Arithmetic cohomology over local fields

在我正在进行的关于henselianddiscrete估值环，有限域和算法格式上的Weil-etale上同调的项目中，我能够完成以下结果的发表：与B.Morin联合，我们概述了局部域上满足Pontrjagin对偶理论的变量的Weil-etale上同调理论的定义。这些群是在局部紧阿贝尔群的派生范畴上t结构的核心对象（这项工作被接受发表并在线发表）。作为应用，我们在局部场上证明了类场论的结果，推广和改进了斋藤和吉田的工作。我们给出了基本群的积分模型，并给出了互易映射核的一些额外信息（已提交预印本）。在与T.Suzuki的联合工作中，我们将我们对伯奇猜想和斯温纳顿-戴尔猜想的韦尔-埃塔尔版本的工作推广到单一动机。特别地，我们的工作给出了Oda的Tamagawa数公式的一个新的证明（已发表）。

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).