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Arithmetic study of regulators using special functions

Research Project

Project/Area Number 19K03391
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 11010:Algebra-related
Research InstitutionHokkaido University

Principal Investigator

Asakura Masanori  北海道大学, 理学研究院, 教授 (60322286)

Project Period (FY) 2019-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2021: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2019: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywordsレギュレーター / 周期積分 / L関数 / p進コホモロジー / p進超幾何関数 / モチーフ / 代数的K理論 / 超幾何関数 / p進L関数 / 特殊関数
Outline of Research at the Start

代数多様体のレギュレーターおよびL関数の数論的な研究を行う。より具体的には次の2つの課題に取り組む。
(1)レギュレーターとL関数の特殊値に関するベイリンソン予想
(2) (1)のp進化であるペランリュウの予想の研究
これまで、多くの研究者たちが、上記の問題に携わり成果をあげてきたが、いまだ十分ではない上、知られている計算例も、非常に限定的である。一方、近年、超幾何関数をはじめとする特殊関数を用いた特色ある研究が盛んになってきた。本研究でも、この特殊関数論からの研究を推進することで、上記課題に対して成果をあげる。

Outline of Final Research Achievements

In this research project, we studied regulators and L-functions of algebraic varieties.
Regulators and p-adic regulators determined from the motivic cohomology of algebraic varieties defined over the field of rational numbers are expected to represent special values of L-functions, but the road to general solutions is still far away. We do not think that the general solution of this conjecture is mature, and then we have focused on various special varieties as a strategy to attack the conjecture. In particular, I studied in detail special varieties related to hypergeometric functions, and achieved some remarkable results, such as describing p-adic regulators using p-adic hypergeometric functions, a numerical approach towards the p-adic Beilinson conjecture for regulators of elliptic curves.

Academic Significance and Societal Importance of the Research Achievements

本研究課題は、L関数の特殊値を幾何学的な不変量として記述できるかという古くからある整数論の問題に対して貢献するものである。
上述の整数論の問題は、19世紀のディリクレの類数公式から始まる。20世紀になり代数的K理論ないしモチヴィックコホモロジーが導入されるに至り、大きく一般化が進んだ。しかしながら、多くの課題が未解決の状態であり、現在に残された難問となっている。本研究課題では、これらを攻略するため、特殊多様体に焦点を当てたさまざまな研究を行い、いくつかの著しい研究成果をあげることができた。これらの成果は、上述の整数論の問題に対する進歩を与えた。

Report

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

    (15 results)

All 2023 2022 2021 2020 2019 2018

All Journal Article (9 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 9 results) Presentation (5 results) (of which Int'l Joint Research: 2 results,  Invited: 5 results) Funded Workshop (1 results)

  • [Journal Article] A numerical approach toward the p-adic Beilinson conjecture for elliptic curves over Q2023

    • Author(s)
      Masanori Asakura, Masataka Chida
    • Journal Title

      Research in the Mathematical Sciences

      Volume: 10 Issue: 1 Pages: 1-58

    • DOI

      10.1007/s40687-023-00374-2

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Regulators of K1 of Hypergeometric Fibrations2021

    • Author(s)
      Asakura Masanori、Otsubo Noriyuki
    • Journal Title

      Progress in Mathematics

      Volume: 338 Pages: 1-30

    • DOI

      10.1007/978-3-030-65203-6_1

    • ISBN
      9783030652029, 9783030652036
    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Hypergeometric functions and L-functions2021

    • Author(s)
      Masanori Asakura
    • Journal Title

      RIMS講究録別冊

      Volume: B86 Pages: 3-20

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Frobenius Action on a Hypergeometric Curve and an Algorithm for Computing Values of Dwork’s p-adic Hypergeometric Functions2021

    • Author(s)
      Asakura Masanori
    • Journal Title

      Springer Proceedings in Mathematics

      Volume: 373 Pages: 1-45

    • DOI

      10.1007/978-3-030-84304-5_1

    • ISBN
      9783030843038, 9783030843045
    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Explicit logarithmic formulas of special values of hypergeometric functions 3F22020

    • Author(s)
      M. Asakura, T. Yabu
    • Journal Title

      Communications in Contemporary Mathematics

      Volume: 22 Issue: 05 Pages: 1950040-1950040

    • DOI

      10.1142/s0219199719500408

    • Related Report
      2020 Research-status Report 2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] 超幾何関数とL関数2020

    • Author(s)
      朝倉政典
    • Journal Title

      RIMS講究録別冊

      Volume: -

    • NAID

      120007145012

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Chern class and Riemann-Roch theorem for cohomology theory without homotopy invariance2019

    • Author(s)
      Asakura Masanori, Sato Kanetomo
    • Journal Title

      J.Math.Sci.Univ.Tokyo

      Volume: 26 Pages: 249-334

    • NAID

      120006942550

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] A functional logarithmic formula for the hypergeometric function ${}_3F_2$2019

    • Author(s)
      Masanori Asakura and Noriyuki Otsubo
    • Journal Title

      Nagoya Math. J.

      Volume: 236 Pages: 29-46

    • DOI

      10.1017/nmj.2018.33

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] An algebro-geometric study of special values of hypergeometric functions ${}_3F_2$2018

    • Author(s)
      Masanori Asakura, Noriyuki Otsubo and Tomohide Terasoma
    • Journal Title

      Nagoya Math. J.

      Volume: First View Pages: 1-16

    • DOI

      10.1017/nmj.2018.36

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Presentation] 超幾何モチーフの1-拡大2022

    • Author(s)
      朝倉政典
    • Organizer
      代数学シンポジウム
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] レギュレーターとL関数の特殊値2021

    • Author(s)
      朝倉政典
    • Organizer
      日本数学会
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] レギュレーターとL関数の特殊値2021

    • Author(s)
      朝倉政典
    • Organizer
      日本数学会・代数分科会
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] An algorithm of computing special values of Dwork's p-adic hypergeometric functions in polynomial time2019

    • Author(s)
      Asakura Masanori
    • Organizer
      Hypergeometric Series, Mahler Measures, and Multiple Zeta Values
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] A generalization of the Ross symbols in higher K-groups and p-adic hypergeometric functions2019

    • Author(s)
      Asakura Masanori
    • Organizer
      p-adic cohomology and arithmetic geometry 2019
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Funded Workshop] Regulators in Niseko 20192019

    • Related Report
      2019 Research-status Report

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Published: 2019-04-18   Modified: 2024-01-30  

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