Study on p-adic Galois representations and p-adic etale local systems over a p-adic field

2023 Fiscal Year Final Research Report

• Project/Area Number: 20H01793
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: The University of Tokyo
• Principal Investigator: Tsuji Takeshi 東京大学, 大学院数理科学研究科, 教授 (40252530)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: p進Hodge理論 / p進Simpson対応 / prismaticコホモロジー / q接続 / 整p進Hodge理論

Outline of Final Research Achievements

We studied p-adic Simpson correspondence, integral p-adic Hodge theory, prismatic cohomology via q-Higgs fields, and multivariable local Iwaswa theory for Lubin-Tate extensions. We obtained a global comparison of cohomologies in integral p-adic Simpson correspondence, a functoriality in p-adic Simpson correspondence, a comparison between Ainf cohomology and prismatic cohomology with coefficients, a description of a prismatic crystal and its cohomology in terms of q-Higgs modules, and a new construction of the description of the Lubin-Tate Iwasawa module of a p-adic representation in terms of the Lubin-Tate (phi,Gamma)-module, via Lubin-Tate generalized Coleman power series.

Free Research Field

数論幾何学

Academic Significance and Societal Importance of the Research Achievements

数体上の代数多様体のエタールコホモロジーとして得られるガロア表現は，数論幾何学の研究における基本的な道具の一つとなっている．p進Hodge理論は，微分形式や微分方程式を用いて，ガロア表現から数論的情報を取り出す有効な手段となっている．本研究では，応用上重要な整コホモロジーの係数理論の基礎（具体的記述やコホモロジーの比較）や，非アーベルp進Hodge理論（p進Simpson対応）の代数多様体の間の写像に関するふるまいなどについての成果を得た．