The Euler system of cyclotomic units and higher Fitting ideals

TL;DR

This paper refines the understanding of the plus-part of the Iwasawa main conjecture for cyclotomic fields by introducing higher cyclotomic ideals and establishing their relation to higher Fitting ideals.

Contribution

It introduces higher cyclotomic ideals defined via Kolyvagin derivatives and proves they bound the higher Fitting ideals of the plus-part Iwasawa module.

Findings

Higher cyclotomic ideals C_i are defined and studied.

C_i provide upper bounds for higher Fitting ideals.

Results refine the plus-part of the Iwasawa main conjecture.

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, we study the higher Fitting ideals of the plus-part of the Iwasawa module associated to the cyclotomic Z_p-extension of Q(\mu_p) for an odd prime number p by similar methods as in Kurihara's work. We define the higher cyclotomic ideals C_i, which are ideals of the Iwasawa algebra defined by the Kolyvagin derivatives of cyclotomic units, and prove that they give upper bounds of the higher Fitting ideals. Our result can be regarded as a refinement of the plus-part of the Iwasawa main conjecture for Q.