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.
