Functions on Irreducible Components of the Emerton-Gee Stack
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866910593032650752 |
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| author | Hjelle, Eivind Otto Jaburi, Louis Knak, Rachel Lee, Hao Wang, Shenrong |
| author_facet | Hjelle, Eivind Otto Jaburi, Louis Knak, Rachel Lee, Hao Wang, Shenrong |
| contents | Let $K / \mathbb{Q}_p$ be a finite unramified extension, and let $\mathcal{X}_n$ denote the Emerton-Gee stack parametrizing étale $(φ,Γ_K)$-modules of rank $n$. It is known since the work of Emerton-Gee that the irreducible components of the reduced special fiber of $\mathcal{X}$ are labeled by Serre weights $σ$ of $\operatorname{GL}_n(k)$. If such a component is denoted $\mathcal{X}(σ)$, we prove that $\mathcal{O}(\mathcal{X}(σ)) \cong \mathbb{F}[x_1,x_2,\dots,x_{n-1},x_n^{\pm 1}]$ when $σ$ is sufficiently generic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_00141 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Functions on Irreducible Components of the Emerton-Gee Stack Hjelle, Eivind Otto Jaburi, Louis Knak, Rachel Lee, Hao Wang, Shenrong Number Theory Let $K / \mathbb{Q}_p$ be a finite unramified extension, and let $\mathcal{X}_n$ denote the Emerton-Gee stack parametrizing étale $(φ,Γ_K)$-modules of rank $n$. It is known since the work of Emerton-Gee that the irreducible components of the reduced special fiber of $\mathcal{X}$ are labeled by Serre weights $σ$ of $\operatorname{GL}_n(k)$. If such a component is denoted $\mathcal{X}(σ)$, we prove that $\mathcal{O}(\mathcal{X}(σ)) \cong \mathbb{F}[x_1,x_2,\dots,x_{n-1},x_n^{\pm 1}]$ when $σ$ is sufficiently generic. |
| title | Functions on Irreducible Components of the Emerton-Gee Stack |
| topic | Number Theory |
| url | https://arxiv.org/abs/2306.00141 |