Monodromy and rigidity of crystalline local systems
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arXiv
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| Format: | Preprint |
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2025
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| author | Diao, Hansheng Yao, Zijian |
| author_facet | Diao, Hansheng Yao, Zijian |
| contents | We study several rigidity properties of $p$-adic local systems on a smooth rigid analytic space $X$ over a $p$-adic field. We prove that the monodromy of the log isocrystal attached to a $p$-adic local system is ''rigid'' along irreducible components of the special fiber. Then we give several applications. First, suppose that $X$ has good reduction. We show that if a family of semistable representations is crystalline at one classical point on $X$, then it is crystalline everywhere. Second, combining with the $p$-adic monodromy theorem recently studied by the authors and their collaborators, we prove the following surprising rigidity result conjectured by Shankar: for any $p$-adic local system on a smooth projective variety with good reduction, if it is potentially crystalline at one classical point, then it is potentially crystalline everywhere. Finally, we show that if a $p$-adic local system on the complement of a reduced normal crossing divisor on a smooth rigid analytic space is crystalline at all classical points, then it extends uniquely to a $p$-adic local system on the entire space. In other words, such a local system cannot have geometric monodromy if it has no arithmetic monodromy everywhere on the complement of a reduced normal crossing divisor. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_19813 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Monodromy and rigidity of crystalline local systems Diao, Hansheng Yao, Zijian Algebraic Geometry Number Theory 14D05, 14F20, 14F30, 14G22 We study several rigidity properties of $p$-adic local systems on a smooth rigid analytic space $X$ over a $p$-adic field. We prove that the monodromy of the log isocrystal attached to a $p$-adic local system is ''rigid'' along irreducible components of the special fiber. Then we give several applications. First, suppose that $X$ has good reduction. We show that if a family of semistable representations is crystalline at one classical point on $X$, then it is crystalline everywhere. Second, combining with the $p$-adic monodromy theorem recently studied by the authors and their collaborators, we prove the following surprising rigidity result conjectured by Shankar: for any $p$-adic local system on a smooth projective variety with good reduction, if it is potentially crystalline at one classical point, then it is potentially crystalline everywhere. Finally, we show that if a $p$-adic local system on the complement of a reduced normal crossing divisor on a smooth rigid analytic space is crystalline at all classical points, then it extends uniquely to a $p$-adic local system on the entire space. In other words, such a local system cannot have geometric monodromy if it has no arithmetic monodromy everywhere on the complement of a reduced normal crossing divisor. |
| title | Monodromy and rigidity of crystalline local systems |
| topic | Algebraic Geometry Number Theory 14D05, 14F20, 14F30, 14G22 |
| url | https://arxiv.org/abs/2509.19813 |