The arithmetic of abelian varieties with complex multiplication and Euler systems.

• Project/Area Number: 19740010
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: Keio University (2008-2009); Nagoya University (2007)
• Principal Investigator: BANNAI Kenichi 慶應大, 理工学部, 講師 (90343201)
• Co-Investigator(Renkei-kenkyūsha): KOBAYASHI Shinichi 東北大学, 大学院・理学研究科, 准教授 (80362226) ; TSUJI Takeshi 東京大学, 大学院・数理科学研究科, 准教授 (40252530)
• Research Collaborator: GUIDO Kings Regensburg University (Germany), Professor
• Project Period (FY): 2007 – 2008
• Project Status: Completed (Fiscal Year 2009)
• Budget Amount: ¥3,930,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥630,000); Fiscal Year 2009: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2008: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2007: ¥1,200,000 (Direct Cost: ¥1,200,000)
• Keywords: ポリログ / 楕円曲線 / アーベル多様体 / 国際研究者交流 / ドイツ / 整数論 / 数論幾何学 / L-関数 / 混合層 / オイラー系

Research Abstract

In this research, we investigated motivic objects such as the polylogarithm and the Eisenstein classes, for the case of elliptic curves, which are one dimensional abelian varieties. We have obtained the following results. Through joint research with Shinichi Kobayashi and Takeshi Tsuji, we gave an explicit description of the p-adic realization of the elliptic polylogarithm for a prime p≧5, even in the case when the elliptic curve has complex multiplication and good supersingular reduction at p, for the Frobenius lifting which is the second power of the absolute Frobenius. Also, through joint research with G. Kings, we gave an explicit description of the p-adic Eisenstein class on the modular curve when p≧5 is a prime of good ordinary reduction.

Research Products

• [Journal Article] Realizations of the elliptic polylogarithm for CM elliptic curves (2009)
  • Author(s): K. Bannai, S. Kobayashi and T. Tsuji
  • Journal Title: 数理解析研究所講究録別冊 (掲載確定)
  • Peer Reviewed
• [Journal Article] Algebraic theta functions and Eisenstein-Kronecker numbers (2007)
  • Author(s): K. Bannai, S. Kobayashi
  • Journal Title: RIMS Kokyuroku Bessatsu B4: Proceedings of the Symposium on Algebraic Number theory and Related Topics, eds. K. Hashimoto, Y. Nakajima and H. Tsunogai Pages: 63-78
  • Peer Reviewed
• [Journal Article] On the p-adic elliptic polylogarithm and the two-variable p-adic L-function for CM elliptic curves (2007)
  • Author(s): K. Bannai
  • Journal Title: Oberwolfach Research Report, Report No.30 Pages: 1768-1770
• [Journal Article] Algebraic Theta Functions and Eisenstein-Kronecker numbers (2007)
  • Author(s): Bannai, K. and Kobayashi, S.
  • Journal Title: RIMS Kokyuryoku Bessatsu B4 Pages: 63-78
  • Peer Reviewed
• [Presentation] Eisenstein類とp進L関数 (2008)
  • Author(s): 坂内健一, 楕円ポリログ
  • Organizer: 日本数学会2008年度秋季総合分科会, 代数分科会・特別講演
  • Place of Presentation: 東京工業大学
  • Year and Date: 2008-09-26
• [Presentation] 楕円ポリログ、Eisenstein類とp進L関数 (2008)
  • Author(s): 坂内健一
  • Organizer: 日本数学会
  • Place of Presentation: 東京工業大学(特別講演)
  • Year and Date: 2008-09-26
• [Presentation] とp進L関数 (2008)
  • Author(s): 坂内健一, 楕円ポリログ
  • Organizer: 代数学シンポジウム
  • Place of Presentation: 岩手県盛岡市
  • Year and Date: 2008-08-07
• [Presentation] 楕円ポリログとp-進L-関数 (2008)
  • Author(s): 坂内健一
  • Organizer: 代数学シンポジウム
  • Place of Presentation: 盛岡
  • Year and Date: 2008-08-07
• [Presentation] Eisenstein-Kronecker数と代数的テータ関数 (2008)
  • Author(s): 坂内健一
  • Organizer: 日本数学会2008年度年会
  • Place of Presentation: 近畿大学
  • Year and Date: 2008-03-24
• [Presentation] 代数的データ関数とEisenstein-Kronecker数 (2008)
  • Author(s): 坂内 健一
  • Organizer: 日本数学会
  • Place of Presentation: 近畿大学
  • Year and Date: 2008-03-24
• [Presentation] 虚数乗法をもつ楕円曲線のp進ポリログと2変数p進L関数, 「代数的整数論とその周辺」 (2007)
  • Author(s): 坂内健一
  • Place of Presentation: 数理解析研究所
  • Year and Date: 2007-12-10
• [Presentation] 1. Introduction, and the case of the Riemann zeta function 2. Realizations of the elliptic polylogarithm for CM elliptic curves (2007)
  • Author(s): 坂内健一
  • Organizer: Number Theory Monthly Mini Workshop
  • Place of Presentation: POSTECH, Korea
  • Year and Date: 2007-10-27
• [Presentation] 夏の学校「多重ゼータ値とモティーフ」 (2007)
  • Author(s): 坂内健一, 楕円ポリログ
  • Place of Presentation: 東北大学
  • Year and Date: 2007-07-27
• [Presentation] On the p-adic Eisenstein- Kronecker series for CM elliptic curves (2007)
  • Author(s): 坂内健一
  • Organizer: Number Theory Seminar
  • Place of Presentation: Regensburg University, Germany
  • Year and Date: 2007-06-29
• [Presentation] On realizations of the elliptic polylogarithm (2007)
  • Author(s): 坂内健一
  • Organizer: Algebraische Zahlentheorie
  • Place of Presentation: Oberwolfach, Germany
  • Year and Date: 2007-06-20
• [Presentation] On the p-adic elliptic polylogarithm and the two-variable p-adic L-function for CM elliptic curves (2007)
  • Author(s): 坂内健一
  • Place of Presentation: Tambara Institute of Mathematical Science
  • Year and Date: 2007-06-08
• [Remarks]
  • URL: http://www.math.keio.ac.jp/~bannai/