FRG: Collaborative Research: Chern classes in Iwasawa Theory

迄今为止所研究的Iwasawa理论的主要猜想将Iwasawa模和Selmer配合的第一类Chern类与p进l级数联系起来。这个FRG项目的目的是将这个理论推广到更高的chen类。这种泛化的一个组成部分涉及如何以一种便于通过l级数研究的方式定义更高的chen类。这将通过将Parshin和Beilinson的adelic方法扩展到Iwasawa理论的背景来完成。推广的另一个组成部分与将更高的陈氏类不变量与l级数联系起来有关。要做到这一点，我们需要在算术问题中有足够的结构来使用l级数来观察它的高协维特征。三种特殊的情况将被考虑：(i)格林伯格猜想在全实场，（ii） Iwasawa理论的虚二次场在分裂素数，和（iii）函数场的情况。关于(i)， Greenberg推测自然Iwasawa模在余维1上有平凡支持；pi将使用l级数来研究它们的余维二支持。关于（ii）， Rubin、Kings和Johnson-Leung的工作表明，应该通过与p进l级数对相关联的K_2群中的符号来研究第二Chern类。关于（iii）， pi将在Iwasawa代数的高相对k群的函数域情况下研究由Witte定义的类的Chern类映射下的图像。这个项目的另一个组成部分是将用于证明第一个陈氏类主要猜想的约简技术推广到更高的陈氏类。这涉及到在群表示理论和研究傅立叶-穆凯函子中使用的倾斜复合体和衍生等价理论推广到Iwasawa代数。这一建议处理关于代数方程的对称群的基本问题。在20世纪50年代，Iwasawa通过考虑它们在无限族中的行为，开始了研究这类方程的新方法。Iwasawa证明了许多这样的族具有良好定义的渐近行为。这导致了关于由这些族产生的对称群的数值增长率的基本猜想。这些“主要猜想”的证明在过去50年中一直是抽象代数的中心目标之一。这一建议与这些猜想的改进有关，这些猜想涉及对增长率的更精确测量。就广泛的影响而言，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。

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.