Higher Chern classes in Iwasawa theory

TL;DR

This paper extends classical Iwasawa theory by developing higher Chern classes and characteristic symbols for modules supported in higher codimension, linking them to p-adic L-functions.

Contribution

It introduces a theory of higher Chern classes for Iwasawa modules and applies it to modules from class groups, connecting to p-adic L-functions.

Findings

Determined the second Chern class of a specific Iwasawa module.

Identified the characteristic symbol with the Steinberg symbol of two Katz p-adic L-functions.

Extended classical characteristic ideal theory to higher codimension modules.

Abstract

We begin a study of m-th Chern classes and m-th characteristic symbols for Iwasawa modules which are supported in codimension at least m. This extends the classical theory of characteristic ideals and their generators for Iwasawa modules which are torsion, i.e., supported in codimension at least 1. We apply this to an Iwasawa module constructed from an inverse limit of p-parts of ideal class groups of abelian extensions of an imaginary quadratic field. When this module is pseudo-null, which is conjecturally always the case, we determine its second Chern class and show that it has a characteristic symbol given by the Steinberg symbol of two Katz p-adic L-functions.