P-adic L-function and P-adic Beilinson conjecture

• Project/Area Number: 21740009
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: Tohoku University
• Principal Investigator: KOBAYASHI Shinichi 東北大学, 大学院・理学研究科, 准教授 (80362226)
• Project Period (FY): 2009 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2011: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2010: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2009: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
• Keywords: ゼータ関数 / 岩澤理論 / p-進L-関数 / p進Beilinson予想 / Birch and Swinnerton-Dyer予想 / 楕円曲線 / 整数論 / p進L-関数 / Beilinson予想 / 岩沢理論 / L-関数の特殊値 / BSD予想 / 佐藤理論 / Birth and Swinnerton-Dyer予想

Research Abstract

I studied the p-adic L-function of elliptic curves defined over the rational number field. I obtained the p-adic Gross-Zagier formula at supersingular prime p that describes the derivative of the p-adic L-function of elliptic curves base changed over an imaginary quadratic field satisfying the Heegner condition in terms of the p-adic height of the Heegner point. As an application, I proved the full Birch and Swinnerton-Dyer conjecture for CM elliptic curves of rank 1 up to bad primes. This formula also gives an important example of the p-adic Beilinson conjecture.

Research Products

• [Journal Article] The p-adic Gross-Zagier formula for elliptic curves at supersingular primes (2012)
  • Author(s): Shinichi Kobayashi
  • Journal Title: Inventiones mathematicae Volume: (掲載決定)
  • Peer Reviewed
• [Journal Article] On the p-adic Gross-Zagier formula for elliptic curves at supersingular primes (2012)
  • Author(s): Shinichi Kobayashi
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: (掲載決定)
  • Peer Reviewed
• [Journal Article] The p-adic Gross-Zagier formula for elliptic curves at supersingular primes (2012)
  • Author(s): S.Kobayashi
  • Journal Title: Inventiones Mathematicae Volume: (未定)(掲載確定)
  • Peer Reviewed
• [Journal Article] On the p-adic Gross-Zagier formula for elliptic curves at supersingular primes (2012)
  • Author(s): S.Kobayashi
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: (未定)(掲載確定)
  • Peer Reviewed
• [Journal Article] On the de Rham and p-adic realizationsof the elliptic polylogarithm for CMelliptic curves (2010)
  • Author(s): Kenichi Bannai, Shinichi Kobayashi, Takeshi Tsuji
  • Journal Title: Les AnnalesScientifiques de l' Ecole NormaleSuperieure Volume: 43. no.2 Pages: 185-234
  • Peer Reviewed
• [Journal Article] Algebraic theta functions and p-adicinterpolation of Eisenstein-Kroneckernumbers (2010)
  • Author(s): Kenichi Bannai, Shinichi Kobayashi
  • Journal Title: Duke mathematical journal Volume: 153. No.2 Pages: 229-295
  • Peer Reviewed
• [Journal Article] On the de Rham and p-adic realizations of the elliptic polylogarithm for CM elliptic curves (2010)
  • Author(s): K.Bannai, S.Kobayashi, T.Tsuji
  • Journal Title: Les Annales Scientifiques de l'Ecole Normale Superieure Volume: 43 No.2 Pages: 185-234
  • Peer Reviewed
• [Journal Article] Algebraic theta functions and p-adic interpolation of Eisenstein-Kronecker numbers (2010)
  • Author(s): K.Bannai, S.Kobayashi
  • Journal Title: Duke Mathematical Journal Volume: 153 No.2 Pages: 229-295
  • Peer Reviewed
• [Journal Article] Realizations of the elliptic polylogarithm for CM elliptic curves (2009)
  • Author(s): Kenichi Bannai, Shinichi Kobayashi, Takeshi Tsuji
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: B12 Pages: 33-50
  • Peer Reviewed
• [Journal Article] Realizations of the elliptic polylogarithm for CM elliptic curves (2009)
  • Author(s): K.Bannai, S.Kobayashi, T.Tsuji
  • Journal Title: RIMS Kokyuroku Bessatsu B12 Pages: 33-50
  • Peer Reviewed
• [Journal Article] On the de Rham and p-adic realizations of the elliptic polylogarithm for CM elliptic curves
  • Author(s): K.Bannai, S.Kobayashi, T.Tsuji
  • Journal Title: Les Annales Scientifiques de l'Ecole Normale Superieure (未定, accepted)
  • Peer Reviewed
• [Journal Article] Algebraic theta functions and p-adic interpolation of Eisenstein-Kronecker numbers
  • Author(s): K.Bannai, S.Kobayashi
  • Journal Title: Duke Mathematical Journal (未定, accepted)
  • Peer Reviewed
• [Presentation] Torsion points on the quotient of a Fermat Jacobian via Anderson's p-adic soliton theory II (2012)
