Deformation theory and local Shimura varieties

2023 Fiscal Year Final Research Report

• Project/Area Number: 22K20332
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Multi-year Fund
• Review Section: 0201:Algebra, geometry, analysis, applied mathematics,and related fields
• Research Institution: Tohoku University (2023); The University of Tokyo (2022)
• Principal Investigator: Ito Kazuhiro 東北大学, 理学研究科, 助教 (90962267)
• Project Period (FY): 2022-08-31 – 2024-03-31
• Keywords: 局所志村多様体 / G ディスプレイ / プリズマティックコホモロジー / プリズマティック F ゲージ

Outline of Final Research Achievements

I have studied the notion of G-displays in the context of the prismatic theory introduced by Bhatt-Scholze. Here G is a reductive group over the ring of p-adic integers. If G is the general linear group, then G-displays can be regarded as Breuil-Kisin modules in the integral p-adic Hodge theory. In particular, I constructed the universal deformation ring for G-displays over a perfect field of characteristic p. As an application, I determined the local structure of the integral model of a local Shimura variety with hyperspecial level structure. Moreover, I studied the relationship between prismatic F-gauges (introduced by Drinfeld and Bhatt-Lurie) and our G-displays, and proved that they are essentially the same objects when the base ring is regular.

Free Research Field

数論幾何学

Academic Significance and Societal Importance of the Research Achievements

局所ラングランズ対応への応用もあり，局所志村多様体と呼ばれるモジュライ空間は重要な対象である．本研究では G-ディスプレイを用いて，局所志村多様体をプリズムの圏上のモジュライ関手として解釈することで局所構造を決定した．局所志村多様体の先行研究は，ほとんどが付加構造付きの p 可除群のモジュライ空間である場合に限定されたものであったが，本研究の手法では任意の局所志村多様体を扱える．局所志村多様体に限らず，数論幾何学における多くのモジュライ空間がプリズムを用いて精密化されることが期待でき，本研究はその第一歩である．