Non-abelian Fundamental Groups and Iwasawa Theory: Completed cohomology – a survey
Non-abelian Fundamental Groups and Iwasawa Theory: Completed cohomology – a survey
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非阿贝尔基本群和岩泽理论：完成的上同调——一项调查
DOI：
10.1017/cbo9780511984440.010
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发表时间：
2011
影响因子：
0.9
通讯作者：
M. Emerton
中科院分区：
文献类型：
--
作者：
Frank Calegari;M. Emerton
本笔记总结了$p$-进完备上同调理论。这种构造首次在文献[4]中被引入（尽管该文献对该理论的积分方面关注不足），然后在文献[2]和[6]中得到进一步发展。文献[4]和[2]可能会给人一种印象，即$p$-进完备上同调是某种辅助构造，可用于证明关于自守形式的（$p$-进性质或经典性质的）定理。然而，我们认为$p$-进完备上同调实际上是一个具有根本重要性的对象，并且它提供了我们所知的对$p$-进自守形式空间的最佳逼近。（特别地，与在模曲线理论中有时通过算术几何方法构造的、或更一般地在志村簇理论中被称为这个名字的空间不同，$p$-进完备上同调允许$p$-进群的一个表示，因此允许将表示论方法引入到自守形式的$p$-进性质的研究中。）该理论的系统阐述以及它（在这一点上主要是推测性的）在自守本征形式和伽罗瓦表示之间的朗兰兹对应之$p$-进方面的应用将在文献[3]中给出。这些笔记总结了该理论的一些基本要点以及文献[3]中的一个主要猜想（下面的猜想6.1）。
This note summarizes the theory of p-adically completed cohomology. This construction was first introduced in paper [4] (although insufficient attention was given there to the integral aspects of the theory), and then further developed in the papers [2] and [6]. The papers [4] and [2] may give the impression that p-adically completed cohomology is some sort of auxiliary construction that can be used to prove theorems (of either a p-adic or classical nature) about automorphic forms. However, we believe that p-adically completed cohomology is in fact an object of fundamental importance, and that it provides the best approximation that we know of to spaces of p-adic automorphic forms. (In particular, unlike the spaces that go by this name that are sometimes constructed by arithmetico-geometric means in the theory of modular curves, or more generally Shimura varieties, p-adically completed cohomology admits a representation of the p-adic group, and thus allows the introduction of representation-theoretic methods into the study of p-adic properties of automorphic forms.) A systematic exposition of the theory, and of its (largely conjectural, at this point) applications to the p-adic aspects of the Langlands correspondence between automorphic eigenforms and Galois representations, will be given in the paper [3]. These notes provide a summary of some of the basic points of the theory, as well as one of the main conjectures of [3] (Conjecture 6.1 below).