A p-adic approach to the special value formula of L-functions

• Project/Area Number: 25707001
• Research Category: Grant-in-Aid for Young Scientists (A)
• Allocation Type: Partial Multi-year Fund
• Research Field: Algebra
• Research Institution: Kyushu University (2016-2017); Tohoku University (2013-2015)
• Principal Investigator: Kobayashi Shinichi 九州大学, 数理学研究院, 教授 (80362226)
• Research Collaborator: OTA Kazuto 慶応大学, 理工学部, 特任助教 (70770775)
• Project Period (FY): 2013-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥9,880,000 (Direct Cost: ¥7,600,000、Indirect Cost: ¥2,280,000); Fiscal Year 2016: ¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000); Fiscal Year 2015: ¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000); Fiscal Year 2014: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000); Fiscal Year 2013: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• Keywords: 整数論 / L-関数 / 岩澤理論 / p進 / BSD予想 / 保型形式 / 代数学 / 数論幾何 / Gross-Zagier公式 / p進L関数 / Heegerサイクル / L-関数の特殊値 / Bloch-Kato予想 / p進L-関数 / p進高さ関数 / L関数 / 数論幾何学 / L関数の特殊値 / Heegner cycle

Outline of Final Research Achievements

We proved the p-adic Gross-Zagier formula for higher weight modular forms at non-ordinary primes. We also showed a Coates-Wiles type theorem and a one-side divisibility of Iwasawa main conjecture for Galois representations of modular forms twisted by anticyclotomic Hecke characters. This research is closely related to the Birch and Swinnerton-Dyer conjecture and gives an important example of the Beilinson-Bloch-Kato conjecture, which is a central theme of the modern number theory containing the Birch and Swinnerton-Dyer conjecture as a special case.

Research Products

• [Journal Article] The radius of convergence of the p-adic sigma function (2016)
  • Author(s): Kenichi Bannai, Shinichi Kobayashi, Seidai Yasuda
  • Journal Title: Mathematische Zeitschrift Volume: 印刷中 Issue: 1-2 Pages: 751-781
  • DOI: 10.1007/s00209-016-1783-x
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Integral structures on pp-adic Fourier theory (2016)
  • Author(s): Kenichi Bannai and Shinichi Kobayashi
  • Journal Title: Annales de L'Institut Fourier Volume: 66 Issue: 2 Pages: 521-550
  • DOI: 10.5802/aif.3018
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Torsion points on Jacobian varieties via Anderson's p-adic Soliton Theory (2015)
  • Author(s): S. Kobayashi, T. Yamazaki
  • Journal Title: Asian Journal of Mathematics Volume: in press
  • Peer Reviewed
• [Journal Article] p-adic Eisenstein-Kronecker series for CM elliptic curves and the Kronecker limit formulas (2015)
  • Author(s): K. Bannai, H. Furusho, S. Kobayashi
  • Journal Title: Nagoya mathematical journal Volume: in press
  • Peer Reviewed
• [Journal Article] The p-adic height pairing on abelian varieties at non-ordinary primes (2014)
  • Author(s): S. Kobayashi
  • Journal Title: Iwasawa Theory 2012, Contributions in Mathematical and Computational Sciences Volume: 7 Pages: 264-290
  • DOI: 10.1007/978-3-642-55245-8_9
  • ISBN: 9783642552441, 9783642552458
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] The p-adic height pairing on abelian varieties at non-ordinary primes (2014)
  • Author(s): Shinichi Kobayashi
  • Journal Title: Iwasawa 2012, Contributions in Mathematical and Computational Sciences, Springer Volume: Vol. 7
  • Peer Reviewed
• [Presentation] Lectures on the p-adic Gross-Zagier formula (2018)
  • Author(s): Shinichi Kobayashi
  • Organizer: Lecture series in Fudan University
  • Int'l Joint Research / Invited
• [Presentation] The p-adic Gross-Zagier formula for higher weight modular forms at non-ordinary primes (2017)
  • Author(s): Shinichi Kobayashi
  • Organizer: Fukuoka International Conference on Arithmetic Geometry in 2017
  • Int'l Joint Research / Invited
• [Presentation] Local Iwasawa theory of modular forms for the anti-cyclotomic Z_p-extension (2017)
  • Author(s): Shinichi Kobayashi
  • Organizer: Iwasawa 2017
  • Int'l Joint Research / Invited
• [Presentation] Iwasawa theory for generalized Heegner cycles at non-ordinary primes (2017)
  • Author(s): Shinichi Kobayashi
  • Organizer: Special Cycles on Shimura Varieties and Iwasawa Theory
  • Int'l Joint Research / Invited
• [Presentation] Iwasawa theory for generalized Heegner cycles at non-ordinary primes (2017)
  • Author(s): Shinichi Kobayashi
  • Organizer: Workshop on p-adic L-functions and algebraic cycles
  • Int'l Joint Research / Invited
• [Presentation] 一般Heegner cycleの岩澤主予想 (2017)
  • Author(s): 小林真一
  • Organizer: 東北大学代数セミナー
  • Invited
• [Presentation] 楕円曲線の反円分拡大の岩澤理論 (2017)
  • Author(s): 小林真一
  • Organizer: 新潟代数セミナー
  • Invited
• [Presentation] 保型形式の反円分岩澤理論 (2017)
  • Author(s): 小林真一
  • Organizer: 東京電機大学 数学講演会
  • Invited
• [Presentation] Anticyclotomic Iwasawa theory for modular forms at non-ordinary primes (2016)
  • Author(s): Shinichi Kobayashi
  • Organizer: Sendai international conference on arithmetic geometry
  • Place of Presentation: Tohoku University
  • Year and Date: 2016-01-29
  • Int'l Joint Research
• [Presentation] Anticyclotomic Iwasawa theory and generalized Heegner cycles (2016)
  • Author(s): 小林 真一
  • Organizer: 代数的整数論とその周辺2016
  • Place of Presentation: 京都大学数理解析研究所
  • Invited
• [Presentation] Iwasawa theory for generalized Heegner cycles at non-ordinary primes (2016)
  • Author(s): Shinichi Kobayashi
  • Organizer: Seminar de theorie des nombres de l'IMJ-PRG
  • Place of Presentation: Paris 6 University
  • Int'l Joint Research / Invited
• [Presentation] The radius of convergence of the p-adic sigma function at supersingular primes (2016)
  • Author(s): Shinichi Kobayashi
  • Organizer: Number theory Seminar
  • Place of Presentation: Padova University
  • Int'l Joint Research / Invited
• [Presentation] 一般Heegner cycleのp進補間について (2015)
  • Author(s): 小林 真一
  • Organizer: Regulators in Niseko 2015
  • Place of Presentation: ヒルトンニセコヴィレッジ
  • Year and Date: 2015-09-07
  • Invited
• [Presentation] A p-adic interpolation of Heegner cycles (2014)
  • Author(s): S. Kobayashi
  • Organizer: Arithmetic Geometry in Tohoku
  • Place of Presentation: 東北大学
  • Year and Date: 2014-10-30
• [Presentation] Heegner cycleのp進高さ (2014)
  • Author(s): 小林真一
  • Organizer: 日本数学会 代数学シンポジウム
  • Place of Presentation: 東京大学
  • Year and Date: 2014-09-10
  • Invited