Vol. 12, No. 10, 2018

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Stark systems over Gorenstein local rings

Ryotaro Sakamoto

Vol. 12 (2018), No. 10, 2295–2326

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.

Stark systems, Euler systems, Selmer groups, Iwasawa theory
Mathematical Subject Classification 2010
Primary: 11R23
Secondary: 11F80, 11S25
Received: 19 April 2017
Revised: 26 February 2018
Accepted: 23 August 2018
Published: 1 February 2019
Ryotaro Sakamoto
Graduate School of Mathematical Sciences
The University of Tokyo