Stable free lattices in residually reducible Galois deformations and control theorem of Selmer groups

TL;DR

This paper investigates the structure of stable free lattices in 2D Galois representations over local UFDs, extending Serre's classical results, and applies these findings to control theorems of Selmer groups in residually reducible Hida families.

Contribution

It generalizes Serre's classical lattice result to local UFDs and applies this to analyze Selmer groups in residually reducible Galois deformations.

Findings

Characterization of homothety classes of stable free lattices

Extension of control theorem for Selmer groups in residually reducible cases

Clarification of the main conjecture in this context

Abstract

In this paper, we study the graph of homothety classes of stable free lattices in a two-dimensional representation over a local UFD. This generalizes a classical result of the case where the base ring is a discrete valuation ring due to Serre. As applications, we consider the case when the representation comes from a residually reducible Hida family and we study the control theorem of Selmer groups. These results enable us to know the precise statement of the main conjecture in residually reducible case as we will remark in section 4 and section 5.