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Study of the local Langlands functoriality via rigid geometry and harmonic analysis on p-adic algebraic groups

Research Project

Project/Area Number 21740022
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeSingle-year Grants
Research Field Algebra
Research InstitutionKyushu University

Principal Investigator

MIEDA Yoichl  九州大学, 大学院・数理学研究院, 助教 (70526962)

Project Period (FY) 2009 – 2011
Project Status Completed (Fiscal Year 2011)
Budget Amount *help
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2011: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2010: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2009: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Keywords数論幾何学 / Rapoport-Zink空間 / エタールコホモロジー / リジッド空間 / 局所ラングランズ対応 / p進群の表現論 / Lefschetz跡公式 / 隣接輪体
Research Abstract

I worked on general theory of etale cohomology of rigid spaces and more generally, adic spaces, and obtained results on the Lefschetz trace formula, the comparison theorem in the case of general basis, and generalization of formal nearby cycles. Applying these results to the Rapoport-Zink space for the symplectic group GSp(4), I have proved the following :・The local Jacquet-Langlands correspondence between GSp(4) and its inner form, which is a kind of the local Langlands functoriality, appears in the alternating sum of the cohomology of the Rapoport-Zink space.・A supercuspidal representation appears in i-th cohomology of the Rapoport-Zink space only if i=2, 3, 4. In the proof of the former result, harmonic analysis on p-adic algebraic groups was used effectively. By adopting methods used in the research above to the classical case such as the Lubin-Tate space and the Drinfeld space, we got some results for GL(n). For example, I found a purely local proof of the local JacquetLanglands correspondence for GL(n) for prime n. These results are expected to be an important step in the geometric study of the local Langlands functoriality.

Report

(4 results)
  • 2011 Annual Research Report   Final Research Report ( PDF )
  • 2010 Annual Research Report
  • 2009 Annual Research Report
  • Research Products

    (21 results)

All 2012 2011 2010 2009 Other

All Journal Article (8 results) (of which Peer Reviewed: 8 results) Presentation (9 results) Remarks (4 results)

  • [Journal Article] Lefschetz trace formula andl-adicc cohomology of Lubin-Tate tower2012

    • Author(s)
      Yoichi Mieda
    • Journal Title

      Mathematical Research Letters

      Volume: 19 Pages: 95-107

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Noncuspidality outside the middle degree ofl-adicc cohomology of the Lubin-Tate tower2010

    • Author(s)
      Yoichi Mieda
    • Journal Title

      Advances in Mathematics

      Volume: 225 Pages: 2287-2297

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Non-cuspidality outside the middle degree of l-adic cohomology of the Lubin-Tate tower2010

    • Author(s)
      Yoichi Mieda
    • Journal Title

      Advances in Mathematics

      Volume: 225 Pages: 2287-2297

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A remark on the geometric Jacquet functor2010

    • Author(s)
      Noriyuki Abe, Yoichi Mieda
    • Journal Title

      Journal of Mathematical Sciences, the University of Tokyo

      Volume: 17 Pages: 243-246

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Cycle classes, Lefschetz trace formula and integrality for p-adic cohomology2009

    • Author(s)
      Yoichi Mieda
    • Journal Title

      RIMS Kokyuroku Besatsu B12

      Pages: 57-66

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Comparison results for etale cohomology in rigid geometry

    • Author(s)
      Yoichi Mieda
    • Journal Title

      Journal of Algebraic Geome tryに採録決定済

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Lefschetz trace formula and 1-adic cohomology of Lubin-Tate tower

    • Author(s)
      Yoichi Mieda
    • Journal Title

      Mathematical Eesearch Letters

      Volume: (掲載決定)

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Journal of Algebraic Geometry

    • Author(s)
      Yoichi Mieda
    • Journal Title

      Comparison results for etale cohomology in rigid geometry

      Volume: (掲載決定)

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Presentation] Supercuspidal representations in l-adic cohomology of the Rapoport-Zink tower for GSp(4)2012

