Zariski density of modular points in the Eisenstein case

TL;DR

This paper investigates the Zariski closure of modular points in a two-dimensional deformation space with reducible residual Galois representations, utilizing advanced compatibility and finiteness techniques to extend understanding in the residually reducible case.

Contribution

It introduces a novel approach based on local-global compatibility and potential pro-modularity to analyze the Zariski closure in the reducible case, extending previous methods.

Findings

Constructs many non-ordinary regular de Rham points with guaranteed modularity.

Proves equidimensionality of certain big Hecke algebras.

Establishes big R= T theorems in the residually reducible case.

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 Gouv\^ea-Mazur, B\"ockle 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.