On the higher Fitting ideals of Iwasawa modules of ideal class groups over real abelian fields

TL;DR

This paper extends Kurihara's refinement of the Iwasawa main conjecture to the plus-part of Iwasawa modules over abelian fields, introducing higher cyclotomic ideals and analyzing their relation to Fitting ideals.

Contribution

It defines higher cyclotomic ideals for the plus-part of Iwasawa modules and establishes bounds and classifications, refining the Iwasawa main conjecture for abelian fields.

Findings

Higher cyclotomic ideals provide bounds for Fitting ideals.

Determination of pseudo-isomorphism classes of plus-part Iwasawa modules.

Partial analogue of Kurihara's results for the plus-part.

Abstract

Kurihara established a refinement of the minus-part of the Iwasawa main conjecture for totally real number fields using the higher Fitting ideals. In this paper, by using Kurihara's methods and Mazur-Rubin theory, we study the higher Fitting ideals of the plus-part of Iwasawa modules associated the cyclotomic Z_p-extension of abelian fields for an odd prime number p. We define the higher cyclotomic ideals C_i for any non-negative integer i, which are ideals of the Iwasawa algebra defined by the Kolyvagin derivative classes of circular units, and prove that they give upper and lower bounds of the higher Fitting ideals in some sense, and determine the pseudo-isomorphism classes of the plus-part of Iwasawa modules. Our result can be regarded as an partial analogue of Kurihara's results and a refinement of the plus-part of the Iwasawa main conjecture for abelian fields.