Arithmetic cohomology over local fields

局部域上的算术上同调

• 批准号: 18K03258
• 负责人: ガイサ トーマス
• 金额: $ 2.75万
• 依托单位: Rikkyo University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2018
• 资助国家: 日本
• 起止时间: 2018-04-01 至 2024-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-18K03258/
• 关键词: Weil etale cohomology; Local class field theory; Duality; Locally compact groups; One-motives; Birch Swinnerton Dyer; Brauer group; Weil-etale cohomology; Tamagawa number formula; BSD conjecture; Brauer Manin obstruction; Arithmetic cohomology

In my ongoing project on Weil-etale cohomology for schemes over henseliandiscrete valuation rings, finite fields, and arithmetic schemes, I was able to finalize publication of the following results:Joint with B.Morin, we outline the definition of a Weil-etale cohomology theory for varieties over local fields which satisfy a Pontrjagin duality theory. The groups are objects of the heart of the t-structure on the derived category of locally compact abelian groups (this work is accepted for publication and published online).As an application we prove results on class field theory over local fields, generalizing and improving work of S.Saito and Yoshida. We give an integral model for the fundamental group, and some extra information on the kernel of the reciprocity map (a preprint is submitted for publication).In joint work with T.Suzuki, we generalized our work on the Weil-etale version of the Birch and Swinnerton-Dyer conjecture to one-motives. In particular, our work gives a new proof of the Tamagawa number formula of Oda (this is published).

In my ongoing project on Weil-etale cohomology for schemes over henseliandiscrete valuation rings, finite fields, and arithmetic schemes, I was able to finalize publication of the following results:Joint with B.Morin, we outline the definition of a Weil-etale cohomology theory for varieties over local fields which satisfy a Pontrjagin duality theory.这些小组是T结构的核心对象，在本地紧凑的Abelian群体的派生类别中（这项工作被接受用于出版并在线发布）。作为一个应用程序，我们证明了对本地领域的类领域理论的结果，从而推广和改善了S.Saito和Yoshida的工作。我们为基本组提供了一个不可或缺的模型，并在互惠图的内核上提供了一些额外的信息（提交了预印本以供出版）。在与T.Suzuki的联合合作中，我们对桦木版本的Weil-Etale版本进行了概括，并将其推广到Birch和Swinnerton-Dyer-Dyer-dyer的猜想。特别是，我们的工作提供了oda的Tamagawa数字公式的新证明（这已发布）。

项目成果

Motives in Tokyo 2023

2023 年东京奥运会的动机

• 发表时间: 2023

PONTRYAGIN DUALITY FOR VARIETIES OVER p-ADIC FIELDS

p-ADIC 领域品种的庞特里亚金二元性

• DOI: 10.1017/s1474748022000469
• 发表时间: 2023
• 期刊: Journal of the Institute of Mathematics of Jussieu
• 影响因子: 0.9
• 作者: Geisser Thomas H.;Morin Baptiste
• 通讯作者: Morin Baptiste

Heidelberg University/Wuppertal Univesity(ドイツ)

海德堡大学/伍珀塔尔大学（德国）

ボルドー大学(フランス)

波尔多大学（法国）

Brauer groups and Neron-Severi groups of surfaces over finite fields

有限域上的表面布劳尔群和 Neron-Severi 群

• 发表时间: 2022
• 作者: Geisser Thomas H.;Schmidt Alexander;Thomas Geisser
• 通讯作者: Thomas Geisser