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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.21249 |
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| _version_ | 1866915703052828672 |
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| author | Zhang, Xinyao |
| author_facet | Zhang, Xinyao |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21249 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Zariski density of modular points in the Eisenstein case Zhang, Xinyao Number Theory 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. |
| title | Zariski density of modular points in the Eisenstein case |
| topic | Number Theory |
| url | https://arxiv.org/abs/2512.21249 |