  • Author(s): Shinichi Kobayashi
  • Organizer: 2012 NCTS Mini-Workshop on Number Theory
  • Place of Presentation: National Tsinghua University, Taiwan
  • Year and Date: 2012-03-07
• [Presentation] The p-adic Gross-Zagier formula for elliptic curves at supersingular primes (2012)
  • Author(s): Shinichi Kobayashi
  • Organizer: East Asia Number Theory Conference
  • Place of Presentation: National Taiwan University. Taiwan
  • Year and Date: 2012-01-16
• [Presentation] The p-adic Gross-Zagier formula for elliptic curves at supersingular primes (2012)
  • Author(s): S.Kobayashi
  • Organizer: East Asia Number Theory Conference
  • Place of Presentation: National Taiwan University, Taiwan
  • Year and Date: 2012-01-16
• [Presentation] The p-adic Gross-Zagier formula for elliptic curves at supersingular primes (2011)
  • Author(s): Shinichi Kobayashi
  • Organizer: PanAsian Number Theory Conference
  • Place of Presentation: Morningside Center of Mathematics, Beijing. China
  • Year and Date: 2011-08-26
• [Presentation] The p-adic Gross-Zagier formula for elliptic curves at supersingular primes (2011)
  • Author(s): S.Kobayashi
  • Organizer: PanAsian Number Theory Conference
  • Place of Presentation: Morningside Center of Mathematics, China
  • Year and Date: 2011-08-26
• [Presentation] 超特異素点におけるp進Gross-Zagier公式について (2011)
  • Author(s): 小林真一
  • Organizer: 代数学コロギウム
  • Place of Presentation: 東京大学数理科学研究科
  • Year and Date: 2011-01-26
• [Presentation] 超特異素点におけるp進Gross-Zagier公式について (2011)
  • Author(s): 小林真一
  • Organizer: 代数学コロキウム
  • Place of Presentation: 東京大学大学院数理科学研究科
  • Year and Date: 2011-01-26
• [Presentation] 超特異素点におけるp進Gross-Zagier公式について (2011)
  • Author(s): 小林真一
  • Organizer: 数論幾何セミナー
  • Place of Presentation: 北海道大学
• [Presentation] 超特異素点におけるp進Gross-Zagier公式について (2010)
  • Author(s): 小林真一
  • Organizer: 代数的整数論とその周辺
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2010-12-07
• [Presentation] 超特異素点におけるp進Gross-Zagier公式について (2010)
  • Author(s): 小林真一
  • Organizer: 大阪大学整数論 & 保型形式セミナー
  • Place of Presentation: 大阪大学
  • Year and Date: 2010-11-12
• [Presentation] 超特異素点におけるp進Gross-Zagier公式について (2010)
  • Author(s): 小林真一
  • Organizer: 大阪大学整数論&保型形式セミナー
  • Place of Presentation: 大阪大学
  • Year and Date: 2010-11-12
• [Presentation] 超特異素点におけるp進Gross-Zagier公式について (2010)
  • Author(s): 小林真一
  • Organizer: L-関数の特殊値と数論幾何
  • Place of Presentation: 美山町自然文化村河鹿荘
  • Year and Date: 2010-10-08
• [Presentation] Mazur-Tateのp-進テータ関数とその周辺 (2010)
  • Author(s): 小林真一
  • Organizer: P進佐藤理論と数論幾何
  • Place of Presentation: 東北大学
  • Year and Date: 2010-09-30
• [Presentation] 超特異素点におけるp進Gross-Zagier公式について (2010)
  • Author(s): 小林真一
  • Organizer: 岩澤セミナー
  • Place of Presentation: 慶応大学
  • Year and Date: 2010-07-31
• [Presentation] 超特異素点における楕円曲線のp進L関数の微分値について (2010)
  • Author(s): 小林真一
  • Organizer: 東北大学整数論セミナー
  • Place of Presentation: 東北大学
  • Year and Date: 2010-05-17
• [Presentation] 超特異素点における楕円曲線のp進L関数の微分値について (2010)
  • Author(s): 小林真一
  • Organizer: 名古屋大学数論幾何セミナー
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-02-09
• [Presentation] Integral structures on p-adic Fourier theory (2010)
  • Author(s): 小林真一
  • Organizer: 2010 Korea-Japan Number Theory
  • Place of Presentation: Seoul National University
  • Year and Date: 2010-01-20
• [Presentation] Integral structures on p-adic Fourier theory (2010)
  • Author(s): 小林真一
  • Organizer: 2010 Korea-Japan Number Theory
  • Place of Presentation: Seoul National University, Korea
  • Year and Date: 2010-01-20
• [Presentation] 超特異素点におけるp進Gross-Zagier公式についてI, II (2010)
  • Author(s): 小林真一
  • Organizer: 数論幾何ワークショップ2010
  • Place of Presentation: 沖縄尚学高等学校
• [Presentation] 超特異素点におけるp進Gross-Zagier公式についてI,II (2010)
  • Author(s): 小林真一
  • Organizer: 数論幾何ワークショップ2010
  • Place of Presentation: 沖縄尚学高等学校
• [Presentation] Mumfordの代数的テータ関数とp-進テータ関数 (2009)
  • Author(s): 小林真一
  • Organizer: P-adic Special functions & Arithmetic Geometry
  • Place of Presentation: 蔵王ゆと森倶楽部
  • Year and Date: 2009-10-31
• [Presentation] Mumford の代数的テータ関数とp-進テータ関数 (2009)
  • Author(s): 小林真一
  • Organizer: P-adic Special functions & Arithmetic Geometry
  • Place of Presentation: 蔵王ゆと森倶楽部
  • Year and Date: 2009-10-31