Surjectivity of the global-to-local map defining a Selmer group
Surjectivity of the global-to-local map defining a Selmer group
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定义 Selmer 群的全局到局部映射的满射性
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发表时间：
2010
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通讯作者：
M. Nagata
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作者：
M. Nagata
“Selmer 群”一词首次在 20 世纪 60 年代使用，指的是某个群，该群被证明可用于研究在数域上定义的椭圆曲线的算术性质。经典定义很容易扩展到在数域上定义的阿贝尔簇。稍后我们会回忆起这个定义。多年来，人们发现可以在更普遍的背景下定义此类对象。这样的定义出现在布洛赫-加藤猜想（在[BK]中）的表述中以及在岩泽猜想（在[Gr1]和[Gr2]中）的概括中。粗略地说，Selmer 群是全局伽罗瓦上同调群的子群，通过对余循环类施加某种局部限制来定义。在上面引用的例子中，这些当地条件采取了相当具体的形式。然而，在本文中，Selmer 群将被简单地定义为一种非常通用的映射类型的核心，我们将其称为“全局到局部”。
The term “Selmer group” was first used in the 1960s to refer to a certain group that proved to be useful in studying the arithmetic properties of an elliptic curve defined over a number field. The classical definition is easily extended to abelian varieties defined over number fields. We will recall that definition later. Over the years, it was found that one could define such objects in a much more general context. Such definitions occur in the formulation of the Bloch-Kato conjecture (in [BK]) as well as in generalizations of a conjecture of Iwasawa (in [Gr1] and [Gr2]). Roughly speaking, a Selmer group is a subgroup of a global Galois cohomology group defined by imposing local restrictions of some kind on the cocycle classes. These local conditions take a rather specific form in the examples cited above. However, in this paper, a Selmer group will be defined simply as the kernel of a very general type of map which we will call “global-to-local”.