p-adic differential equations on p-adic analytic spaces

• Project/Area Number: 17K14161
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Nagoya University
• Principal Investigator: Ohkubo Shun 名古屋大学, 多元数理科学研究科, 講師 (20755160)
• Project Period (FY): 2017-04-01 – 2022-03-31
• Project Status: Completed (Fiscal Year 2021)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2020: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2017: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: p進微分方程式 / 対数的増大度 / ピカールフックス方程式 / Gauss-Manin接続 / Picard-Fuchs equation

Outline of Final Research Achievements

As a result of our research, we affirmatively prove Chiarellotto-Tsuzuki conjecture of logarithmic growth of solutions of p-adic differential equations. A paper containing this result appeared in Composition Mathematica in 2021. We also prove that a variant of Chiarellotto-Tsuzuki conjecture in a way compatible with p-adic local monodromy conjecture. We also extend a decomposition theorem of p-adic differential equations over complete valuation fields proved by Kedlaya-Xiao.

Academic Significance and Societal Importance of the Research Achievements

p進微分方程式は、2010年以降に、Kedlaya, Baldassarri, Poineau, Pulitaらによる解の収束半径の理論の完成によって大きく進歩した。p進微分方程式の局所理論における残る大きな課題は、解の対数的増大度の研究であった。本研究では、その基本予想であるChiarellotto-Tsuzuki予想を肯定的に解決し、p進微分方程式の理論の応用への道を開くことができた。本予想は、フロベニウス構造という代数的情報をp進微分方程式の解の対数的増大度という解析的情報を研究をつなぐ橋である。今後は、この橋を使って、代数体上の微分方程式の大域的性質の研究が進展することが期待される。

Research Products

• [Journal Article] Logarithmic growth filtrations for -modules over the bounded Robba ring (2021)
  • Author(s): Ohkubo Shun
  • Journal Title: Compositio Mathematica Volume: 157 Issue: 6 Pages: 1265-1301
  • DOI: 10.1112/s0010437x21007107
  • Peer Reviewed
• [Journal Article] Logarithmic growth filtrations for (\varphi,\nabla)-modules over the bounded Robba ring (2021)
  • Author(s): Shun Ohkubo
  • Journal Title: Compositio Mathematica Volume: －
  • Peer Reviewed
• [Journal Article] A note on logarithmic growth of solutions of p-adic differential equations without solvability (2019)
  • Author(s): Shun Ohkubo
  • Journal Title: Mathematical Research Letters Volume: 26 Issue: 5 Pages: 1527-1557
  • DOI: 10.4310/mrl.2019.v26.n5.a13
  • Peer Reviewed
• [Presentation] A note on the convergence Newton polygons of p-adic differential equations in the regular singular case (2021)
  • Author(s): 大久保俊
  • Organizer: Arithmetic geometry research report meeting 2021
  • Invited
• [Presentation] Logarithmic growth f.iltrations for (φ, ∇)-modules over the bounded Robba ring (2019)
  • Author(s): Shun Ohkubo
  • Organizer: RIMS Workshop Algebraic Number Theory and Related Topics
  • Invited
• [Presentation] Logarithmic growth filtrations for $(\varphi,\nabla)$-modules over the bounded Robba ring (2018)
  • Author(s): Shun Ohkubo
  • Organizer: p-adic cohomology and arithmetic geometry 2018
  • Int'l Joint Research / Invited
• [Presentation] On the logarithmic growth of solutions of p-adic differential equations (2018)
  • Author(s): Shun Ohkubo
  • Organizer: UK-Japan Winter School 2018 on Number Theory
  • Int'l Joint Research / Invited
• [Presentation] On the logarithmic growth of solutions of p-adic differential equations (2017)
  • Author(s): Shun Ohkubo
  • Organizer: p-adic cohomology and arithmetic geometry
  • Invited