Perrods and cogruences of modular forms

• Project/Area Number: 24540005
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Muroran Institute of Technology
• Principal Investigator: KATSURADA Hidenori 室蘭工業大学, 工学(系)研究科(研究院), 教授 (80133792)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥5,200,000 (Direct Cost: ¥4,000,000、Indirect Cost: ¥1,200,000); Fiscal Year 2014: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2013: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2012: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
• Keywords: congruence / lift / L-values / Koecher-Maass series / Gross-Keating invariant / Siegel series / Hermitian Ikeda lift / Kima-Shahidi lift / period congruence / 周期 / 合同 / ジーゲル保型形式 / 特殊値

Outline of Final Research Achievements

We proposed conjectures on prime ideals giving congruence between various lifts (eg. Ikeda-Miyawaki lift, Kim-Shahidi lift) and modular forms not coming from the lifts. I also gave numerical examples which support the conjectures. (Joint works with T. Ibukiyama, C. Poor, D. Yuen and S. Takemori) I gave an explicit formula for the twisted Koecher-Maass series of the Duke-Imamoglu-Ikeda lift.We also applied this to special values of the Rankin-Selberg series of half-integral weight modular form.
I investigated the properties of the Gross-Keating invariants　of quadratic forms, and as an application we gave an explicit formula of the Siegel series of a quadartic forms over any non-archimedian local field.

Research Products

• [Journal Article] Koecher-Maass series of the Ikeda lift for U(m,m) (2015)
  • Author(s): H. Katsurada
  • Journal Title: Kyoto J. Math. Volume: 55
  • NAID: 120006459738
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] On Fourier coefficients of Siegel modular forms of degree two with respect to congruence subgroups (2014)
  • Author(s): Masataka Chida, Hidenori Katsurada and Kohji Matsumoto
  • Journal Title: Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg Volume: 84 Issue: 1 Pages: 31-47
  • DOI: 10.1007/s12188-013-0087-x
  • NAID: 120006459682
  • Peer Reviewed
• [Journal Article] Explicit formulas for the twisted Koecher-Maass seres of the Duke-Imamoglu-Ikeda lift and their applications (2014)
  • Author(s): Hidenori Katsurada
  • Journal Title: Math. Zeitschrift Volume: 276 Issue: 3-4 Pages: 1049-1075
  • DOI: 10.1007/s00209-013-1232-z
  • Peer Reviewed
• [Journal Article] On the special values of certain L-series related to half-integral weight modular forms (2014)
  • Author(s): H. Katsurada
  • Journal Title: Automorphic Forms, Springer Poceedings in Math. and Stat. Volume: 115
  • NAID: 120006459739
  • Peer Reviewed
• [Journal Article] On the space of Eisenstein series for Γo(p) (2014)
  • Author(s): S. Boecherer (with appendix by H. Katsurada)
  • Journal Title: Comment. Math. Univ. St. Pauli Volume: 63
  • Peer Reviewed
• [Journal Article] Koecher-Maass series of the adelic Hermitian Eisenstein series and the adlic Hermitian Ikeda lift (2014)
  • Author(s): H. Katsurada
  • Journal Title: Comment. Math.Univ. St. Pali Volume: 63
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Koecher-Maass series of a certain half-integral weight modular form related with the Duke-Imamoglu-Ikeda lift (2014)
  • Author(s): H. Katsurada and H. Kawamura
  • Journal Title: Acta Arithmetica Volume: 162 Issue: 1 Pages: 1-42
  • DOI: 10.4064/aa162-1-1
  • Peer Reviewed / Open Access
• [Journal Article] Exact critical values of the symmetric fourth <i>L</i> function and vector valued Siegel modular forms (2014)
  • Author(s): Tomoyoshi Ibukiyama and Hidenori Katsurada
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 66 Issue: 1 Pages: 139-160
  • DOI: 10.2969/jmsj/06610139
  • NAID: 130003394350
  • ISSN: 0025-5645, 1881-1167, 1881-2333
  • Peer Reviewed
• [Journal Article] Cogruences to Ikeda-Miyawaki lifts and triple L-values of elliptic modular forms (2014)
  • Author(s): T. Ibukiyama, H. Katsurada, C. Poor, and D. S. Yuen
  • Journal Title: J. Number Theory Volume: 134 Pages: 142-180
  • Peer Reviewed
• [Journal Article] Explicit formulas for the twisted Koecher-Maass seres of the Duke-Imamoglu-Ikeda lift and their applications (2014)
  • Author(s): H. Katsurada
  • Journal Title: Math. Z. Volume: 276 Pages: 1049-1075
  • Peer Reviewed
• [Journal Article] Linear dependence of certain L-values of half-integral weight modular forms (2012)
  • Author(s): Hidenori Katsurada
  • Journal Title: Journal of the London Mathematical Society Volume: 85 Issue: 2 Pages: 455-471
  • DOI: 10.1112/jlms/jdr057
  • NAID: 120006459638
  • Peer Reviewed
• [Journal Article] Congruences for Hecke Eigenvalues of Siegel modular forms (2012)
  • Author(s): H. Katsurada and S. Mizumoto
  • Journal Title: Abhand. Math. Semi.Univ. Hamburg Volume: 82 Issue: 2 Pages: 129-152
  • DOI: 10.1007/s12188-012-0069-4
  • NAID: 120006459643
  • Peer Reviewed
• [Presentation] 二次形式のGross-Keating不変量とその応用 (2015)
  • Author(s): 池田保，桂田英典
  • Organizer: RIMS研究集会「Modular形式と保型表現」
  • Place of Presentation: 京都府京都市 京大数理解析研究所
  • Year and Date: 2015-02-06
• [Presentation] Periods and congruences of various lifts of modular forms (2015)
  • Author(s): H. Katsurada
  • Organizer: Workshp on automorphic forms at Kumamoto
  • Place of Presentation: 熊本県熊本市 熊本大学
  • Year and Date: 2015-01-07
  • Invited
• [Presentation] On thespecial values of certain L-series related to half-integral weight modular forms (2012)
  • Author(s): H. Katsurada
  • Organizer: Workshop on Modular Forms
  • Place of Presentation: 九州大学
  • Invited
• [Presentation] Refined version of Ikeda's conjecture on the period of the Hermitian Ikeda lift
  • Author(s): H. Katsurada
  • Organizer: Explicit Theory of Automorphic Forms
  • Place of Presentation: Tongji University(Shanghai, China)
  • Invited