Multivariable period rings of $p$-adic false Tate curve extension
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866913934687076352 |
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| author | Yuan, Yijun |
| author_facet | Yuan, Yijun |
| contents | Let $p\geq 3$ be a prime number and $K$ be a finite extension of $\mathbf{Q}_p$ with uniformizer $π_K$. In this article, we introduce two multivariable period rings $\mathbf{A}_{\mathfrak{F},K}^{\operatorname{np}}$ and $\mathbf{A}_{\mathfrak{F},K}^{\operatorname{np},\operatorname{c}}$ for the étale $(φ,Γ_{\mathfrak{F},K})$-modules of $p$-adic false Tate curve extension $K\left(π_K^{1/p^\infty},ζ_{p^\infty}\right)$. Various properties of these rings are studied and as applications, we show that $(φ,Γ_{\mathfrak{F},K})$-modules over these rings bridge $(φ,Γ)$-modules and $(φ,τ)$-modules over imperfect period rings in both classical and cohomological sense, which answers a question of Caruso. Finally, we construct the $ψ$ operator for false Tate curve extension and discuss the possibility to calculate Iwasawa cohomology for this extension via $(φ,Γ_{\mathfrak{F},K})$-modules over these rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_24064 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multivariable period rings of $p$-adic false Tate curve extension Yuan, Yijun Number Theory 14G45, 11F80, 11E95, 11S25, 11S15 Let $p\geq 3$ be a prime number and $K$ be a finite extension of $\mathbf{Q}_p$ with uniformizer $π_K$. In this article, we introduce two multivariable period rings $\mathbf{A}_{\mathfrak{F},K}^{\operatorname{np}}$ and $\mathbf{A}_{\mathfrak{F},K}^{\operatorname{np},\operatorname{c}}$ for the étale $(φ,Γ_{\mathfrak{F},K})$-modules of $p$-adic false Tate curve extension $K\left(π_K^{1/p^\infty},ζ_{p^\infty}\right)$. Various properties of these rings are studied and as applications, we show that $(φ,Γ_{\mathfrak{F},K})$-modules over these rings bridge $(φ,Γ)$-modules and $(φ,τ)$-modules over imperfect period rings in both classical and cohomological sense, which answers a question of Caruso. Finally, we construct the $ψ$ operator for false Tate curve extension and discuss the possibility to calculate Iwasawa cohomology for this extension via $(φ,Γ_{\mathfrak{F},K})$-modules over these rings. |
| title | Multivariable period rings of $p$-adic false Tate curve extension |
| topic | Number Theory 14G45, 11F80, 11E95, 11S25, 11S15 |
| url | https://arxiv.org/abs/2505.24064 |