On the Euler-Poincaré characteristics of finite dimensional p-adic galois representations
On the Euler-Poincaré characteristics of finite dimensional p-adic galois representations
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有限维p进伽罗瓦表示的欧拉-庞加莱特性
DOI：
10.1007/s10240-001-8189-x
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发表时间：
2001
期刊：
影响因子：
--
通讯作者：
J.
中科院分区：
文献类型：
--
作者：
J. Coates;R. Sujatha;J.
设p是素数，V是p进数域Qp上的有限维向量空间.设GL（V）是V的Qp-线性自同构群，GV表示GL（V）的一个紧子群，则GV是一个p-adic李群。我们记Hi（GV，V）为GV作用在V上的上同调群，它们由连续上链定义，其中V具有p-adic拓扑.我们可以说，如果对所有i ∈ 0，Hi（GV，V）= 0，则GV的表示V具有零GV-上同调。更一般地说，如果V是GV在Qp上的任何有限维连续表示，那么如果Hk（GV，V）= 0（对于所有k 0），我们可以说V具有消失的GV-上同调。第一个有趣的例子，这样的V与消失GV-上同调出现在算术几何是由于塞尔[26]，其中GV是伽罗瓦的形象，在泰特模的自同构群的阿贝尔簇定义在一个有限的扩张Q。本文的目的之一是建立一个广泛的新的例子所产生的光滑真代数簇的étale上同调定义在有限的扩展Qp，并具有潜在的良好的减少。在本文中，F将始终表示Qp的有限扩张。设Y是定义在这样的域F上的光滑真簇。像往常一样，我们写YQp是为了将Y的标量扩展到Qp的代数闭包Qp。对于每个i ≠ 0，令
Let p be a prime number, and let V be a finite dimensional vector space over the field Qp of p-adic numbers. We write GL (V) for the group of Qp-linear automorphisms of V. Let GV denote a compact subgroup of GL (V), so that GV is a p-adic Lie group. We write Hi (GV, V) for the cohomology groups of GV acting on V, which are defined by continuous cochains, where V is endowed with the p-adic topology. We shall say that our representation V of GV has vanishing GV-cohomology if Hi (GV, V)= 0 for all i⩾ 0. More generally, if V is any finite dimensional continuous representation of GV over Qp, we shall say that V has vanishing GV-cohomology if Hk (GV, V)= 0 for all k⩾ 0. The first interesting example of such V with vanishing GV-cohomology which occur in arithmetic geometry is due to Serre [26], where GV is the image of Galois in the automorphism group of the Tate module of an abelian variety defined over a finite extension of Q. One of the aims of the present paper is to establish a broad class of new examples arising from the étale cohomology of smooth proper algebraic varieties defined over a finite extension of Qp, and having potential good reduction. Throughout this paper, F will always denote a finite extension of Qp. Let Y be a smooth proper variety defined over such a field F. As usual, we write YQp for the extension of scalars of Y to the algebraic closure Qp of Qp. For each i⩾ 0, let