Special values of automorphic L-functions

• Project/Area Number: 16K05069
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Osaka City University
• Principal Investigator: FURUSAWA Masaaki 大阪市立大学, 大学院理学研究科, 教授 (50294525)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Project Status: Completed (Fiscal Year 2019)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2018: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2017: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2016: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 保型L函数 / L函数の特殊値 / テータ対応 / 代数学

Outline of Final Research Achievements

In a joint work with Kazuki Morimoto of Kobe University, we proved Boecherer's conjecture. It is a conjecture concerning degree two cuspidal Siegel modular forms which are Hecke eigenforms. More precisely, it predicted a relationship between the finite sum of Fourier coefficients over the binary quadratic forms corresponding to the ideal class group of an imaginary quadratic field and the central special value of the spinor L-function twisted by the quadratic character corresponding to the imaginary quadratic field. Siegfried Boecherer proclaimed the conjecture in the 1980's and it has been open till now.

Academic Significance and Societal Importance of the Research Achievements

算術的なL函数の特殊値は、対応する数論的対象物の重要な情報を含んでいると予想されている。Birch & Swinnerton-Dyer予想及びその一般化にみられるように、函数等式の中心における特殊値は特に興味深い。本研究の成果であるベッヘラー予想は、GL(2)に関するWaldspurgerの定理の自然な一般化であるとみなすことができる。Waldspurgerの結果はこれまでに、楕円曲線及びGL(2)の保型形式の数論において、重要な局面で応用されてきた。我々の結果に対しても、今後、様々な応用が期待できる。

Research Products

• [Journal Article] On special Bessel periods and the Gross-Prasad conjecture for SO(2n+1)xSO(2) (2017)
  • Author(s): Masaaki Furusawa and Kazuki Morimoto
  • Journal Title: Mathematische Annalen Volume: 印刷中 Issue: 1-2 Pages: 561-586
  • DOI: 10.1007/s00208-016-1440-z
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] On Boecherer's conjecture (2019)
  • Author(s): Masaaki Furusawa
  • Organizer: Modular Forms on Higher Rank Groups
  • Int'l Joint Research / Invited
• [Presentation] Refined global Gross-Prasad conjecture on special Bessel periods and Boecherer's conjecture (2018)
  • Author(s): Masaaki Furusawa
  • Organizer: CMO Workshop: Special Values of Automorphic L-functions and Associated p-adic L-Functions (18w5053)
  • Int'l Joint Research / Invited
• [Presentation] On special Bessel periods for SO(2n+1) (2017)
  • Author(s): 古澤昌秋
  • Organizer: RIMS 研究集会「保型形式とその周辺」
  • Place of Presentation: 京都大学数理解析研究所（京都府京都市）
  • Year and Date: 2017-02-06
  • Int'l Joint Research
• [Presentation] 保型L函数の特殊値の代数性について - 極私的総括 - (2017)
  • Author(s): 古澤昌秋
  • Organizer: 第62回代数学シンポジウム
  • Invited