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The geometry of Shimura varieties over positive characteristic and the development of Galois representations

Research Project

Project/Area Number 15K04787
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionTohoku University (2016-2018)
Kagoshima University (2015)

Principal Investigator

Yamauchi Takuya  東北大学, 理学研究科, 准教授 (90432707)

Research Collaborator Chida Masataka  
Miyauchi Michitaka  
Project Period (FY) 2015-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000)
Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywordsガロア表現 / 法pガロア表現 / セール予想 / 保型性問題 / テータ作用素 / ジーゲル保型形式 / 楕円保型形式 / 法p保型形式 / 等分布性 / ジーゲルモジュラー形式 / ジーゲル形式 / テータサイクル / 二次形式 / グロスキーティング不変量 / アイゼンシュタイ級数 / 法pジーゲル形式 / アーサーセルバーグ跡公式 / 等分布定理 / ヘッケ体
Outline of Final Research Achievements

The study conducted during the project was analysis of Serre weight. The Serre weight is a concept necessary to formulate modularity problem of the residual Galois representation, and specifies the possibility of the weights of the corresponding automorphic forms. In this research, we introduce the theta operators as one of the tools to analyze the Serre weight, and when the algebraic group is GSp4, we succeeded in giving a concrete expression of the theta operators. We use the modular lifting theorems to prove the Serre weight theorem in a fairly general case for GSp4 over totally real fields, and further a list of the possible Serre weight of corresponding automorphic forms was described completely in terms of the local properties of a given mod p Galois representation.

Academic Significance and Societal Importance of the Research Achievements

保型性問題とは幾何的ガロア表現と保型表現という代数的対象と解析的対象の間にある種の関係を問う問題である。GL2の場合にはWiles が楕円曲線に付随するガロア表現に対して保型性問題を解決したが、その応用として350年も未解決であったフェルマー予想が解決された。またそれと同時に周辺分野の大きな発展をもたらした。今回の研究も目標は保型性問題の解決のための基礎を発展させることに主眼が置かれ、特に正標数体上での志村多様体上の幾何を用いて法p保型形式の性質(テータ作用や重さの還元)に関する当該研究者の成果はセール重さを調べることには必要不可欠な道具であると期待する。

Report

(5 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Research-status Report
  • 2016 Research-status Report
  • 2015 Research-status Report
  • Research Products

    (32 results)

All 2019 2018 2017 2016 2015 Other

All Int'l Joint Research (4 results) Journal Article (10 results) (of which Int'l Joint Research: 5 results,  Peer Reviewed: 10 results,  Acknowledgement Compliant: 6 results,  Open Access: 2 results) Presentation (17 results) (of which Int'l Joint Research: 2 results,  Invited: 17 results) Remarks (1 results)

  • [Int'l Joint Research] Army Cyber Institute United States(米国)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] トロント大学(カナダ)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] トロント大学(カナダ)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] KIAS(韓国)

    • Related Report
      2016 Research-status Report
  • [Journal Article] On the class numbers of the fields of the pn-torsion points of elliptic curves over Q2018

    • Author(s)
      F. Sairaiji and T. Yamauchi
    • Journal Title

      Journal de Theorie des Nombres de Bordeaux

      Volume: 30 Pages: 893-915

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A Miyawaki type lift for GSpin(2,10)2018

    • Author(s)
      Henry H. Kim and T. Yamauchi
    • Journal Title

      Mathematische Zeitschrift

      Volume: 288 Issue: 1-2 Pages: 415-437

    • DOI

      10.1007/s00209-017-1895-y

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] An equidistribution theorem for holomorphic Siegel modular forms for GSp_42018

    • Author(s)
      Henry H. Kim, S. Wakatsuki and T. Yamauchi
    • Journal Title

      Journal of the Institute of Mathematics of Jussieu

      Volume: 未定 Issue: 2 Pages: 351-419

    • DOI

      10.1017/s147474801800004x

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] On Prym varieties for the coverings of some singular plane curves2018

    • Author(s)
      Lubjana Beshaj and Takuya Yamauchi
    • Journal Title

      Manuscripta mathematica

      Volume: 未定 Issue: 1-2 Pages: 1-18

    • DOI

      10.1007/s00229-018-1018-z

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] A uniform structure on subgroups of GLn(Fq) and its application to a conditional construction of Artin representations of GLn2017

    • Author(s)
      Henry H. Kim and T. Yamauchi
    • Journal Title

      J. Ramanujan Math. Soc

      Volume: 32 Pages: 75-99

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] On intermediate Jacobians of cubic threefolds admitting an automorphism of order five2016

    • Author(s)
      B. van Geeman and T. Yamauchi
    • Journal Title

      Pure Appl. Math. Q.

