 9 Pages: 1-70

    • DOI

      10.1017/fms.2021.24

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Vanishing theorems for the mod p cohomology of some simple Shimura varieties2020

    • Author(s)
      Koshikawa Teruhisa
    • Journal Title

      Forum of Mathematics, Sigma

      Volume: 8

    • DOI

      10.1017/fms.2020.36

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Open Access
  • [Presentation] 志村多様体のコホモロジーの消滅定理2024

    • Author(s)
      越川皓永
    • Organizer
      日本数学会2024年度年会
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Around logarithmic primsatic cohomology2023

    • Author(s)
      越川皓永
    • Organizer
      IAS/Princeton Arithmetic Geometry Seminar
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Vanishing range beyond the generic case2023

    • Author(s)
      越川皓永
    • Organizer
      Trimester Program, The Arithmetic of the Langlands Program, Trimester Seminar Series
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Some computations in categorical local Langlands2023

    • Author(s)
      越川皓永
    • Organizer
      Arithmetic and Cohomology of Algebraic Varieties 2023
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Genericity and cohomology of locally symmetric spaces2023

    • Author(s)
      越川皓永
    • Organizer
      Oberwolfach workshop, Arithmetic of Shimura varieties
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Vanishing theorems for local ang global Shimura varieties2022

    • Author(s)
      越川皓永
    • Organizer
      30eradecaen : 30e Rencontres arithmetiques de Caen
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Some cases of the Hodge standard conjecture2022

    • Author(s)
      越川皓永
    • Organizer
      Franco-Asian Summer School on Arithmetic Geometry
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On the cohomology of unitary Shimura varieties2022

    • Author(s)
      越川皓永
    • Organizer
      Community-building in the Langlands Program
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] K3曲面の自己積のHodge標準予想2022

    • Author(s)
      越川皓永
    • Organizer
      北海道大学数論セミナー
    • Related Report
      2022 Research-status Report
  • [Presentation] Logarithmic prismatic cohomology2022

    • Author(s)
      越川皓永
    • Organizer
      p-adic cohomology and arithmetic geometry 2022
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On the generic part of the cohomology of locally symmetric spaces2022

    • Author(s)
      越川皓永
    • Organizer
      Mini-workshop on the geometrization of the local Langlands correspondences and related topics
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Heights of pure motives2022

    • Author(s)
      越川皓永
    • Organizer
      The Andre-Oort conjecture
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] ユニタリ型志村多様体のコホモロジーの消滅定理2021

    • Author(s)
      越川皓永
    • Organizer
      金沢整数論オータムワークショップ2021 兼 北陸数論セミナー
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Vanishing theorems for unitary Shimura varieties2021

    • Author(s)
      越川皓永
    • Organizer
      KAIST Number Theory seminar
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Logarithmic prismatic site2021

    • Author(s)
      越川皓永
    • Organizer
      Arithmetic geometry - Takeshi 60
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited

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Published: 2020-04-28   Modified: 2025-01-30  

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