On higher Fitting ideals of certain Iwasawa modules associated with Galois representations and Euler systems

Abstract

By using “Gauss sum-type” Kolyvagin systems, Kurihara studied the higher Fitting ideals of Iwasawa modules arising from Greenberg Selmer groups of p-adic Galois representations, and proved a refinement of the Iwasawa main conjecture. In this article, we study the higher Fitting ideals of Iwasawa modules arising from the dual fine Selmer groups of general Galois representations which have rank one Euler systems of “Rubin-type” circular units or Beilinson–Kato elements. By using Kolyvagin derivatives, we construct an ascending filtration ${\{{\mathfrak{C}}_i(\mathbf{c})\}}_{i\ge 0}$ of the Iwasawa algebra, and we show that the filtration ${\{{\mathfrak{C}}_i(\mathbf{c})\}}_{i\ge 0}$ gives good approximation of the higher Fitting ideals of the Iwasawa module under the assumption analogous to the Iwasawa main conjecture.