Euler systems for Galois deformations and the pseudo-isomorphism class of the dual of fine Selmer groups

In this article, we study the pseudo-isomorphism class of the dual fine Selmer group $X$ attached to a $p$-adic Galois deformation whose deformation ring $\mathrm{\Lambda }$ is isomorphic to the ring of formal power series. By using the “Kolyvagin system” arising from a given Euler system $c$, we shall construct ideals $C_i(c)$ of $\mathrm{\Lambda }$, and prove that the ideals $C_i(c)$ approximate the higher Fitting ideals of $X$ under suitable hypothesis. In particular, we shall prove that the ideals $C_i(c)$ arising from the Euler system of Beilinson-Kato elements determine the pseudo-isomorphism classes of the dual fine Selmer groups attached to ordinary and nearly ordinary Hida deformations satisfying certain conditions.