On higher Fitting ideals of Iwasawa modules of ideal class groups over imaginary quadratic fields and Euler systems of elliptic units II

TL;DR

This paper investigates the structure of Iwasawa modules over imaginary quadratic fields, establishing bounds for higher Fitting ideals using elliptic units and Euler systems, and relates these to conjectures like ETNC.

Contribution

It proves that certain constructed ideals serve as both upper and lower bounds for higher Fitting ideals, determining the modules' pseudo-isomorphism class, even when ETNC is unproven.

Findings

Ideals C_i bound higher Fitting ideals from above and below.

C_i determine the pseudo-isomorphism class of the Iwasawa module.

In cases where ETNC is proved, C_i coincide with ideals from Rubin--Stark elements.

Abstract

In our previous work, by using Kolyvagin derivatives of elliptic units, we constructed ideals C_i of the Iwasawa algebra, and proved that the ideals C_i become "upper bounds" of the higher Fitting ideals of the one and two variable p-adic unramified Iwasawa module X over an abelian extension field K_0 of an imaginary quadratic field K. In this article, by using "non-arithmetic" specialization arguments, we prove that the ideals C_i also become "lower bounds" of the higher Fitting ideals of X. In particular, we show that the ideals C_i determine the pseudo-isomorphism class of X. Note that in this article, we also treat the cases when the p-part of the equivariant Tamagawa number conjecture (ETNC)_p is not proved yet. In the cases when (ETNC)_p is proved, stronger results have already been obtained by Burns, Kurihara and Sano: under the assumption of (ETNC)_p and certain conditions on the character $\psi$ on the Galois group of K_0/K, they have given a complete description of the higher Fitting ideals of the \psi-component of X by using Rubin--Stark elements. In our article, we also prove that the $\psi$-part of C_i coincide with the ideals constructed by Burns, Kurihara and Sano in certain cases when (ETNC)_p is proved. As a corollary of this comparison results, we also deduce that the annihilator ideal of the $\psi$-part of the maximal pseudo-null submodule of X coincides with the initial Fitting ideal in certain situations.