On the relative theory of Tamagawa numbers
On the relative theory of Tamagawa numbers
复制标题
玉川数的相对论
DOI：
10.1090/s0002-9904-1964-11140-x
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发表时间：
1964
影响因子：
1.3
通讯作者：
T. Ono
中科院分区：
文献类型：
--
作者：
T. Ono
本文概述了作者最近在半单代数群的玉川数的相关理论方面的一些工作。细节和应用将在其他地方发布。设k是Q上的有限次代数数域，G是定义在k上的连通半单代数群。G允许一个且仅一个定义在k上的单连通覆盖（G，ir）（k上的同构除外）。用g表示k/ky的伽罗瓦群，其中k表示k的代数闭包。然后有限交换群Ker T得到一个g-模的结构。我们的目的是用模Kerw的一些不变量来表示Isogawa的Tamagawa数，即数r（γ r）=r（G）/r（G）.关于玉川数的概念，参见[ó]，[2]。我们用z表示k在k的一个位置v的完备化，如果v是非阿基米德的，则使用z> = p。我们也使用代数群的Galois上同调的标准符号[1]。我们说定义在k上的代数群A是（K）型的，如果对所有p H（k^y ^4）= 0且映射H（k，A）->Hv H（k，A）是单射的。Kneser证明了定义在k上的每个单连通半单群都是（K）型的，并对许多经典群证明了这一点[4]。
This note is an outline of some of the author's recent work on the relative theory of Tamagawa numbers of semisimple algebraic groups. The details and applications will be published elsewhere. Let k be an algebraic number field of finite degree over Q, let G be a connected semisimple algebraic group defined over k. G admits one and only one simply connected covering (G, ir) defined over k (except for isomorphisms over k). Denote by g the Galois group of k/ky where k means the algebraic closure of k. Then the finite commutative group Ker T obtains a structure of a g-module. Our purpose is to express the Tamagawa number of the isogeny w, i.e. the number r(7r)=r(G)/r(G), in terms of some invariants of the module Ker w. For the notion of the Tamagawa number, see [ó], [2]. We denote by kv the completion of k at a place v of k and use z> = p if v is nonarchimedean. We also use the standard notation in the Galois cohomology of algebraic groups [ l ] . We say that an algebraic group A defined over k is of type (K) if H(k^y ^4) = 0 for all p and the map H (k, A) —>Hv H(kv, A) is injective. Kneser has conjectured that every simply connected semisimple group defined over k is of type (K) and has verified it for many classical groups [4].