Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Xiaoyu Huang

TL;DR

The paper extends deformation theory of Galois representations to the case with three Jordan--Holder factors, proving that under precise Selmer-group conditions the Fontaine--Laffaille at $p$ deformation ring $R$ is a DVR and establishing an $R = T$ theorem in a broad setting. The approach harnesses pseudorepresentations and generalized matrix algebras to analyze the totally reducible locus via the reducibility ideal $I^{ ext{tot}}$, and uses lattice techniques to guarantee non-split extensions in prescribed directions. It then applies these structural results to two substantial arithmetic applications: (i) abelian surfaces with a rational $p$-isogeny, where $R$ is DVR and a unique isogeny class emerges, and (ii) automorphic forms congruent to Ikeda lifts, where the $oldsymbol{oldlambda}$-part of Bloch--Kato conjecture yields modularity and an $R=T$ theorem. Collectively, the work provides a scalable framework for modularity and deformation questions beyond two-Jordan–Holder-factor cases, with concrete consequences for abelian surfaces and Ikeda-type automorphic lifts.

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.

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

TL;DR

The paper extends deformation theory of Galois representations to the case with three Jordan--Holder factors, proving that under precise Selmer-group conditions the Fontaine--Laffaille at deformation ring is a DVR and establishing an theorem in a broad setting. The approach harnesses pseudorepresentations and generalized matrix algebras to analyze the totally reducible locus via the reducibility ideal , and uses lattice techniques to guarantee non-split extensions in prescribed directions. It then applies these structural results to two substantial arithmetic applications: (i) abelian surfaces with a rational -isogeny, where is DVR and a unique isogeny class emerges, and (ii) automorphic forms congruent to Ikeda lifts, where the -part of Bloch--Kato conjecture yields modularity and an theorem. Collectively, the work provides a scalable framework for modularity and deformation questions beyond two-Jordan–Holder-factor cases, with concrete consequences for abelian surfaces and Ikeda-type automorphic lifts.

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.
Paper Structure (42 sections, 35 theorems, 108 equations, 2 tables)

This paper contains 42 sections, 35 theorems, 108 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Context
  3. Main Results
  4. Methods
  5. Organization
  6. Acknowledgement
  7. Notation and Selmer Groups
  8. Notation
  9. Local Cohomology
  10. $\ell = p$ and Fontaine--Laffaille
  11. $\ell \ne p$
  12. Global Selmer Groups
  13. Deformation Problem and Main Assumptions
  14. Residual Representation
  15. Main Assumptions
  16. ...and 27 more sections

Key Result

Theorem 1.1

Consider the deformations $\sigma: G_{\Sigma} \to \text{GL}_{n}(\overline{\mathbf{Q}_{p}})$ of $\overline{\sigma}$ with the following deformation conditions: Let $R$ be the reduced quotient of the related universal deformation ring. Suppose If $r = 1$ and $\overline{\sigma}$ admits a deformation to $\text{GL}_n(\mathcal{O})$ that is Fontaine-Laffaille and $\tau$-self-dual, then $R$ is a DVR. If

Theorems & Definitions (94)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Definition 3.1
  • ...and 84 more