Study on p-adic perverse sheaves and p-adic representations of fundamental groups

2020 Fiscal Year Final Research Report

• Project/Area Number: 15H02050
• Research Category: Grant-in-Aid for Scientific Research (A)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• 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): 2015-04-01 – 2020-03-31
• Keywords: p進Hodge理論 / p進Simpson対応 / p進表現 / 整p進Hodge理論 / (φ,Γ)加群

Outline of Final Research Achievements

We studied p-adic Hodge theory for p-adic perverse sheaves，p-adic Simpson correspondence, local Iwasawa theory for Lubin-Tate extensions, and integral p-adic Hodge theory. We obtained a comparison theorem for p-adic perverse sheaves with the stratification along a simple normal crossing divisor, an analogue of local p-adic correspondence for arithmetic fundamental group, and a formulation of a conjecture on multivariable explicit reciprocity law in local Iwasawa theory for Lubin-Tate extensions (under certain restrictions on Galois representations). We also developed a theory of relative Breuil-Kisin-Fargues modules as coefficients of the integral p-adic Hodge theory of Bhatt-Morrow-Scholze.

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である．p進Hodge理論は，微分形式や微分方程式を足がかりとして，このガロア表現から数論的情報を取り出す有効な手段を与えている．本研究では，このp進Hodge理論の適用範囲を拡大（局所系からperverse層へ，円分拡大からLubin-Tate拡大へ）あるいは精密化（有理係数から整係数へ）する成果を得た．