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Zariski density of modular points in the Eisenstein case

Xinyao Zhang

TL;DR

The paper advances the understanding of Zariski density for modular points in the two-dimensional universal deformation space when residual Galois representations are reducible. It harnesses local-global compatibility, a potential pro-modularity framework, and a non-ordinary finiteness mechanism to bound the global deformation space and establish density results for modular (and crystalline) points on appropriate irreducible components. The authors also derive equidimensionality results for big Hecke algebras and prove big R = T theorems in the residually reducible setting, with applications to Fontaine–Mazur conjecture instances and Serre-type modularity conjectures. Collectively, these results extend automorphy-lifting techniques beyond the residually irreducible regime and provide new tools for relating local deformation theory to global automorphic phenomena.

Abstract

In this article, we study the Zariski closure of modular points in the two-dimensional universal deformation space when the residual Galois representation is reducible. Unlike the previous approaches in the residually irreducible case from Gouvêa-Mazur, Böckle and Allen, our method relies on local-global compatibility results, potential pro-modularity arguments and a non-ordinary finiteness result between the local deformation ring at $p$ and the global deformation ring. This allows us to construct sufficiently many non-ordinary regular de Rham points whose modularity is guaranteed by the recent progress on the Fontaine-Mazur conjecture. Also, we will discuss some applications of our main results, including the equidimensionality of certain big Hecke algebras and big $R=\mathbb{T}$ theorems in the residually reducible case.

Zariski density of modular points in the Eisenstein case

TL;DR

The paper advances the understanding of Zariski density for modular points in the two-dimensional universal deformation space when residual Galois representations are reducible. It harnesses local-global compatibility, a potential pro-modularity framework, and a non-ordinary finiteness mechanism to bound the global deformation space and establish density results for modular (and crystalline) points on appropriate irreducible components. The authors also derive equidimensionality results for big Hecke algebras and prove big R = T theorems in the residually reducible setting, with applications to Fontaine–Mazur conjecture instances and Serre-type modularity conjectures. Collectively, these results extend automorphy-lifting techniques beyond the residually irreducible regime and provide new tools for relating local deformation theory to global automorphic phenomena.

Abstract

In this article, we study the Zariski closure of modular points in the two-dimensional universal deformation space when the residual Galois representation is reducible. Unlike the previous approaches in the residually irreducible case from Gouvêa-Mazur, Böckle and Allen, our method relies on local-global compatibility results, potential pro-modularity arguments and a non-ordinary finiteness result between the local deformation ring at and the global deformation ring. This allows us to construct sufficiently many non-ordinary regular de Rham points whose modularity is guaranteed by the recent progress on the Fontaine-Mazur conjecture. Also, we will discuss some applications of our main results, including the equidimensionality of certain big Hecke algebras and big theorems in the residually reducible case.
Paper Structure (27 sections, 53 theorems, 38 equations)

This paper contains 27 sections, 53 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. The previous work
  3. Main results
  4. Notations
  5. Acknowledgements
  6. Galois deformation theory
  7. Preliminaries in commutative algebra
  8. Pseudo-representations
  9. Local deformation rings
  10. Global deformation rings
  11. Local-global compatibility
  12. Completed cohomology and Hecke algebra
  13. Paškūnas theory
  14. Local-global compatibility
  15. Pro-modularity
  16. ...and 12 more sections

Key Result

Theorem 1.2.1

Assume that $\bar{\chi}|_{G_{F_v}} \ne \omega$ for any $v|p$ if $p=3$. 1) The Zariski closure of the set of all modular points in $\mathop{\mathrm{Spec}}\nolimits R_F^{\mathrm{ps}, \chi}$ is the union of all irreducible components of dimension $1+2[F:\mathbb{Q}]$. If further $\chi$ is crystalline at

Theorems & Definitions (105)

  • Theorem 1.2.1: Theorem \ref{['main thm']} & Theorem \ref{['main thm without det']}
  • Theorem 1.2.2: Theorem \ref{['dim of T']}
  • Theorem 1.2.3: Proposition \ref{['finite same dim']}
  • Theorem 1.2.5: Theorem \ref{['main thm for emerton']}
  • Theorem 1.2.8: Theorem \ref{['big R=T modular curve']}
  • Definition 2.1.1
  • Proposition 2.1.2
  • proof
  • Proposition 2.1.3
  • proof
  • ...and 95 more