Higher Chern classes in Iwasawa theory

Abstract

We begin a study of mth Chern classes and mth 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.In the language of the classical main conjectures, an element f ∈ Q × such that ν 1 (f ) = c 1 (M ) is a characteristic power series for M when R is a formal power series ring. A main conjecture for M posits that there is such an f which can be constructed analytically, e.g. via p-adic L-functions.The key to generalizing this is to observe that Q × is the first Quillen K-group K 1 (Q) and ν 1 is a tame symbol map. To try to relate c m (M ) to analytic invariants for arbitrary m, one can consider elements of K m (Q) which can be described by symbols involving m-tuples of elements of Q associated to L-functions. The homomorphism ν 1 is replaced by a homomorphism ν m involving compositions of tame symbol maps. We now describe one way to do this.Suppose that η = (η 0 , . . . , η m ) is a sequence of points of Y with codim(η i ) = i and such that η i+1 lies in the closure η i of η i for all i < m. Denote by P m (Y ) the set of

Codimension Two Cycles in Iwasawa Theory and Elliptic Curves With Supersingular Reduction

“…We would like to thank Ted Chinburg, Ralph Greenberg and Mahesh Kakde for answering various questions related to the work [3]. We are also grateful towards Rob Pollack for sharing the Sage code implementing his algorithms and answering our questions related to it.…”

“…Since our approach towards proving Theorem 1 only involves studying the module theory of Galois cohomology groups, we have not defined p-adic L-functions in the general setup. One must view the statements in Assumption MC simply as abstract formulations of Iwasawa main conjectures (just as in Section 3 of the seven author paper [3]). The content of equation (3.4) in Assumption MC is significant only when the ring R is not a UFD.…”

“…The crucial insight in the seven-author paper [3], to obtain such an association, is to study a situation when the Galois representation ρ d,n satisfies two distinct Panchishkin conditions. We will use this insight and obtain a result by considering the restriction to an imaginary quadratic field, where an odd prime p splits, of an elliptic curve defined over Q with good supersingular reduction at p.…”

Higher codimension behavior in equivariant Iwasawa theory for CM-fields

“…For example, Iwasawa main conjecture is often formulated as a relation between the characteristic ideals of Iwasawa modules and p-adic L-functions, and in general characteristic ideals ignore pseudo-null modules. On the other hand, Greenberg's conjecture claims that the unramified Iwasawa modules are pseudo-null in certain situations (see [1,Conjecture 3.4.1]). In that case, therefore, there was not satisfactory research on more detailed structure of unramified Iwasawa modules.…”

“…Against this background, F. M. Bleher, T. Chinburg, R. Greenberg, M. Kakde, G. Pappas, R. Sharifi, and M. J. Taylor [1] began studying unramified Iwasawa modules that are assumed to be pseudo-null. More concretely, when the base field is an imaginary quadratic field, in [1, Theorem 5.2.1], they obtained a relation between unramified Iwasawa modules and pairs of p-adic L-functions (see Subsection 6.2 for detail).…”