p进表示的伽罗瓦上同调

项目名称： p进表示的伽罗瓦上同调

项目编号： No.10871183

项目类型： 面上项目

立项/批准年度： 2009

项目学科： 轻工业、手工业

项目作者： 欧阳毅

作者单位： 中国科学技术大学

项目金额： 28万元

中文摘要： 本项目是对p进伽罗瓦表示的伽罗瓦上同调的集中研究。我们采用Fontaine的最新思想，重新诠释了p进Hodge理论。具体说, 我们首先证明Colmez关于p进表示的基本引理，证明Be是主理想整环，由此构造p进基本曲线，并由Harder-Narasimhan定理，确定其稳定向量丛，再给出伽罗瓦不变向量丛和p进伽罗瓦表示的对应，从而再次给出p进Hodge理论两大基本定理"wealkly admissible is admissible"和"de Rham is potentially log-crystalline"的最新证明。我们进一步讨论了由Fontaine理论表述的p进Galois表示的伽罗瓦上同调理论和p进zeta函数，p进L函数及Iwasawa理论的关系。我们还给出了φ27169;的Dieudonne-Manin分类定理的新证明，确定了完备离散赋值环上的Laurent级数环的素谱，证明它是主理想整环。

中文关键词： p进伽罗瓦表示; p进Hodge理论; φ27169;; p进zeta函数; Iwasawa理论

英文摘要： This project is a study of p-adic Galois representations and their Galois cohomology. Based on the latest idea of Fontaine, we rewrite p-adic Hodge theory. Namely, we give a new proof of Colmez's fundamental lemma on p-adic Galois representations, show the key fact that Be is a principal ideal domain, and from this construct the fundamental curve of p-adic representations. We then use Harder-Narasimhan's theorem to study the vector bundles of the p-adic fundamental curve and decide the stable objects of the vector bundles. We give the categorical equivalence of Galois equivariant vector bundles of p-adic fundamental curve and the p-adic Galois representations, and use this to give new proofs of two famous and fundamental results in this field:"weakly admissible is admissible" and "de Rham is potentially log-crystalline". We furthermore study the relation of the p-adic Galois representations in the language of Fontaine to p-adic zeta functions, p-adic L-functions and Iwasawa theory. We also give a new proof of the classification theorem of Dieudonne-Manin on φodules, and decide the spectrum of the ring of Laurent series of given annular convergence over a complete discrete valuation ring and prove that it is a principal ideal domain.

英文关键词： p-adic Galois representation; p-adic Hodge theory; φodule; p-adic zeta function; Iwasawa theory