FRG: Collaborative Research: Chern classes in Iwasawa Theory

FRG：合作研究：岩泽理论中的陈省身课程

• 批准号: 1360733
• 负责人: Georgios Pappas
• 金额: $ 28万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1360733&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Chern; classes; FRG; Collaborative; Research; Chern; classes

The Main Conjectures of Iwasawa theory which have been studied up to now relate the first Chern classes of Iwasawa modules and Selmer complexes to p-adic L-series. The object of this FRG project is to generalize this theory to higher Chern classes. One component of this generalization concerns how to define higher Chern classes in a way that facilitates studying them by L-series. This will be done by extending to the context of Iwasawa theory the adelic methods of Parshin and Beilinson. Another component of the generalization has to do with connecting higher Chern class invariants to L-series. To do this, one needs enough structure in the arithmetic problem to see into its higher codimension features using L-series. Three particular cases will be considered are (i) Greenberg's conjecture over totally real fields, (ii) Iwasawa theory for imaginary quadratic fields at split primes, and (iii) the function field case. Concerning (i), Greenberg has conjectured that the natural Iwasawa modules have trivial support in codimension one; the PIs will study their codimension two support using L-series. Concerning (ii), work of Rubin, and of Kings and Johnson-Leung, suggests that one should study second Chern classes via symbols in K_2 groups associated to pairs of p-adic L-series. Concerning (iii), the PIs will study the images under Chern class maps of classes defined by Witte in the function field case inside the higher relative K-groups of Iwasawa algebras. One further component of this project has to do with generalizing to higher Chern classes the reduction techniques used in proving first Chern class Main Conjectures. This involves generalizing to Iwasawa algebras the theory of tilting complexes and derived equivalences which is used in group representation theory and in studying Fourier-Mukai functors.This proposal deals with fundamental questions about the groups of symmetries of algebraic equations. In the 1950's, Iwasawa began a new approach to the study of such equations by considering their behavior in infinite families. Iwasawa showed that many such families have well defined asymptotic behavior. This led to fundamental conjectures concerning the numerical growth rate of the symmetry groups arising from such families. The proof of such "Main Conjectures" has been one of the central goals of abstract algebra over the last 50 years. This proposal has to do with the refinements of these conjectures which deal with more precise measures of rates of growth. Concerning broad impacts, work on algebraic questions of this kind has led to the development of technology essential to society, such as the improved compression and secure transmission of data.

现在已经研究的岩泽理论的主要猜想将iwasawa模块和Selmer Complexse的最初的Chern类别与P-Adic L系列有关。该FRG项目的目的是将该理论推广到更高的Chern类。这种概括的一个组成部分涉及如何以促进L系列研究它们的方式定义更高的Chern类。这将通过扩展到岩泽理论的上下文来完成。概括的另一个组成部分与将较高的Chern类不变性连接到L系列有关。 为此，在算术问题中需要足够的结构，才能使用L系列介绍其更高的consimension特征。将考虑三种特殊情况是（i）格林伯格对完全真实的领域的猜想，（ii）在分裂素数中对假想二次领域的伊瓦萨岛理论，以及（iii）功能场案例。 关于（i），格林伯格（Greenberg）猜想天然的伊瓦苏瓦模块在编辑中具有微不足道的支持。 PI将使用L系列研究他们的编码两个支持。关于（ii），鲁宾的工作以及国王和约翰逊 - 皇帝的工作，表明应该通过与P-Adic L系列成对相关的K_2组中的符号研究第二个Chern类。 关于（iii），PIS将研究iwasawa代数的较高相对K群中Witte定义的类别的Chern类图像下的图像。 该项目的另一个组成部分与对更高的Chern类概述有关的还原技术证明了Chern Class主要猜想。这涉及将倾斜复合物和得出的等价理论推广到小组表示理论和研究傅立叶穆凯函数时。该建议涉及有关代数方程对称性组的基本问题。 在1950年代，伊瓦沙瓦（Iwasawa）通过考虑无限家庭的行为开始了一种新方法来研究此类方程式。 Iwasawa表明，许多这样的家庭具有明确的渐近行为。这导致了有关此类家庭产生的对称群体的数值增长率的基本猜想。 在过去的50年中，这种“主要猜想”的证明一直是抽象代数的核心目标之一。 该提议与这些猜想的改进有关，这些猜想涉及更精确的增长率度量。 关于广泛的影响，此类代数问题的工作导致了对社会必不可少的技术发展的发展，例如改进的压缩和安全传输数据。