Generalization of Iwasawa theory through Galois doformation and search for new phenomena

• Project/Area Number: 26287005
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Osaka University
• Principal Investigator: Ochiai Tadashi 大阪大学, 理学研究科, 准教授 (90372606)
• Co-Investigator(Kenkyū-buntansha): 原 隆 東京電機大学, 未来科学部, 助教 (40722608) ; 下元 数馬 日本大学, 文理学部, 教授 (70588780) ; 安田 正大 大阪大学, 理学研究科, 准教授 (90346065)
• Project Period (FY): 2014-04-01 – 2020-03-31
• Project Status: Completed (Fiscal Year 2019)
• Budget Amount: ¥16,250,000 (Direct Cost: ¥12,500,000、Indirect Cost: ¥3,750,000); Fiscal Year 2018: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000); Fiscal Year 2017: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2016: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000); Fiscal Year 2015: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000); Fiscal Year 2014: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
• Keywords: 岩澤理論 / 肥田理論 / p進モジュラー形式 / p進L関数 / Euler形式 / Selmer群 / Euler系 / 保型L関数 / ジーゲルモジュラー / ジーゲルモジュラー形式 / p進L函数 / Coleman写像 / Coleman変形

Outline of Final Research Achievements

With this grant, I executed the following projects
(I) Euler system thheory over deformation rings with singularity, (II) Iwasawa theory for GSp(4), (III) Iwasawa theory for Coleman families, (IV) Iwasawa theory for CM fields and CM modular forms, (V) functional equation of Selmer group in noncommutative Iwasawa theory, (VI) Euler system theory for higher rank Galois representation. Also, I organized an international workshop for the generalization of p-adic L-function.

Academic Significance and Societal Importance of the Research Achievements

現代の整数論においてゼータ関数は最も重要な中心的研究対象であり、そのゼータ関数の研究において現在最も有望で最も重要な理論である岩澤理論の研究に向き合っている。ガロワ表現の変形の一般化が大事だという視点で切り込んでおり、その方向で理論に大きなインパクトがある現象を研究してきた。Euler系を使ってSelmer群を抑える理論を特異点のある変形で確率したこと、Euler系からp進L関数を構成するColeman写像の理論の複数の方向性での一般化を得たことなどをはじめとして本研究ではいくつかの学術的意義の深い結果が得られた。

