On variation of Hodge-Tate structures
On variation of Hodge-Tate structures
复制标题
关于 Hodge-Tate 结构的变异
DOI：
10.1007/bf01443501
复制
发表时间：
1989
影响因子：
1.4
通讯作者：
Osamu Hyodo
中科院分区：
文献类型：
--
作者：
Osamu Hyodo
（0.0）在p进域的伽罗瓦群的p进表示理论中，对于某些“好的”p进表示，例如那些几何起源的p进表示，人们对证明Hodge-Tate结构的存在性很感兴趣。Hodge-Tate结构的概念是由Tare[11]提出的，最近Faltings[-4]证明了所谓的Hodge-Tate猜想，即在p进域上一个变元的p进上同调中存在Hodge-Tate结构。霍奇-塔特猜想处理常数束的上同群。为了进一步的进展，我们认为考虑某些“好”局部系统的上同群是很重要的。例如，Faltings[3]研究了p可分群对应的局部系统，并推导了模形式对应的p进表示的Hodge-Tate结构的存在性。本文的目的是给出Hodge-Tate结构的“变分”的形式：我们引入了在p进域上定义的光滑变分上的p进局部系统的“Hodge-Tate”概念。我们还给出了关于稳定性的部分结果；Hodge-Tate局部系统的上同群继承了Hodge-Tate结构。（0.1）我们将在（0.2）和式（0.3）中解释我们的结果。在此之前，我们先回顾一下经典理论（Tate Ill[参见Serre bb2010]）。设K为具有完全残差域的混合特征（0,p）的完全离散估值域。我们用GK= Gal （K/K）表示K的绝对伽罗瓦群，用IEp=/~表示K的代数闭包的p进补全。设V为具有连续gr作用的有限维qp向量空间。定义每个i~ Z的k向量空间D*(V)
(0.0) In the theory of p-adic representation of the Galois group of a p-adic field, there has been great interest in showing the existence of Hodge-Tate structures for certain" nice" p-adic representations, eg those of geometric origin. The notion of Hodge-Tate structure was introduced by Tare [I1] and recently Faltings [-4] proved the so-called Hodge-Tate conjecture, the existence of Hodge-Tate structure in the p-adic etale cohomology of a variety over a p-adic field. The Hodge-Tate conjecture treats the cohomology group of the constant sheaf. For further progress, we think that it is important to consider the cohomology groups of certain" nice" local systems. For example, Faltings [3] studied local systems corresponding to p-divisible groups and deduced the existence of Hodge-Tate structure for p-adic representations corresponding to modular forms. The aim of this paper is to give the formalism of the" variation" of Hodge-Tate structure: We introduce the notion of" Hodge-Tate" for p-adic local systems on smooth varieties defined over a p-adic field. We also give a partial result concerning the stability; the cohomology groups of Hodge-Tate local systems inherit a Hodge-Tate structure.(0.1) We shall explain our result in (0.2) and (0.3). Before doing this, we review the classical theory (Tate Ill] cf. also Serre [10]). Let K be a complete discrete valuation field of mixed characteristics (0, p) with perfect residue field. We denote by GK= Gal (K/K) the absolute Galois group of K and by IEp=/~ the p-adic completion of an algebraic closure of K. Let V be a finite dimensional Qp-vector space with continuous Gr-action. Define a K-vector space D*(V) for each i~ Z by