Galois Representations in Arithmetic Algebraic Geometry: An introduction to Kato's Euler systems
Galois Representations in Arithmetic Algebraic Geometry: An introduction to Kato's Euler systems
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算术代数几何中的伽罗瓦表示：加藤欧拉系统简介
DOI：
10.1017/cbo9780511662010.011
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发表时间：
1998
期刊：
影响因子：
--
通讯作者：
A. Scholl
中科院分区：
文献类型：
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作者：
A. Scholl
简介 在会议中，进行了一系列专门讨论加藤在附加模形式的伽罗瓦表示的岩泽理论方面的工作的演讲。本笔记主要致力于解释关键成分，即加藤构建的欧拉系统，首先在 K 2 群模曲线中，然后在伽罗瓦上同调中使用陈省级图。本材料主要基于加藤和作者在研讨会上的演讲，以及加藤 1993 年在剑桥的一系列讲座。在一篇配套论文 [29] 中，鲁宾解释了如何在给定有关欧拉系统的足够信息的情况下，只要适当的 L 函数非零，就可以证明 Selmer 群的非常一般的有限性定理（有关椭圆的精确结果，请参阅他的论文第 §8 节） 曲线）。部分是由于篇幅的原因，部分是由于作者缺乏理解，这些注释的范围是有限的。有两个特殊限制。首先，我们只证明关键的互易律（下面的定理 3.2.3），它允许人们在素数 p 良好约简的情况下计算欧拉系统在双指数映射下的图像（实际上，由于第 2.1 节末尾解释的愚蠢原因，我们还必须假设 p 是奇数）。其次，我们对大于 2 的权重形式所附加的伽罗瓦表示的情况只字不提。对于最一般的结果，读者需要查阅预印本 [17] 和 Kato 未来的论文。 Kato 的 K 2 Euler 系统起源于 Beilinson [1] 的工作（另请参阅 [30] 了解初学者的处理方法）。
Introduction In the conference there was a series of talks devoted to Kato's work on the Iwasawa theory of Galois representations attached to modular forms. The present notes are mainly devoted to explaining the key ingredient, which is the Euler system constructed by Kato, first in the K 2 -groups of modular curves, and then using the Chern class map, in Galois cohomology. This material is based mainly on the talks given by Kato and the author at the symposium, as well as a series of lectures by Kato in Cambridge in 1993. In a companion paper [29] Rubin explains how, given enough information about an Euler system, one can prove very general finiteness theorems for Selmer groups whenever the appropriate L -function is non-zero (see §8 of his paper for precise results for elliptic curves). Partly because of space, and partly because of the author's lack of understanding, the scope of these notes is limited. There are two particular restrictions. First, we only prove the key reciprocity law (Theorem 3.2.3 below), which allows one to compute the image of the Euler system under the dual exponential map, in the case of a prime p of good reduction (actually, for stupid reasons explained at the end of §2.1, we also must assume p is odd). Secondly, we say nothing about the case of Galois representations attached to forms of weight greater than 2. For the most general results, the reader will need to consult the preprint [17] and Kato's future papers. Kato's K 2 Euler system has its origins in the work of Beilinson [1] (see also [30] for a beginner's treatment).