Research Products

• [Int'l Joint Research] Universite de Paris(フランス)
• [Int'l Joint Research] University colledge Dublin(アイルランド)
• [Int'l Joint Research] IIT Kanpur(インド)
• [Int'l Joint Research] パリ第７大学(フランス)
• [Int'l Joint Research] University colledge Dublin(アイルランド)
• [Int'l Joint Research] IIT Kanpur(インド)
• [Int'l Joint Research] Koc University(トルコ)
• [Int'l Joint Research] IIT Kanpur(インド)
• [Int'l Joint Research] University of Paris 7(フランス)
• [Int'l Joint Research] Koc University(トルコ)
• [Int'l Joint Research] IIT Kanpur(インド)
• [Int'l Joint Research] Paris University 7(フランス)
• [Int'l Joint Research] Indian Institutt of Technology Kanpur(インド)
• [Int'l Joint Research] University of Paris 7(フランス)
• [Int'l Joint Research] University of Saint=Etienne(フランス)
• [Int'l Joint Research] Koch University(トルコ)
• [Journal Article] Specialization Method in Krull Dimension Two and Euler System Theory over Normal Deformation Rings (2019)
  • Author(s): Kazuma Simomoto, Tadashi Ochiai
  • Journal Title: Annales mathematiques du Quebec Volume: 43 Issue: 2 Pages: 357-409
  • DOI: 10.1007/s40316-018-0099-0
  • Peer Reviewed
• [Journal Article] An elementary proof of Cohen-Gabber theorem in the equal characteristic p>0 case (2018)
  • Author(s): Kazuma Shimomoto and K. Kurano
  • Journal Title: Tohoku Math Journal Volume: 70 Pages: 377-389
  • Peer Reviewed
• [Journal Article] The cyclotomic Iwasawa Main Conjecture for Hilbert cuspforms with complec multiplication (2018)
  • Author(s): Tadasih Ochiai and Takashih Hara
  • Journal Title: Kyoto journal of Mathematics Volume: 58 Pages: 1-100
  • Peer Reviewed
• [Journal Article] On twists of modules over non-commutative Iwasawa algebras (2016)
  • Author(s): Somnath Jha, Tadash Ochiai, Gergely Zabradi
  • Journal Title: Algebra and Number Theory Journal) Volume: 未定
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] 岩澤理論の高次元をとりまく風景 (2016)
  • Author(s): 落合理
  • Journal Title: 第60回代数学シンポジウム報告集 Volume: 60
• [Journal Article] p-adic L-function and Coleman map for Hida families of GSp_4 (2014)
  • Author(s): Francesco Lemma, Tadashi Ochiai
  • Journal Title: American Journal of Mathematics Volume: 136 Issue: 3 Pages: 729-760
  • DOI: 10.1353/ajm.2014.0022
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] Endoscopic congruences and adjoint L-values for GSp(4) (2020)
  • Author(s): Tadashi Ochiai
  • Organizer: 保型形式とL関数の解析的, 幾何的, p進的研究
  • Int'l Joint Research / Invited
• [Presentation] Endoscopic congruences and adjoint L-values for GSp(4) (2019)
  • Author(s): Tadashi Ochiai
  • Organizer: Recent advances in the arithmetic of Galois representations
  • Int'l Joint Research / Invited
• [Presentation] Congruence of modular forms and Special values of adjoint L-functions (2019)
  • Author(s): Tadashi Ochiai
  • Organizer: Number theory colloquium at Indian Statistical Institute (Kolkata)
  • Invited
• [Presentation] GSp(4) の内視的な保型表現の mod p 合同とadjoint L 関数の特殊値 (2019)
  • Author(s): 落合理
  • Organizer: 2019大分佐賀整数論研究集会
  • Invited
• [Presentation] Iwasawa Main Conjecture for Galois deformation spaces and glueing of Euler systems I, II (2019)
  • Author(s): Tadashi Ochiai
  • Organizer: CMC special weeks on number theory
  • Int'l Joint Research / Invited
• [Presentation] Congruences between endoscopic representations and non-endoscopic representationsof GSp(4)modulo adjoint L-values (2019)
  • Author(s): Tadashi Ochiai
  • Organizer: Workshop on the congruence ideals and p-adic L-functions
  • Int'l Joint Research / Invited
• [Presentation] Congruences endoscopiques et fonction L adjointe pour GSp(4) (2019)
  • Author(s): Tadashgi Ochiai
  • Organizer: Seminaire: Groupes Reductifs et Formes Automorphes (University of Paris 7)
  • Invited
• [Presentation] Congruences of modular forms and special values of L-functions (2019)
  • Author(s): Tadashgi Ochiai
  • Organizer: Algebra and Number Theory Seminar (University College Dublin)
  • Invited
• [Presentation] Overview of p-adic L-function (2018)
  • Author(s): Tadashgi Ochiai
  • Organizer: Special Values of Automorphic L-functions and Associated p-adic L-Functions (Casa Mathematica Oxaca)
  • Int'l Joint Research
• [Presentation] Endoscopic congruences modulo adjoint L-values for GSp(4) (2018)
  • Author(s): Tadashgi Ochiai
  • Organizer: Workshop on L-functions, Langlands functoriality and Trace formula, including relative aspects
  • Invited
• [Presentation] La conjecture principale d'Iwasawa pour les families p-adiques de formes modulaires (2018)
  • Author(s): Tadashgi Ochiai
