p-adic cohomologies of arithmetic varieties

2018 Fiscal Year Final Research Report

• Project/Area Number: 25400023
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Tokyo Denki University
• Principal Investigator: NAKAJIMA Yukiyoshi 東京電機大学, 工学部, 教授 (80287440)
• Project Period (FY): 2013-04-01 – 2019-03-31
• Keywords: p進コホモロジー / 無限小コホモロジー / 重みフィルトレーション / Weil-Deligne群 / フロベニウス作用素 / モノドロミー作用素 / 固有半安定多様体

Outline of Final Research Achievements

There is a theory which solves arithmetic problems by geometric methods. This is called theory of arithmetic geometry. In this theory, there is a notion of the infinitesimal cohomology. Concretely speaking, we show that there exists a well-defined increasing filtration on the infinitesimal cohomology, which we call the limit weight filtration. We also prove that the Weil-Deligne group (this is generated by a Frobenius and a monodoromy operator) acts compatibly with the weight filtration. Moreover, we prove that this action commutes with the induced morphism on the infinitesimal cohomology by a morphism of geometry objects.

Free Research Field

数論幾何

Academic Significance and Societal Importance of the Research Achievements

有理数体や有限体や有理数体を素数pによって決まるp進距離で完備化したp進体を係数とする代数方程式の零点で定義される幾何的対象である代数多様体には数論的手法で得られる様々なコホモロジーと呼ばれる線型空間がある. それらのコホモロジーを使って, 元々の多様体の性質を調べる方法があるが, 本研究はp進体上の代数多様体の無限小コホモロジーと呼ばれるコホモロジーには極限重みフィルトレーションという意味深い構造があることを解明し, さらに基本的性質を調べた.