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Fourier coefficients and zeros of modular forms

Research Project

Project/Area Number 19F19318
Research Category

Grant-in-Aid for JSPS Fellows

Allocation TypeSingle-year Grants
Section外国
Review Section Basic Section 11010:Algebra-related
Research InstitutionKyushu University

Principal Investigator

金子 昌信  九州大学, 数理学研究院, 教授 (70202017)

Co-Investigator(Kenkyū-buntansha) PAUL BIPLAB  九州大学, 数理(科)学研究科(研究院), 外国人特別研究員
Project Period (FY) 2019-11-08 – 2022-03-31
Project Status Discontinued (Fiscal Year 2021)
Budget Amount *help
¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 2021: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2020: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2019: ¥500,000 (Direct Cost: ¥500,000)
KeywordsHecke 固有形式 / 符号変化 / Siegel モジュラー形式 / Ramanujan-Petersson 予想 / ジーゲルモジュラー形式 / ラマヌジャン・ピーターソン予想 / デデキントゼータ関数 / 零点の個数評価
Outline of Research at the Start

サルナックらによる尖点形式の零点とフーリエ係数の符号変化に関する研究,特別研究員による符号変化と重複度1定理の関係に関する研究,そして研究代表者のモジュラー形式の零点に関する研究を統合し推し進めることで,モジュラー形式の重複度1定理と零点布を結びつけられる可能性を探り,モジュラー形式の零点の全く新しい意味づけを見いだす.また,代表者によるモジュラー形式と多重ゼータ値の関係についての研究を,多重 L 値版に拡張することを図る.

Outline of Annual Research Achievements

サルナックらの尖点形式の零点とフーリエ係数の符号変化に関する研究を,受入研究者金子によるモジュラー形式の零点に関する研究とPaul氏のフーリエ係数の符号変化に関する研究の統合のためのプロトタイプとして,その統合を推し進める方向で研究を進めた.Paul氏は金子による零点研究の知見をもとに,やや意外な方向,すなわち,フーリエ係数の符号変化と,フーリエ係数の非零性を結びつけるという,古典的なLehmer予想に一つの進展をもたらす結果を得た.他にも二つの new forms を Hecke 作用素の固有値で区別するという,いわゆる重複度1の問題,また Siegel モジュラー形式に関する Ramanujan-Petersson 予想に関する研究についても多く議論を行い,一定の進展があった.重複度1問題についての成果は多岐にわたっており,技術的になるので詳述はしないが,例えばCM型の二つの Hecke 固有形式に対し,素数べきで符号変化が無限に起こるような素数の密度を考え,それによってCM体の区別がされるといった興味深い成果を得ている. Ramanujan-Petersson 予想については,Duke-Imamoglu-Ikeda リフトになっているような Siegel 尖点形式に対して予想が正しいことを証明した.
任意の偶数次数で成り立つこの結果は,この予想について大きな進展をもたらすもので,高く評価できるものである.さらには,Hecke 固有形式に付随するL関数の対数微分の1での値の上から,および下からの評価について,先行のIhara-Murty-Shimuraの結果を一般化する成果を得た.

Research Progress Status

令和3年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

令和3年度が最終年度であるため、記入しない。

Report

(3 results)
  • 2021 Annual Research Report
  • 2020 Annual Research Report
  • 2019 Annual Research Report
  • Research Products

    (15 results)

All 2021 2020 2019 Other

All Int'l Joint Research (1 results) Journal Article (5 results) (of which Int'l Joint Research: 4 results,  Peer Reviewed: 5 results,  Open Access: 2 results) Presentation (9 results) (of which Int'l Joint Research: 3 results,  Invited: 8 results)

  • [Int'l Joint Research] Indian Institute of Technology Ropar/Tata Institute of Fundamental Research(インド)

    • Related Report
      2019 Annual Research Report
  • [Journal Article] Distinguishing newforms by their Hecke eigenvalues2021

    • Author(s)
      Sanoli Gun、V. Kumar Murty、Biplab Paul
    • Journal Title

      Research in Number Theory

      Volume: 7

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Ramanujan-Petersson conjecture for Fourier-Jacobi coefficients of Siegel cusp forms2021

    • Author(s)
      Kumar Balesh、Paul Biplab
    • Journal Title

      Bulletin of the London Mathematical Society

      Volume: 53 Issue: 1 Pages: 274-284

    • DOI

      10.1112/blms.12419

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] On values of logarithmic derivative of L-functions2020

    • Author(s)
      Biplab Paul
    • Journal Title

      RIMS Kokyuroku

      Volume: 2196

    • NAID

      120007165847

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the Petersson inner products of Fourier-Jacobi coecients and Hecke eigenvalues of Siegel cusp forms2020

    • Author(s)
      Balesh Kumar、Biplab Paul
    • Journal Title

      Acta Arith

      Volume: -

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] On the error term and zeros of the Dedekind zeta function2020

    • Author(s)
      Paul Biplab、Sankaranarayanan Ayyadurai
    • Journal Title

      Journal of Number Theory

      Volume: - Pages: 98-119

    • DOI

      10.1016/j.jnt.2020.02.006

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Presentation] Joint distribution and simultaneous sign change of Hecke eigenvalues of cusp forms2021

    • Author(s)
      Biplab Paul
    • Organizer
      Joint Number Theory Seminar
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] On analogues of Ramanujan conjecture for Siegel modular forms2021

    • Author(s)
      Biplab Paul
    • Organizer
      SNU-Kyushu Workshop in Number Theory
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Hecke eigenvalues and Fourier coefficients of Siegel cusp forms of degree 22021

    • Author(s)
      Biplab Paul
    • Organizer
      Analytic Number Theory Seminar
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] Hecke eigenvalues and Fourier coefficients of Siegel cusp forms of degree 22021

    • Author(s)
      Biplab Paul
    • Organizer
      Kanazawa Number Theory Autumn Workshop 2021
    • Related Report
      2021 Annual Research Report
  • [Presentation] Growth of Petersson inner products of Fourier-Jacobi coefficients of Siegel cusp forms2020

    • Author(s)
      Biplab Paul
    • Organizer
      IMSc Number Theory Web-seminar
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Logarithmic derivatives of L-functions attached to modular forms2020

    • Author(s)
      Biplab Paul
    • Organizer
      RIMS Conference: "Problems and prospects in analytic number theory"
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Growth of Petersson inner products of Fourier-Jacobi coecients of Siegel cusp forms2020

    • Author(s)
      Biplab Paul
    • Organizer
      RIMS conference, Analytic, geometric and p-adic aspects of automorphic forms and L-functions"
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Values of logarithmic derivatives of modular L-functions at s=12020

    • Author(s)
      Biplab Paul
    • Organizer
      The 13th young mathematicians conference on zeta functions
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Arithmetic behavior of Hecke eigenvalues of modular forms2019

    • Author(s)
      Biplab Paul
    • Organizer
      Algebra Seminar
    • Related Report
      2019 Annual Research Report
    • Invited

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Published: 2019-11-29   Modified: 2024-03-26  

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