Theory on singular homology of cyclyes using sheafy Banach algebras

2023 Fiscal Year Final Research Report

• Project/Area Number: 21K13763
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: University of Tsukuba
• Principal Investigator: Mihara Tomoki 筑波大学, 数理物質系, 助教 (90827106)
• Project Period (FY): 2021-04-01 – 2024-03-31
• Keywords: リジッド幾何 / p進解析 / ガロア表現 / 積分

Outline of Final Research Achievements

A main target of the research of number theory is the algebraic object called Galois groups, which controls the symmetry of solutions of an equation of integrs. It is typical to use a tool called Galois representation, which is an application of linear algebra, in order to study Galois groups, For this purpose, one needs to construct various Galois representartions and verify their non-triviality.
This research is devoted to a construction of Galis representations realisedas a geometric invariant called singular homology defined for adic spaces through the construction of a geometric object called a cosimplicial perfectoid space.
In addition, we developed a theory on integration along the singular homology in order to verify the non-triviality of the Galois representations.

Free Research Field

数論幾何

Academic Significance and Societal Importance of the Research Achievements

先述したように新たにガロア表現を豊富に得られ、また積分論により非自明性の検証方法が与えられている。期待したようにこれらがガロア群の構造に関する研究に実際に役立てられれば、整数論の目標の１つである整数の方程式の解の対称性の記述に貢献することとなる。
また今回はパーフェクトイド空間という非常に良い無限次元空間を完備群環という道具を用いて構成できたため、リジッド幾何学の壁の１つである無限次元空間のsheafy性の判定の複雑さという問題にも今後の応用を期待している。