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, and he obtained a refinement of the minus part of the Iwasawa main conjecture over totally real fields ([Ku]). In this paper, we study the higher Fitting ideals of Iwasawa modules arising from the dual fine Selmer groups of general Galois representations which have Euler systems of “Rubintype”, like circular units or Beilinson–Kato elements. By using Kolyvagin derivatives, we construct an ascending filtration {Ci(c)}i≥0 of the Iwasawa algebra, and show that the filtration {Ci(c)}i≥0 gives good approximation of the higher Fitting ideals of the Iwasawa module under the assumption of “Iwasawa main conjecture”. Our results can be regarded as analogues of Kurihara’s results, and a refinement of “Iwasawa main conjecture” and Mazur–Rubin theory in certain cases.