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Study on arithmetic theory of automorphic forms of several variables

Research Project

Project/Area Number 25400031
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionKinki University

Principal Investigator

NAGAOKA Shoyu  近畿大学, 理工学部, 教授 (20164402)

Project Period (FY) 2013-04-01 – 2016-03-31
Project Status Completed (Fiscal Year 2015)
Budget Amount *help
¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2014: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2013: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Keywords保型形式 / 代数学 / 整数論 / モジュラー形式 / p進理論
Outline of Final Research Achievements

The aim of this study is to extend Serre’s theory of mod p or p-adic modular forms to the case of modular forms of several variables. In the previous study we determined the structure of the algebra of mod p modular forms in the case of Siegel modular forms which is a typical example of such modular forms. Moreover we determined the structure of such algebra in the case of Hermitian modular forms whose based field is the Gaussian field. Based on these results, we extended the notion of the theta operator and got some results on mod p kernel of such operator. This is defined as the image of the theta operator is congruent zero modulo some prime p. We could get a lot of such examples, for example, Igusa’s weight 35 cusp form and theta series for the Leech lattice are mod 23 kernel of the theta operator. We also clarified the relation between the weight and the congruence prime.

Report

(4 results)
  • 2015 Annual Research Report   Final Research Report ( PDF )
  • 2014 Research-status Report
  • 2013 Research-status Report
  • Research Products

    (16 results)

All 2015 2014 2013 Other

All Journal Article (5 results) (of which Peer Reviewed: 5 results) Presentation (8 results) (of which Invited: 6 results) Book (3 results)

  • [Journal Article] On the mod p kernel of the theta operator2015

    • Author(s)
      Shoyu Nagaoka
    • Journal Title

      Proceedings of the American Mathematical Society

      Volume: 143 Pages: 4237-4244

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Note on Igusa's cusp form of weight 352015

    • Author(s)
      Toshiyuki Kikuta, Hirotaka Kodama, Shoyu Nagaoka
    • Journal Title

      Rocky Mountain Journal of Mathematics

      Volume: 45 Issue: 3 Pages: 963-972

    • DOI

      10.1216/rmj-2015-45-3-963

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On p-adic properties of Siegel modular forms2014

    • Author(s)
      S. Boecherer and S. Nagaoka
    • Journal Title

      Springer Proceedings in Math. and Statistics

      Volume: 115 Pages: 47-66

    • DOI

      10.1007/978-3-319-11352-4_4

    • ISBN
      9783319113517, 9783319113524
    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Journal Article] On p-adic quaternionic Eisenstein series2013

    • Author(s)
      Toshiyuki Kikuta, Shoyu Nagaoka
    • Journal Title

      Abh. Math. Semin. Univ. Hamburg

      Volume: 83 Issue: 2 Pages: 147-157

    • DOI

      10.1007/s12188-013-0084-0

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Ramanujan type congruences for modular forms of several variables2013

    • Author(s)
      T. Kikuta, S. Nagaoka
    • Journal Title

      The Ramanujan Journal

      Volume: Published online Issue: 1 Pages: 143-157

    • DOI

      10.1007/s11139-012-9423-5

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Presentation] On Hermitian-Eisenstein series2015

    • Author(s)
      Shoyu Nagaoka
    • Organizer
      Mathematisches Seminar RWTH
    • Place of Presentation
      ドイツ・アーヘン工科大学
    • Year and Date
      2015-12-15
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] Siegel modular形式の合同2015

    • Author(s)
      長岡昇勇
    • Organizer
      第60回代数学シンポジウム
    • Place of Presentation
      静岡大学
    • Year and Date
      2015-09-02
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] テータ作用素のmod p核2015

    • Author(s)
      長岡昇勇
    • Organizer
      早稲田大学整数論研究集会
    • Place of Presentation
      早稲田大学(東京都新宿区大久保)
    • Year and Date
      2015-03-19
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] On the mod p kernel of the theta operator2014

    • Author(s)
      S. Nagaoka
    • Organizer
      Oberseminar Colloquium in Leherstuhl A fuer Math.
    • Place of Presentation
      アーヘン工科大学(ドイツ・ノルトライン=ヴェストファーレン州、アーヘン)
    • Year and Date
      2014-12-01
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] On the mod p kernel of the theta operator2014

    • Author(s)
      兒玉浩尚、長岡昇勇
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      広島大学(広島県東広島市鏡山)
    • Year and Date
      2014-09-25
    • Related Report
      2014 Research-status Report
  • [Presentation] ある奇数ウエイトのジーゲルカスプ形式のなす合同式

    • Author(s)
      兒玉浩尚、長岡昇勇
    • Organizer
      日本数学会総合分科会(代数学分科会)
    • Place of Presentation
      愛媛大学(愛媛県松山市)
    • Related Report
      2013 Research-status Report
  • [Presentation] On p-adic Siegel-Eisenstein series

    • Author(s)
      Shoyu Nagaoka
    • Organizer
      ドイツ、マンハイム大学数学講演会
    • Place of Presentation
      ドイツ、マンハイム大学
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] On p-adic quaternionic Eisenstein series

    • Author(s)
      Shoyu Nagaoka
    • Organizer
      ドイツ、アーヘン工科大学数学講演会
    • Place of Presentation
      ドイツ、アーヘン工科大学
    • Related Report
      2013 Research-status Report
    • Invited
  • [Book] ジーゲルモジュラー関数論2014

    • Author(s)
      E. Freitag (長岡昇勇訳)
    • Total Pages
      385
    • Publisher
      共立出版株式会社
    • Related Report
      2014 Research-status Report
  • [Book] 学校数学事典2014

    • Author(s)
      F.Reinhardt
    • Total Pages
      281
    • Publisher
      共立出版株式会社
    • Related Report
      2014 Research-status Report
  • [Book] カラー図解学校数学事典2014

    • Author(s)
      長岡昇勇、長岡由美子
    • Total Pages
      296
    • Publisher
      共立出版株式会社
    • Related Report
      2013 Research-status Report

URL: 

Published: 2014-07-25   Modified: 2019-07-29  

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