Stark systems over Gorenstein local rings

Abstract

In this paper, we define a Stark system over a complete Gorenstein local ring with a finite residue field. Under some standard assumptions, we show that the module of Stark systems is free of rank 1 and that these systems control all the higher Fitting ideals of the Pontryagin dual of the dual Selmer group. This is a generalization of the theory, developed by B. Mazur and K. Rubin, on Stark (or Kolyvagin) systems over principal ideal local rings. Applying our result to a certain Selmer structure over the cyclotomic Iwasawa algebra, we propose a new method for controlling Selmer groups using Euler systems.