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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2408.10546 |
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| _version_ | 1866914917678841856 |
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| author | Huang, Shanxiao |
| author_facet | Huang, Shanxiao |
| contents | Analogue to Fontaine's computation for $Ω_{\bar{\mathbb{Z}}_p/\mathbb{Z}_p}$, we compute the structure of $Ω_{\mathcal{O}_{\bar{K}_0}/\mathcal{O}_{K_0}}$ (here $K_0$ is the completion of $\mathbb{Q}_p(T)$ at place $p$) and prove that $p^{1-1/p^n}\mathrm{d}p^{1/p^n}$, $T^{1-1/p^n}\mathrm{d}T^{1/p^n}$ and $S^{1-1/p^n}\mathrm{d}S^{1/p^n}$ are linearly dependent (Here $S := 1-T$). The main aim of this article is to find the linear equations for these three differential forms. Then we define a map which is called "differential version" of Fontaine's map to express the equations in a computable way. Finally, we prove that the coefficients in the equation can be expressed in some polynomial forms and compute some examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_10546 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Coordinate Transformation in Faltings' Extension Huang, Shanxiao Representation Theory Algebraic Geometry Number Theory Analogue to Fontaine's computation for $Ω_{\bar{\mathbb{Z}}_p/\mathbb{Z}_p}$, we compute the structure of $Ω_{\mathcal{O}_{\bar{K}_0}/\mathcal{O}_{K_0}}$ (here $K_0$ is the completion of $\mathbb{Q}_p(T)$ at place $p$) and prove that $p^{1-1/p^n}\mathrm{d}p^{1/p^n}$, $T^{1-1/p^n}\mathrm{d}T^{1/p^n}$ and $S^{1-1/p^n}\mathrm{d}S^{1/p^n}$ are linearly dependent (Here $S := 1-T$). The main aim of this article is to find the linear equations for these three differential forms. Then we define a map which is called "differential version" of Fontaine's map to express the equations in a computable way. Finally, we prove that the coefficients in the equation can be expressed in some polynomial forms and compute some examples. |
| title | Coordinate Transformation in Faltings' Extension |
| topic | Representation Theory Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2408.10546 |