  • Organizer: Seminaire de la theorie des nombres（Universite de Caen)
  • Invited
• [Presentation] La conjecture principale d'Iwasawa pour les families de formes modulaires p-adiques (2018)
  • Author(s): Tadashgi Ochiai
  • Organizer: Seminaire de la theorie des nombres （Universite de Bordeaux)
  • Invited
• [Presentation] Iwasawa theory for p-adic family of modular forms (2017)
  • Author(s): Tadasih Ochiai
  • Organizer: Tata Institute of Fundamental Research
  • Invited
• [Presentation] Iwasawa Main conjecture for p-adic family of elliptic modular forms (2017)
  • Author(s): Tadasih Ochiai
  • Organizer: Iwasawa 2017
  • Int'l Joint Research / Invited
• [Presentation] 高階数のp擬通常的ガロワ変形の岩澤主予想について (2017)
  • Author(s): Tadashi Ochiai
  • Organizer: Regulators in Niseko 2017
  • Int'l Joint Research / Invited
• [Presentation] Iwasawa Main Conjecture for nearly ordinary deformations of higher rank (2017)
  • Author(s): Tadashih Ochiai
  • Organizer: workshop on p-adic L-functions and algebraic cycles
  • Int'l Joint Research / Invited
• [Presentation] La conjecture principale d'Iwasawa pour les deformations galoisiennes p-adiques et le systeme d'Euler (2017)
  • Author(s): Tadashi Ochiai
  • Organizer: Seminaire: Groupes Reductifs et Formes Automorphes (Universite Paris 7)
  • Invited
• [Presentation] Iwasawa theory for Coleman families (2017)
  • Author(s): Tadashih Ochiai
  • Organizer: number theory seminar at Heidelberg University
  • Invited
• [Presentation] La conjecture principale d'Iwasawa pour les families p-adiques des formes paraboliques (2017)
  • Author(s): Tadashi Ochiai
  • Organizer: Seminaire de la theorie des nombres ( Universite de Grenoble)
  • Invited
• [Presentation] La conjecture principale d'Iwasawa pour les deformations galoisiennes p-adiques de rang general (2017)
  • Author(s): Tadashi Ochiai
  • Organizer: Seminaire de la theorie des nombres (Universite Paris 13)
  • Invited
• [Presentation] 岩澤理論の高次元化と関連する諸問題 (2017)
  • Author(s): 落合理
  • Organizer: 京都大学談話会
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] p進モジュラー形式のColeman変形族とゼータ元 (2016)
  • Author(s): 落合理
  • Organizer: 早稲田大学整数論研究集会 2016
  • Place of Presentation: 早稲田大学
  • Year and Date: 2016-03-22
  • Invited
• [Presentation] local Bertini theorem and Euler system bound over Cohen-Macaulay deformation (2016)
  • Author(s): Tadashi Ochiai
  • Organizer: Sendai International Conference on Arithmetic Geometry 2016
  • Place of Presentation: 東北大学
  • Year and Date: 2016-01-29
  • Int'l Joint Research / Invited
• [Presentation] special values of L-funcctions and p-adic L-function for modular forms (2016)
  • Author(s): Tadashi Ochiai
  • Organizer: IIT Kanpur department seminar
  • Place of Presentation: IIT Kanpur (India)
  • Invited
• [Presentation] 岩澤理論の高次元をとりまく風景 (2015)
  • Author(s): 落合理
  • 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
• [Presentation] Remark on the cyclotomic Iwasawa Main Conjecture for ordinary Hilbert cusp forms (2014)
  • Author(s): Tadashi Ochiai
  • Organizer: Automorphic Forms and Arithmetic (a satellite to the ICM, conference number 28)
  • Place of Presentation: Pohang Mathematics Institute/Korea
  • Year and Date: 2014-08-25
  • Invited
• [Presentation] Coleman map for non-ordinary families of modular Galois representations (2014)
  • Author(s): Tadashi Ochiai
  • Organizer: The sixth annual Pan Asian Number Theory Conference (a satellite to the ICM, conference number 22)
  • Place of Presentation: Pohang Mathematics Institute/Korea
  • Year and Date: 2014-08-05
  • Invited
• [Presentation] On the Iwasawa Main Conjecture for CM Hilbert cuspforms (2014)
  • Author(s): Tadashi Ochiai
  • Organizer: p-adic aspects of modular forms
  • Place of Presentation: IISER Pune /India
  • Year and Date: 2014-06-20
  • Invited
• [Presentation] On the cyclotomic Iwasawa Main Conjecture for ordinary Hilbert cusp forms (2014)
  • Author(s): Tadashi Ochiai
  • Organizer: Mini-Workshop on Iwasawa theory
  • Place of Presentation: 京都大学理学部
  • Year and Date: 2014-04-04
  • Invited
• [Book] 現代整数論の風景 (2019)
  • Author(s): 落合理
  • Total Pages: 208
  • Publisher: 日本評論社
• [Book] 岩澤理論とその展望（下） (2016)
  • Author(s): 落合理
  • Total Pages: 384
  • Publisher: 岩波書店
• [Book] 岩澤理論とその展望（上） (2014)
  • Author(s): 落合理
  • Total Pages: 208
  • Publisher: 岩波書店
• [Remarks] 研究活動およびその他の仕事
  • URL: http://www4.math.sci.osaka-u.ac.jp/~ochiai/work.html
• [Remarks] 研究活動およびその他の仕事
  • URL: http://www.math.sci.osaka-u.ac.jp/%7Eochiai/work.html
• [Remarks] 研究活動およびその他の仕事
  • URL: http://www.math.sci.osaka-u.ac.jp/~ochiai/work.html