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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2502.05528 |
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| _version_ | 1866917916880928768 |
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| author | Wang, Peiduo |
| author_facet | Wang, Peiduo |
| contents | In this paper, we study coherent locally free (logarithmic-)$\nabla$-modules on relative $p$-adic polyannuli satisfying the Robba condition and prove several criteria for decomposition of such (logarithmic-)$\nabla$-modules. Firstly we prove the $p$-adic Fuchs theorem for absolute logarithmic $\nabla$-modules where the exponents have non-Liouville differences, which generalizes a result of Shiho. Secondly, we prove a generalized $p$-adic Fuchs theorem for relative $\nabla$-modules which are semi-constant on fibers. We also prove a generalized $p$-adic Fuchs theorem for absolute $\nabla$-modules, when the derivation on the base has some specific form. In the appendix, we prove the coincidence of two definitions of exponents due to Christol-Mebkhout and Dwork and prove that the set of exponents forms exactly one weak equivalence class. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_05528 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On generalized Fuchs theorem over relative $p$-adic polyannuli Wang, Peiduo Number Theory In this paper, we study coherent locally free (logarithmic-)$\nabla$-modules on relative $p$-adic polyannuli satisfying the Robba condition and prove several criteria for decomposition of such (logarithmic-)$\nabla$-modules. Firstly we prove the $p$-adic Fuchs theorem for absolute logarithmic $\nabla$-modules where the exponents have non-Liouville differences, which generalizes a result of Shiho. Secondly, we prove a generalized $p$-adic Fuchs theorem for relative $\nabla$-modules which are semi-constant on fibers. We also prove a generalized $p$-adic Fuchs theorem for absolute $\nabla$-modules, when the derivation on the base has some specific form. In the appendix, we prove the coincidence of two definitions of exponents due to Christol-Mebkhout and Dwork and prove that the set of exponents forms exactly one weak equivalence class. |
| title | On generalized Fuchs theorem over relative $p$-adic polyannuli |
| topic | Number Theory |
| url | https://arxiv.org/abs/2502.05528 |