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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2509.25461 |
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| _version_ | 1866910103957929984 |
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| author | Bakker, Benjamin Oswal, Abhishek Shankar, Ananth N. Yao, Zijian |
| author_facet | Bakker, Benjamin Oswal, Abhishek Shankar, Ananth N. Yao, Zijian |
| contents | We prove that Shimura varieties and geometric period images satisfy a $p$-adic extension property for large enough primes $p$. More precisely, let $\mathsf{D}^{\times}\subset \mathsf{D}$ denote the inclusion of the closed punctured unit disc in the closed unit disc. Let $X$ be either a Shimura variety or a geometric period image with torsion-free level structure. Let $F$ be a discretely valued $p$-adic field containing the number field of definition of $X$, where $p$ is a large enough prime. Then, any rigid-analytic map $f: (\mathsf{D}^{\times})^a \times \mathsf{D}^b \rightarrow X_F^{\textrm{an}}$ defined over $F$ whose image intersects the good reduction locus of $X_F^{\textrm{an}}$ (with respect to an integral canonical model) extends to a map $\mathsf{D}^{a+b}\rightarrow X_F^{\textrm{an}}$. We note that this hypothesis is vacuous if $X$ is proper. We also deduce an application to algebraicity of rigid-analytic maps. Our methods also apply to the more general situation of the rigid generic fiber of formal schemes admitting Fontaine-Laffaile modules which satisfy certain positivity conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_25461 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $p$-adic hyperbolicity for Shimura varieties and period images Bakker, Benjamin Oswal, Abhishek Shankar, Ananth N. Yao, Zijian Number Theory We prove that Shimura varieties and geometric period images satisfy a $p$-adic extension property for large enough primes $p$. More precisely, let $\mathsf{D}^{\times}\subset \mathsf{D}$ denote the inclusion of the closed punctured unit disc in the closed unit disc. Let $X$ be either a Shimura variety or a geometric period image with torsion-free level structure. Let $F$ be a discretely valued $p$-adic field containing the number field of definition of $X$, where $p$ is a large enough prime. Then, any rigid-analytic map $f: (\mathsf{D}^{\times})^a \times \mathsf{D}^b \rightarrow X_F^{\textrm{an}}$ defined over $F$ whose image intersects the good reduction locus of $X_F^{\textrm{an}}$ (with respect to an integral canonical model) extends to a map $\mathsf{D}^{a+b}\rightarrow X_F^{\textrm{an}}$. We note that this hypothesis is vacuous if $X$ is proper. We also deduce an application to algebraicity of rigid-analytic maps. Our methods also apply to the more general situation of the rigid generic fiber of formal schemes admitting Fontaine-Laffaile modules which satisfy certain positivity conditions. |
| title | $p$-adic hyperbolicity for Shimura varieties and period images |
| topic | Number Theory |
| url | https://arxiv.org/abs/2509.25461 |