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

Abstract

Kurihara established a refinement of the minus-part of the Iwasawa main conjecture for totally real number fields by using the higher Fitting ideals [Ku]. In this paper, by using Kuriharaʼs methods and Mazur–Rubin theory, we study the higher Fitting ideals of the plus-part of Iwasawa modules associated with the cyclotomic Zp-extension of abelian fields for an odd prime number p. We define the higher cyclotomic ideals, which are ideals of the Iwasawa algebra defined by the Kolyvagin derivative classes of circular units. Then, we prove that the higher cyclotomic ideals 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 results can be regarded as analogues of Kuriharaʼs results and a refinement of the plus-part of the Iwasawa main conjecture for abelian fields.