A stacky $p$-adic Riemann--Hilbert correspondence on Hitchin-small locus
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909548411879424 |
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| author | Liu, Yudong Ma, Chenglong Nie, Xiecheng Qu, Xiaoyu Wang, Yupeng |
| author_facet | Liu, Yudong Ma, Chenglong Nie, Xiecheng Qu, Xiaoyu Wang, Yupeng |
| contents | Let $C$ be an algebraically closed perfectoid field over $\mathbb{Q}_p$ with the ring of integer $\mathcal{O}_C$ and the infinitesimal thickening $\Ainf$. Let $\mathfrak X$ be a semi-stable formal scheme over $\mathcal{O}_C$ with a fixed flat lifting $\widetilde{\mathfrak X}$ over $\Ainf$. Let $X$ be the generic fiber of $\mathfrak{X}$ and $\widetilde X$ be its lifting over $\BdRp$ induced by $\widetilde{\mathfrak X}$. Let $\MIC_r(\widetilde X)^{{\rm H}\text{-small}}$ and $\rL\rS_r(X,\BBdRp)^{{\rm H}\text{-small}}$ be the $v$-stacks of rank-$r$ Hitchin-small integrable connections on $X_{\et}$ and $\BBdRp$-local systems on $X_{v}$, respectively. In this paper, we establish an equivalence between these two stacks by introducing a new period sheaf with connection $(\calO\bB_{\dR,\pd}^+,\rd)$ on $X_{v}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_08785 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A stacky $p$-adic Riemann--Hilbert correspondence on Hitchin-small locus Liu, Yudong Ma, Chenglong Nie, Xiecheng Qu, Xiaoyu Wang, Yupeng Algebraic Geometry Number Theory Let $C$ be an algebraically closed perfectoid field over $\mathbb{Q}_p$ with the ring of integer $\mathcal{O}_C$ and the infinitesimal thickening $\Ainf$. Let $\mathfrak X$ be a semi-stable formal scheme over $\mathcal{O}_C$ with a fixed flat lifting $\widetilde{\mathfrak X}$ over $\Ainf$. Let $X$ be the generic fiber of $\mathfrak{X}$ and $\widetilde X$ be its lifting over $\BdRp$ induced by $\widetilde{\mathfrak X}$. Let $\MIC_r(\widetilde X)^{{\rm H}\text{-small}}$ and $\rL\rS_r(X,\BBdRp)^{{\rm H}\text{-small}}$ be the $v$-stacks of rank-$r$ Hitchin-small integrable connections on $X_{\et}$ and $\BBdRp$-local systems on $X_{v}$, respectively. In this paper, we establish an equivalence between these two stacks by introducing a new period sheaf with connection $(\calO\bB_{\dR,\pd}^+,\rd)$ on $X_{v}$. |
| title | A stacky $p$-adic Riemann--Hilbert correspondence on Hitchin-small locus |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2409.08785 |