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

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.