Studies on symmetries for automorphic forms and Borcherds products

• Project/Area Number: 26400027
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Kyoto Sangyo University
• Principal Investigator: MURASE Atsushi 京都産業大学, 理学部, 教授 (40157772)
• Co-Investigator(Renkei-kenkyūsha): NARITA Hiroaki 熊本大学, 大学院自然科学研究科, 准教授 (70433315)
• Research Collaborator: SUGANO Takashi 金沢大学, 理工研究域数物科学系, 教授 (30183841) ; Bernhard Heim German University of Technology, 教授
• Project Period (FY): 2014-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2016: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2015: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2014: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 保型形式 / 対称性 / Borcherds積 / アフィンリー環 / ヤコビ形式 / 無限積 / Kac-Moodyリー代数 / ベクトル系 / affine Lie algebra / 完全再生可能性

Outline of Final Research Achievements

We investigated on a condition for Siegel modular forms on congruence subgroups to have infinite product expansions. We define a notion of “generalized multiplicative symmetries” for a family of automorphic forms on various levels. We furthermore show that a family of automorphic forms with infinite product expansions satisfies generalized multiplicative symmetries. We also considered a similar problem for Jacobi forms and investigated a relation between infinite product expansions and generalized multiplicative symmetries.
We show that a Siegel modular form of degree 2 of level 1 which is simultaneously anSaito-Kurokawa lift and a Borchers product is a constant multiple of the Igusa modular form.

Research Products

• [Int'l Joint Research] ドイツ工科大学（オマーン）(オマーン)
• [Journal Article] Fourier expansion of Arakawa lifting II: Relation with central L-values (2016)
  • Author(s): A. Murase and H. Narita
  • Journal Title: International Journal of Mathematics Volume: 27 Issue: 01 Pages: 50001-50032
  • DOI: 10.1142/s0129167x16500014
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] A Characterization of Holomorphic Borcherds Lifts by Symmetries (2015)
  • Author(s): Berhard Heim and Atsushi Murase
  • Journal Title: International Mathematics Research Notices Volume: 2015 Issue: 21 Pages: 11150-11185
  • DOI: 10.1093/imrn/rnv021
  • Peer Reviewed / Int'l Joint Research
• [Presentation] Symmetries characterizing Borchers products (2017)
  • Author(s): A. Murase
  • Organizer: The XXVth International Conference on Integrable Systems and Quantum symmetries
  • Place of Presentation: チェコ工科大学（チェコ共和国プラハ市）
  • Year and Date: 2017-06-06
  • Int'l Joint Research
• [Presentation] Symmetries characterizing Borcherds lifts (2015)
  • Author(s): 村瀬 篤
  • Organizer: Workshop on automorphic forms at Kumamoto
  • Place of Presentation: 熊本大学（熊本県熊本市）
  • Year and Date: 2015-01-07 – 2015-01-08
  • Invited