      Volume: 12 Pages: 141-164

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Cusp forms on the exceptional group of type E72016

    • Author(s)
      Henry H Kim and Takuya Yamauchi
    • Journal Title

      Compositio Mathematica

      Volume: 152 Issue: 2 Pages: 223-254

    • DOI

      10.1112/s0010437x15007538

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] A conditional construction of Artin representations for real analytic Siegel cusp forms of weight (2; 1)2016

    • Author(s)
      Henry H Kim and Takuya Yamauchi
    • Journal Title

      Contemporary Mathematics

      Volume: 掲載予定

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] A uniform structure on subgroups of $GL_n(\F_q)$ and its application to a conditional construction of Artin representations of GL_n2016

    • Author(s)
      Henry H Kim and Takuya Yamauchi
    • Journal Title

      Journal of Ramanujan mathematical Society.

      Volume: 掲載予定

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] On the class numbers of the elds of the pn-torsion points of certain elliptic curves over Q2015

    • Author(s)
      Sairaiji, Fumio and Yamauchi, Takuya
    • Journal Title

      Journal of Number theory 156 (2015), 277-289

      Volume: 156 Pages: 277-289

    • DOI

      10.1016/j.jnt.2015.04.004

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Presentation] 宮脇型リフトの非消滅性について2019

    • Author(s)
      山内卓也
    • Organizer
      京都大学第談話会
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] On Ikeda type and Miyawaki type construction of holomorphic cusp forms on Hermitian symmetric domains2019

    • Author(s)
      山内卓也
    • Organizer
      NSU (National Singapore university) における招待講演
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] On theta operator2018

    • Author(s)
      山内卓也
    • Organizer
      マンハイム大(ドイツ) における整数論セミナーでの招待講演
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] The weight part of Serre conjecture for GSp4/Q in terms of the classical weights2018

    • Author(s)
      山内卓也
    • Organizer
      研究集会Galois representations, ramification theory, and related topics
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 次数2の正則ジーゲル形式に関する等分布問題2017

    • Author(s)
      山内卓也
    • Organizer
      第43 回愛媛大学代数学セミナー
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] 高次代数群に対する保型形式とその周辺2017

    • Author(s)
      山内卓也
    • Organizer
      第62 回代数学シンポジウム
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] An equidistribution theorem for holomorphic Siegel modular formsfor GSp_42017

    • Author(s)
      山内卓也
    • Organizer
      京都大学RIMS研究集会「保型形式とその周辺」における発表
    • Place of Presentation
      京都大学RIMS
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] The weight reduction theorem of Serre conjecture for GSp42016

    • Author(s)
      山内 卓也
    • Organizer
      日韓整数論セミナー
    • Place of Presentation
      POSTECH(浦項工科大学)(慶尚北道浦項市)
    • Year and Date
      2016-02-04
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Arthur-Selberg trace formula とその応用2016

    • Author(s)
      山内卓也
    • Organizer
      東北大学理学研究科数学専攻の談話会における発表
    • Place of Presentation
      東北大学
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] GSp4 に関するセール予想と重さ還元定理2016

    • Author(s)
      山内卓也
    • Organizer
      日本数学会
    • Place of Presentation
      関西大学
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] A survey for Ikeda type constructions of holomorphic automorphic cusp forms of level one2016

    • Author(s)
      山内卓也
    • Organizer
      台北(台湾)・国立台湾大学 数論セミナーにおける発表
    • Place of Presentation
      台北(台湾)・国立台湾大学
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Ikeda type lift for E_{7,3} /Q2016

    • Author(s)
      山内卓也
    • Organizer
      台北(台湾)・国立台湾大学 数論セミナーにおける発表
    • Place of Presentation
      台北(台湾)・国立台湾大学
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] 保型形式に付随するモチーフ2015

    • Author(s)
      山内 卓也
    • Organizer
      金沢大学数理学談話会
    • Place of Presentation
      金沢大学(石川県金沢市)
    • Year and Date
      2015-08-05
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] rank 3 の例外群E7 に関するIkeda type lift とMiyawaki type lift について2015

    • Author(s)
      山内 卓也
    • Organizer
      大阪大学整数論および保型形式セミナー
    • Place of Presentation
      大阪大学(大阪府豊中市)
    • Year and Date
      2015-07-24
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] Artin 表現に付随する保型表現について2015

    • Author(s)
      山内 卓也
    • Organizer
      松江セミナー
    • Place of Presentation
      島根大学(島根県松江市)
    • Year and Date
      2015-06-22
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] 位数5 の自己同型をもつsmooth cubic 3-fold の中間ヤコビ多様体について2015

    • Author(s)
      山内 卓也
    • Organizer
      早稲田整数論セミナー
    • Place of Presentation
      早稲田大学(東京都新宿区大久保)
    • Year and Date
      2015-06-19
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] 法p テータ作用素について2015

    • Author(s)
      山内 卓也
    • Organizer
      近畿大学数学講演会
    • Place of Presentation
      近畿大学(大阪府東大阪市)
    • Year and Date
      2015-05-14
    • Related Report
      2015 Research-status Report
    • Invited
  • [Remarks]

    • URL

      http://www.sci.tohoku.ac.jp/news/20160920-8035.html

    • Related Report
      2018 Annual Research Report

URL: 

Published: 2015-04-16   Modified: 2022-02-22  

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