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Arithmetic of values at real quadratics of the j-function

Research Project

Project/Area Number 18K18712
Research Category

Grant-in-Aid for Challenging Research (Exploratory)

Allocation TypeMulti-year Fund
Review Section Medium-sized Section 11:Algebra, geometry, and related fields
Research InstitutionKyushu University

Principal Investigator

Kaneko Masanobu  九州大学, 数理学研究院, 教授 (70202017)

Project Period (FY) 2018-06-29 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥6,370,000 (Direct Cost: ¥4,900,000、Indirect Cost: ¥1,470,000)
Fiscal Year 2020: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
Fiscal Year 2019: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Fiscal Year 2018: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
Keywords虚二次体の類数 / 実二次無理数の連分数展開 / 楕円モジュラー j-関数 / テータ関数 / 実二次無理数 / 超越性 / 類数 / Kronecker 極限公式 / データ関数 / マルコフ二次無理数 / j-関数 / 実二次体 / 虚数乗法論 / モックモジュラー形式
Outline of Final Research Achievements

For the class number of an imaginary quadratic field generated by the square root of negative of a prime, there is a beautiful formula expressing it in terms of the continued fraction expansion of the square root of that prime. We generalized this formula to the case of “remaining” half of the primes.
We also obtained some results on the algebraic independence and transcendence of the values of the theta function, a kind of modular form.

Academic Significance and Societal Importance of the Research Achievements

類数は整数論において古くから研究され,かつ未だに謎の多い重要な対象である.その中でも基本的な虚二次体の類数について,それを実二次無理数の連分数展開から計算するという意外な公式の,未だ知られていなかった場合を明らかにした意義は大きい.またその手法は,類似の公式を望むだけ生み出せるようなもので,応用も広い.整数論は暗号などで社会的にも大きな役割を果たしつつあるが,虚二次体の類数は直接的には超特異楕円曲線と関係が深く,暗号研究にも直接繋がるものである.

Report

(6 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • Research Products

    (16 results)

All 2023 2022 2021 2020 2019 2018 Other

All Int'l Joint Research (2 results) Journal Article (2 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 2 results,  Open Access: 1 results) Presentation (12 results) (of which Int'l Joint Research: 7 results,  Invited: 12 results)

  • [Int'l Joint Research] Max Planck Institute for Mathematics(ドイツ)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] ハワイ大学(米国)

    • Related Report
      2018 Research-status Report
  • [Journal Article] A criterion of algebraic independence of values of modular functions and an application to infinite products involving Fibonacci and Lucas numbers2022

    • Author(s)
      Daniel Duverney, Carsten Elsner, Masanobu Kaneko, and Yohei Tachiya
    • Journal Title

      Reseach in Number Theory

      Volume: 8, no. 2 Paper No. 31 Issue: 2

    • DOI

      10.1007/s40993-022-00328-7

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] A generalized regularization theorem and Kawashima’s relation for multiple zeta values2021

    • Author(s)
      Masanobu Kaneko, Ce. Xu and Shuji Yamamoto
    • Journal Title

      Journal of Algebra

      Volume: 580 Pages: 247-263

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] On some formulas for quadratic class numbers2023

    • Author(s)
      Masanobu Kaneko
    • Organizer
      Number Theory in Tokyo
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 虚2次体の類数をめぐる2,3の話題2022

    • Author(s)
      金子昌信
    • Organizer
      愛知数論セミナー
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] Multiple L-values of conductor 4, Entringer numbers, and period polynomials2022

    • Author(s)
      金子昌信
    • Organizer
      第 16 回多重ゼータ研究集会& 第 58 回関西多重ゼータ研究会
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] レベル2有限多重ゼータ値について2021

    • Author(s)
      金子昌信
    • Organizer
      九州代数的整数論2021夏
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Genus character L-functions of quadratic orders and class number formulas2021

    • Author(s)
      Masanobu Kaneko
    • Organizer
      NCTS Seminar on Number Theory
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Caliber2021

    • Author(s)
      金子昌信
    • Organizer
      香川セミナー
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] Genus character L-functions of quadratic orders and class numbers (joint with Y.Mizuno)2020

    • Author(s)
      Masanobu Kaneko
    • Organizer
      Number Theory Seminar
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On a variant of multiple zeta values of level two2019

    • Author(s)
      Masanobu Kaneko
    • Organizer
      Multiple zeta values and related topics
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Genus character L-functions of quadratic orders and class number formulas2019

    • Author(s)
      Masanobu Kaneko
    • Organizer
      Indo-Japan conference on number theory
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Genus character L-functions of quadratic orders and class number formulas2019

    • Author(s)
      Masanobu Kaneko
    • Organizer
      Hawaii Number Theory 2019 (HINT)
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 楕円モジュラー関数の実二次点における「値」について2018

    • Author(s)
      金子昌信
    • Organizer
      紀尾井町数理セミナー
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] On the ``value" of the elliptic modular function at real quadratics2018

    • Author(s)
      Masanobu Kaneko
    • Organizer
      New developments in the theory of modular forms over function fields
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited

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Published: 2018-07-25   Modified: 2024-01-30  

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