    • Author(s)
      Yoichi Mieda
    • Organizer
      East Asia Number Theory Conference
    • Place of Presentation
      National Taiwan University(台湾)
    • Year and Date
      2012-01-18
    • Related Report
      2011 Final Research Report
  • [Presentation] Supercuspidal representations in 1-adic cohomology of the Rapoport-Zinktower for GSp(4)2012

    • Author(s)
      三枝洋一
    • Organizer
      East Asia Number Theory Conference
    • Place of Presentation
      National Taiwan University(台湾)
    • Year and Date
      2012-01-18
    • Related Report
      2011 Annual Research Report
  • [Presentation] Lefschetz trace formula and l-adic cohomology of the Rapoport-Zink spaces for GSp(4)2011

    • Author(s)
      Yoichi Mieda
    • Organizer
      The Third PanAsian Number Theory Conference
    • Place of Presentation
      Morningside Center of Mathematics, Chinese Academy of Sciences
    • Year and Date
      2011-08-22
    • Related Report
      2011 Final Research Report
  • [Presentation] Lefschetz trace formula and 1-adic cohomology of the Rapoport-Zink spaces for GSp(4)2011

    • Author(s)
      三枝洋一
    • Organizer
      The Third PanAsian Number Theory Conference (PANT-Beijing 2011)
    • Place of Presentation
      Morningside Center of Mathematics, Chinese Academy of Sciences(中国)
    • Year and Date
      2011-08-22
    • Related Report
      2011 Annual Research Report
  • [Presentation] Cuspidal representations in the l-adic cohomology of the Rapoport-Zink space for GSp(4)2010

    • Author(s)
      Yoichi Mieda
    • Organizer
      Workshop on Arithmetic Geometry and Related Topics
    • Place of Presentation
      National Taiwan University(台湾)
    • Year and Date
      2010-07-14
    • Related Report
      2011 Final Research Report
  • [Presentation] Cuspidal representations in the l-adic cohomology of the Rapoport-Zink space for GSp(4)2010

    • Author(s)
      三枝洋一
    • Organizer
      Workshop on Arithmetic Geometry and Related Topics
    • Place of Presentation
      台湾国立大学(中国)
    • Year and Date
      2010-07-14
    • Related Report
      2010 Annual Research Report
  • [Presentation] Adic空間における隣接輪体関手2009

    • Author(s)
      三枝洋一
    • Organizer
      代数的整数論とその周辺
    • Place of Presentation
      東京大学数理科学研究科
    • Year and Date
      2009-12-07
    • Related Report
      2011 Final Research Report
  • [Presentation] Adic空間のエタールコホモロジーについて2009

    • Author(s)
      三枝洋一
    • Organizer
      第54回代数学シンポジウム
    • Place of Presentation
      明治大学リバティタワー
    • Year and Date
      2009-08-05
    • Related Report
      2011 Final Research Report
  • [Presentation] Adic空間のエタールコホモロジーについて2009

    • Author(s)
      三枝洋一
    • Organizer
      第54回代数学シンポジウム
    • Place of Presentation
      明治大学リバティータワー
    • Year and Date
      2009-08-05
    • Related Report
      2009 Annual Research Report
  • [Remarks]

    • URL

      http://www2.math.kyushuu.ac.jp/hmieda/index.html

    • Related Report
      2011 Final Research Report
  • [Remarks]

    • URL

      http://www2.math.kyushu-u.ac.jp/-mieda/index-j.html

    • Related Report
      2011 Annual Research Report
  • [Remarks]

    • URL

      http://www2.math.kyushu-u.ac.jp/~mieda/index-j.html

    • Related Report
      2010 Annual Research Report
  • [Remarks]

    • URL

      http://www2.math.kyushu-u.ac.jp/~mieda/index-j.html

    • Related Report
      2009 Annual Research Report

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Published: 2009-04-01   Modified: 2016-04-21  

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