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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.20232 |
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| _version_ | 1866918194667585536 |
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| author | Conti, Andrea Torti, Emiliano |
| author_facet | Conti, Andrea Torti, Emiliano |
| contents | For a profinite group $G$ and a rigid analytic space $X$, we study when an $\mathcal O_X(X)$-linear representation $V$ of $G$ admits a lattice, i.e. an $\mathcal O_{\mathcal X(\mathcal X)}$-linear model for a suitable formal model $\mathcal X$ of $X$ in the sense of Berthelot. We give a positive answer, under mild assumptions, when $X$ is strictly quasi-Stein. As a consequence, we are able to describe explicit open rational subdomains of $X$ over which $V$ is constant after reduction modulo a power of $p$. We give applications in two different directions. First, we prove explicit results on the reduction modulo powers of $p$ of sheaves of crystalline and semistable representations of fixed weight. Second, we deduce a result on the pseudorepresentation carried by the Coleman--Mazur eigencurve, which can be made explicit whenever equations for a rational subdomain of the eigencurve are given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_20232 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lattices in rigid analytic representations Conti, Andrea Torti, Emiliano Number Theory 11F80, 14G22 For a profinite group $G$ and a rigid analytic space $X$, we study when an $\mathcal O_X(X)$-linear representation $V$ of $G$ admits a lattice, i.e. an $\mathcal O_{\mathcal X(\mathcal X)}$-linear model for a suitable formal model $\mathcal X$ of $X$ in the sense of Berthelot. We give a positive answer, under mild assumptions, when $X$ is strictly quasi-Stein. As a consequence, we are able to describe explicit open rational subdomains of $X$ over which $V$ is constant after reduction modulo a power of $p$. We give applications in two different directions. First, we prove explicit results on the reduction modulo powers of $p$ of sheaves of crystalline and semistable representations of fixed weight. Second, we deduce a result on the pseudorepresentation carried by the Coleman--Mazur eigencurve, which can be made explicit whenever equations for a rational subdomain of the eigencurve are given. |
| title | Lattices in rigid analytic representations |
| topic | Number Theory 11F80, 14G22 |
| url | https://arxiv.org/abs/2403.20232 |