Stark Systems over Gorenstein Local Rings

TL;DR

This paper introduces Stark systems over Gorenstein local rings, proving their structure and their role in controlling Fitting ideals of dual Selmer groups, extending prior work from principal ideal rings.

Contribution

It generalizes the theory of Stark systems from principal ideal rings to Gorenstein local rings, establishing their freeness and control over Fitting ideals.

Findings

Stark systems form a free module of rank one over Gorenstein local rings.

They control all Fitting ideals of the Pontryagin dual of the dual Selmer group.

The results extend previous work by Mazur and Rubin.

Abstract

In this paper, we define a module of Stark systems over a complete Gorenstein local ring with a finite residue field of odd characteristic. Under some mild assumptions, we show that it is free of rank one and that these systems control all the Fitting ideals of the Pontryagin dual of the dual Selmer group. These results generalize the work of B. Mazur and K. Rubin on Stark systems over a principal ideal ring.