On deformation rings of residual Galois representations with three Jordan-Holder factors and modularity

TL;DR

This paper investigates deformation rings of certain non-semisimple Galois representations with three Jordan-Holder factors, establishing conditions for their structure and applications to automorphic forms and abelian surfaces.

Contribution

It proves that under specific Selmer group conditions, the universal deformation ring is a DVR and establishes an R = T theorem in a broad context, with applications to automorphic and geometric objects.

Findings

Universal deformation ring is a DVR under certain conditions

Established R = T theorem for specific Galois representations

Connected deformation theory to automorphic forms and abelian surfaces

Abstract

In this paper, we study Fontaine-Laffaille, self-dual deformations of a mod p non-semisimple Galois representation of dimension n with its Jordan-Holder factors being three mutually non-isomorphic absolutely irreducible representations. We show that under some conditions regarding the orders of certain Selmer groups, the universal deformation ring is a discrete valuation ring. Given enough information on the Hecke algebra, we also prove an R = T theorem in the general context. We then apply our results to abelian surfaces with cyclic rational isogenies and certain 6-dimensional representations arising from automorphic forms congruent to Ikeda lifts. Assuming the Bloch-Kato conjecture, our result identifies special L-value conditions for the existence of a unique abelian surface isogeny class and an R = T theorem for certain 6-dimensional Galois representations.