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arXiv · 2512.21249

Zariski density of modular points in the Eisenstein case

Xinyao Zhang
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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.

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Xinyao Zhang. 2025-12-24. Zariski density of modular points in the Eisenstein case. https://arxiv.org/abs/2512.21249

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