Lubin-Tate and multivariable $(φ,\mathcal{O}_K^{\times})$-modules in dimension 2
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917626963296256 |
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| author | Wang, Yitong |
| author_facet | Wang, Yitong |
| contents | Let $p$ be a prime number, $K$ a finite unramified extension of $\mathbb{Q}_p$ and $\mathbb{F}$ a finite extension of $\mathbb{F}_p$. For $\overlineρ$ any reducible two-dimensional representation of $\operatorname{Gal}(\overline{K}/K)$ over $\mathbb{F}$, we compute explicitly the associated étale $(φ,\mathcal{O}_K^{\times})$-module $D_A^{\otimes}(\overlineρ)$ defined by Breuil-Herzig-Hu-Morra-Schraen. Then we let $π$ be an admissible smooth representation of $\operatorname{GL}_2(K)$ over $\mathbb{F}$ occurring in some Hecke eigenspaces of the mod $p$ cohomology and $\overlineρ$ be its underlying two-dimensional representation of $\operatorname{Gal}(\overline{K}/K)$ over $\mathbb{F}$. Assuming that $\overlineρ$ is maximally non-split, we prove under some genericity assumption that the associated étale $(φ,\mathcal{O}_K^{\times})$-module $D_A(π)$ defined by Breuil-Herzig-Hu-Morra-Schraen is isomorphic to $D_A^{\otimes}(\overlineρ)$. This extends the results of Breuil-Herzig-Hu-Morra-Schraen, where $\overlineρ$ was assumed to be semisimple. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_00396 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lubin-Tate and multivariable $(φ,\mathcal{O}_K^{\times})$-modules in dimension 2 Wang, Yitong Number Theory Let $p$ be a prime number, $K$ a finite unramified extension of $\mathbb{Q}_p$ and $\mathbb{F}$ a finite extension of $\mathbb{F}_p$. For $\overlineρ$ any reducible two-dimensional representation of $\operatorname{Gal}(\overline{K}/K)$ over $\mathbb{F}$, we compute explicitly the associated étale $(φ,\mathcal{O}_K^{\times})$-module $D_A^{\otimes}(\overlineρ)$ defined by Breuil-Herzig-Hu-Morra-Schraen. Then we let $π$ be an admissible smooth representation of $\operatorname{GL}_2(K)$ over $\mathbb{F}$ occurring in some Hecke eigenspaces of the mod $p$ cohomology and $\overlineρ$ be its underlying two-dimensional representation of $\operatorname{Gal}(\overline{K}/K)$ over $\mathbb{F}$. Assuming that $\overlineρ$ is maximally non-split, we prove under some genericity assumption that the associated étale $(φ,\mathcal{O}_K^{\times})$-module $D_A(π)$ defined by Breuil-Herzig-Hu-Morra-Schraen is isomorphic to $D_A^{\otimes}(\overlineρ)$. This extends the results of Breuil-Herzig-Hu-Morra-Schraen, where $\overlineρ$ was assumed to be semisimple. |
| title | Lubin-Tate and multivariable $(φ,\mathcal{O}_K^{\times})$-modules in dimension 2 |
| topic | Number Theory |
| url | https://arxiv.org/abs/2404.00396 |