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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.03994 |
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| _version_ | 1866912391248216064 |
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| author | Han, Seongjae Park, Chol |
| author_facet | Han, Seongjae Park, Chol |
| contents | Let $p$ be an odd prime, and $\mathbf{Q}_{p^f}$ the unramified extension of $\mathbf{Q}_p$ of degree $f$. In this paper, we reduce the problem of constructing strongly divisible modules for $2$-dimensional semi-stable non-crystalline representations of $\mathrm{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_{p^f})$ with Hodge--Tate weights in the Fontaine--Laffaille range to solving systems of linear equations and inequalities. We also determine the Breuil modules corresponding to the mod-$p$ reduction of the strongly divisible modules. We expect our method to produce at least one Galois-stable lattice in each such representation for general $f$. Moreover, when the mod-$p$ reduction is an extension of distinct characters, we further expect our method to provide the two non-homothetic lattices. As applications, we show that our approach recovers previously known results for $f=1$ and determine the mod-$p$ reduction of the semi-stable representations with some small Hodge--Tate weights when $f=2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_03994 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On families of strongly divisible modules of rank 2 Han, Seongjae Park, Chol Number Theory 11F80 Let $p$ be an odd prime, and $\mathbf{Q}_{p^f}$ the unramified extension of $\mathbf{Q}_p$ of degree $f$. In this paper, we reduce the problem of constructing strongly divisible modules for $2$-dimensional semi-stable non-crystalline representations of $\mathrm{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_{p^f})$ with Hodge--Tate weights in the Fontaine--Laffaille range to solving systems of linear equations and inequalities. We also determine the Breuil modules corresponding to the mod-$p$ reduction of the strongly divisible modules. We expect our method to produce at least one Galois-stable lattice in each such representation for general $f$. Moreover, when the mod-$p$ reduction is an extension of distinct characters, we further expect our method to provide the two non-homothetic lattices. As applications, we show that our approach recovers previously known results for $f=1$ and determine the mod-$p$ reduction of the semi-stable representations with some small Hodge--Tate weights when $f=2$. |
| title | On families of strongly divisible modules of rank 2 |
| topic | Number Theory 11F80 |
| url | https://arxiv.org/abs/2503.03994 |