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: 1360621
• 负责人: Frauke Bleher
• 金额: $ 18万
• 依托单位: University of Iowa
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1360621&HistoricalAwards=false
• 关键词: 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.

迄今为止，已研究的岩泽理论的主要猜想将第一类岩泽模和塞尔默复形与p-adic L-级数联系起来。这个联邦德国项目的目的是推广这一理论，以更高的陈类。这种推广的一个组成部分是如何定义更高的陈类的方式，有利于研究他们的L-系列。这将通过扩展到岩泽理论的背景下，Parshin和Beilinson的adelic方法来完成。推广的另一个组成部分是将更高的陈类不变量连接到L-级数。 要做到这一点，需要在算术问题中有足够的结构，以便使用L-级数来查看其更高的余维特征。三个特殊的情况下，将被认为是（一）格林伯格的猜想完全真实的领域，（二）岩泽理论的虚二次领域分裂素数，和（三）功能领域的情况。 关于（i），Greenberg已经证明了自然岩泽模在余维1上有平凡支集，PI将利用L-级数研究它们的余维2支集。关于（ii），Rubin，Kings和Johnson-Leung的工作建议我们应该通过与p-adic L-级数对相关的K_2群中的符号来研究第二类陈省身。 关于（iii），PI将研究由Witte定义的类在岩泽代数的高相对K-群内的函数域情况下在Chern类映射下的像。 这个项目的另一个组成部分是与推广到更高的陈省身类的减少技术用于证明第一陈省身类主要猜想。这涉及到推广到岩泽代数理论的倾斜复杂和衍生的等价物，这是用于在群表示理论和研究傅立叶-Mukai函子。这一建议涉及的基本问题群的对称性的代数方程。 在20世纪50年代，岩泽开始了一种新的方法来研究这样的方程，考虑他们的行为在无限的家庭。岩泽表明，许多这样的家庭有很好的定义渐近行为。这导致了关于从这样的家庭所产生的对称群的数值增长率的基本假设。 在过去的50年里，证明这些“主要猜想”一直是抽象代数的中心目标之一。 这一建议涉及对这些结构的改进，这些结构涉及对增长率的更精确的衡量。 关于广泛的影响，对这类代数问题的研究导致了对社会至关重要的技术的发展，例如改进了数据的压缩和安全传输。