Higher Chern Classes in Iwasawa Theory
HIGHER CHERN CLASSES IN IWASAWA THEORY F. M. BLEHER, T. CHINBURG, R. GREENBERG, M. KAKDE, G. PAPPAS, R. SHARIFI, AND M. J. TAYLOR Abstract. We begin a study of mth Chern classes and mth characteristic symbols for Iwasawa modules which are supported in codimension at least m.Thisextends 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. Contents 1. Introduction 2 2. Chern classes and characteristic symbols 10 3. Some conjectures in Iwasawa theory 15 4. Unramified Iwasawa modules 19 5. Reflection-type theorems for Iwasawa modules 29 6. A non-commutative generalization 37 Appendix A. Results on Ext-groups 41 References 46 Date: April 20, 2017. 2010 Mathematics Subject Classification. 11R23, 11R34, 18F25. 1 2BLEHER,CHINBURG,GREENBERG,KAKDE,PAPPAS,SHARIFI,ANDTAYLOR 1. Introduction The main conjecture of Iwasawa theory in its most classical form asserts the equality of two ideals in a formal power series ring. The first is defined through the action of the abelian Galois group of the p-cyclotomic tower over an abelian base field on a limit of p-parts of class groups in the tower.
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