Study of deformations of Galois representations and special values of p-adic L-functions

• Project/Area Number: 15K17509
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Tohoku University
• Principal Investigator: Chida Masataka 東北大学, 理学研究科, 助教 (00451518)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000); Fiscal Year 2017: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2015: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: p進L関数 / 岩澤理論 / 一般化Heegnerサイクル / Chow-Heegnerサイクル / p進Gross-Zagier公式 / p進Abel-Jacobi写像 / 反円分的p進L関数 / 一般Heegnerサイクル / Hilbert保型形式 / CM楕円曲線 / 保型形式 / 代数的サイクル / 肥田理論

Outline of Final Research Achievements

In this research, I studied a relation between algebraic cycles and special values of (p-adic) L-functions and I showed an explicit formula between Chow-Heegner cycles and special values of L-functions associated to Hecke characters. I also constructed an algebraic cycle on the product of a Kuga-Sato variety over a Shimura curve associated to an unitary group over a totally real filed and a CM Abelian variety and gave an explicit description of the image of the algebraic cycle under the p-adic Abel-Jacobi map using Coleman integration.

Research Products

• [Int'l Joint Research] 中央研究院/国立台湾大学(台湾)
• [Int'l Joint Research] 国立台湾大学(台湾)
• [Int'l Joint Research] ENS de Lyon(フランス)
• [Int'l Joint Research] National Taiwan University(台湾)
• [Journal Article] Selmer groups and central values of L-functions for modular forms (2017)
  • Author(s): Masataka Chida
  • Journal Title: Annales de l'Institut Fourier Volume: 67 Issue: 3 Pages: 1231-1276
  • DOI: 10.5802/aif.3108
  • Peer Reviewed / Open Access
• [Journal Article] Selmer groups and central values of L-functions for modular forms (2017)
  • Author(s): Masataka Chida
  • Journal Title: Annales de l'Institut Fourier Volume: 印刷中
  • Peer Reviewed / Open Access
• [Journal Article] Special values of anticyclotomic L-functions for modular forms (2016)
  • Author(s): Masataka Chida and Ming-Lun Hsieh
  • Journal Title: Journal fur die reine und angewandte Mathematik (Crelle's Journal) Volume: - Issue: 741 Pages: 87-131
  • DOI: 10.1515/crelle-2015-0072
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Regulators for Rankin-Selberg products of modular forms (2016)
  • Author(s): Francois Brunault and Masataka Chida
  • Journal Title: Annales mathematiques du Quebec Volume: - Issue: 2 Pages: 221-249
  • DOI: 10.1007/s40316-015-0043-5
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Rankin-Selberg L関数の特殊値に関するBeilinson予想について (2016)
  • Author(s): 千田雅隆
  • Journal Title: 第60回代数学シンポジウム報告集 Volume: - Pages: 195-208
• [Journal Article] RegulatorとRankin-Selberg L関数の特殊値 (2015)
  • Author(s): 千田雅隆
  • Journal Title: 数理解析研究所講究録 Volume: 1973 Pages: 115-119
• [Presentation] Arithmetic diagonal cycles and special values of p-adic L-functions (2018)
  • Author(s): Masataka Chida
  • Organizer: UK-Japan Winter School 2018 ``Galois Representations and Automorphic forms''
  • Int'l Joint Research / Invited
• [Presentation] Chow-Heegner cycles on products of CM elliptic curves (2017)
  • Author(s): Masataka Chida
  • Organizer: Fukuoka International Conference on Arithmetic Geometry in 2017
  • Int'l Joint Research / Invited
• [Presentation] Chow-Heegner cycles on self-products of CM elliptic curves (2017)
  • Author(s): Masataka Chida
  • Organizer: Workshop on p-adic L-functions and algebraic cycles
  • Int'l Joint Research / Invited
• [Presentation] Selmer groups for Rankin-Selberg products of modular forms (2016)
  • Author(s): Masataka Chida
  • Organizer: p-adic methods in arithmetic geometry at Sendai
  • Place of Presentation: 東北大学片平キャンパス 知の館
  • Year and Date: 2016-11-01
  • Int'l Joint Research / Invited
• [Presentation] モチーフの円分p進L函数の存在予想 (2016)
  • Author(s): 千田雅隆
  • Organizer: 保型L函数の特殊値と付随するp進L函数
  • Place of Presentation: 美山町自然文化村 河鹿荘(京都府南丹市)
  • Year and Date: 2016-09-19
  • Invited
• [Presentation] CM cycles and Bloch-Kato conjecture for modular forms (2015)
  • Author(s): Masataka Chida
  • Organizer: Small workshop on number theory
  • Place of Presentation: Fudan University（中国）
  • Year and Date: 2015-12-14
  • Int'l Joint Research / Invited
• [Presentation] Beilinson conjecture for Rankin-Selberg products of modular forms (2015)
  • Author(s): 千田雅隆
  • Organizer: レギュレーター in ニセコ 2015
  • Place of Presentation: ヒルトンニセコビレッジ
  • Year and Date: 2015-09-07
  • Invited
• [Presentation] Rankin-Selberg L関数の特殊値に関するBeilinson予想について (2015)
  • Author(s): 千田雅隆
  • Organizer: 第60回代数学シンポジウム
  • Place of Presentation: 静岡大学
  • Year and Date: 2015-09-02
  • Invited
• [Funded Workshop] Arithmetic Geometry and Related Topics (2017)
• [Funded Workshop] Japan-Taiwan Joint conference on Number Theory 2016 (2016)
  • Place of Presentation: 国立台湾大学(台湾)
  • Year and Date: 2016-09-08
• [Funded Workshop] Workshop on Polylogarithms, MZVs and Mahler measures (2016)
  • Place of Presentation: 東北大学理学部
  • Year and Date: 2016-03-28