Study on overconvergent isocrystals

• Project/Area Number: 25400008
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: The University of Tokyo
• Principal Investigator: Shiho Atsushi 東京大学, 大学院数理科学研究科, 教授 (30292204)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000); Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2015: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2014: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2013: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: アイソクリスタル / エタール基本群 / ドラーム基本群 / 対数的代数多様体 / p進微分方程式 / チャーン類 / 収束アイソクリスタル / 過収束アイソクリスタル

Outline of Final Research Achievements

We checked the compatibility of the algebraization functor from the category of log overconvergent isocrystals to that of modules with integrable connection with the formation of tensor products under certain assumption. We proved a partial generalization of Ogus-Vologodsky correspondence on schemes in which p is nilpotent. We proved under certain assumption a conjecture of de Jong on isocrystals on algebraic varieties with trivial etale fundamental group. We constructed in purely algebraic way the homotopy exact sequence (including the first injection) for certain quotient of de Rham fundamental groups of certain log algebraic varieties. We studied in purely algebraic way the relative pro-unipotent de Rham fundamental groups for certain morphisms of log algebraic varieties, and gave a p-adic application of it. We studied to fix errors in the proof of fundamental theorems in the theory of p-adic differential equations.

Research Products

• [Int'l Joint Research] ベルリン自由大学(ドイツ)
• [Int'l Joint Research] パドヴァ大学(イタリア)
• [Int'l Joint Research] エクセター大学(英国)
• [Int'l Joint Research] カリフォルニア大学サンディエゴ校(米国)
• [Int'l Joint Research] ベルリン自由大学(ドイツ)
• [Int'l Joint Research] パドヴァ大学(イタリア)
• [Journal Article] Notes on Generalizations of Local Ogus-Vologodsky Correspondence (2015)
  • Author(s): Atsushi Shiho
  • Journal Title: Journal of Mathematical Sciences The University of Tokyo Volume: 22 Pages: 793-875
  • NAID: 120006906142
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] On p-adic differential equation on semi-stabke variety II (2015)
  • Author(s): DiProietto, Valentina, Shiho,Atsushi
  • Journal Title: Manuscripta Mathematica Volume: 146 Issue: 1-2 Pages: 179-199
  • DOI: 10.1007/s00229-014-0691-9
  • Peer Reviewed
• [Presentation] Comparison of relatively unipotent log de Rham fundamental groups (2016)
  • Author(s): Atsushi Shiho
  • Organizer: Guest Seminar
  • Place of Presentation: ベルリン(ドイツ)
  • Year and Date: 2016-11-24
  • Invited
• [Presentation] Isocrystals on simply connected varieties (2016)
  • Author(s): Atsushi Shiho
  • Organizer: East Asia Number Theory Conference
  • Place of Presentation: 天津(中国)
  • Year and Date: 2016-08-24
  • Int'l Joint Research / Invited
• [Presentation] Isocrystals on simply connected varieties (2016)
  • Author(s): 志甫 淳
  • Organizer: p進コホモロジーと数論幾何学
  • Place of Presentation: 東京電機大学（東京都足立区）
  • Year and Date: 2016-07-29
  • Invited
• [Presentation] Isocrystals on simply connected varieties (2016)
  • Author(s): Atsushi Shiho
  • Organizer: 2016 Seoul-Tokyo Conference on Number Theory
  • Place of Presentation: ソウル(韓国)
  • Year and Date: 2016-06-16
  • Int'l Joint Research / Invited
• [Presentation] Convergent isocrystals on simply connected varieties (2015)
  • Author(s): Atsushi Shiho
  • Organizer: Workshop on recent trends in p-adic cohomology
  • Place of Presentation: Imperial College London
  • Year and Date: 2015-03-25
  • Invited
• [Presentation] On differential Artin conductor of overconvergent iscocrystals
  • Author(s): 志甫 淳
  • Organizer: Arithmetic and Algebraic Geometry 2014
  • Place of Presentation: 東京大学
  • Invited
• [Presentation] On homotopy exact sequence for log de Rham fundamental groups
  • Author(s): 志甫 淳
  • Organizer: p-adic cohomology and its applications 2014
  • Place of Presentation: 東北大学
  • Invited
• [Presentation] On the differential Artin conductor of overconvergent isocrystals
  • Author(s): Atsushi Shiho
  • Organizer: Seminario Padova "geometria algebrica aritmetica"
  • Place of Presentation: パドヴァ大学(イタリア)
  • Invited
• [Presentation] p進微分方程式と係数つきリジッドコホモロジー
  • Author(s): 志甫 淳
  • Organizer: 談話会
  • Place of Presentation: 大阪大学
  • Invited
• [Presentation] On a generalization of local Ogus-Vologodsky correspondence
  • Author(s): 志甫 淳
  • Organizer: 豊田中央研数学コロキウム
  • Place of Presentation: 豊田中央研究所
  • Invited
• [Presentation] On restriction of overconvergent isocrystalｓ
  • Author(s): 志甫 淳
  • Organizer: 豊田中央研数学コロキウム
  • Place of Presentation: 豊田中央研究所
  • Invited
• [Presentation] p進数とp進微分方程式
  • Author(s): 志甫 淳
  • Organizer: 豊田中央研数学コロキウム
  • Place of Presentation: 豊田中央研究所
  • Invited