On the existence of analytic families of G-stable lattices and their reductions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929595939291136 |
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| author | Torti, Emiliano |
| author_facet | Torti, Emiliano |
| contents | In this article, we prove the existence of rigid analytic families of $G$-stable lattices with locally constant reductions inside families of representations of a topologically compact group $G$, extending a result of Hellman obtained in the semi-simple residual case. Implementing this generalization in the context of Galois representations, we prove a local constancy result for reductions modulo prime powers of trianguline representations of generic dimension $d$. Moreover, we present two explicit applications. First, in dimension two, we extend to a prime power setting and to the whole rigid projective line a recent result of Bergdall, Levin and Liu concerning reductions of semi-stable representations of $\text{Gal}(\overline{\mathbb{Q}}_p / \mathbb{Q}_p)$ with fixed Hodge-Tate weights and large $\mathcal{L}$-invariant. Second, in dimension $d$, let $V_n$ be a sequence of crystalline representations converging in a certain geometric sense to a crystalline representation $V$. We show that for any refined version $(V, σ)$ of $V$ (or equivalently for any chosen triangulation of its attached $(φ, Γ)$-module $D_{\text{rig}} (V)$ over the Robba ring), there exists a sequence of refinement $σ_n$ of each of the $V_n$ such that the limit as refined representations $(V_n , σ_n )$ converges to the $(V, σ)$. This result does not hold under the weaker assumption that $V_n$ converges only uniformly $p$-adically to $V$ (in the sense of Chenevier, Khare and Larsen). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_00462 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the existence of analytic families of G-stable lattices and their reductions Torti, Emiliano Number Theory 11F80, , 11F85, 14G20, 11F03 In this article, we prove the existence of rigid analytic families of $G$-stable lattices with locally constant reductions inside families of representations of a topologically compact group $G$, extending a result of Hellman obtained in the semi-simple residual case. Implementing this generalization in the context of Galois representations, we prove a local constancy result for reductions modulo prime powers of trianguline representations of generic dimension $d$. Moreover, we present two explicit applications. First, in dimension two, we extend to a prime power setting and to the whole rigid projective line a recent result of Bergdall, Levin and Liu concerning reductions of semi-stable representations of $\text{Gal}(\overline{\mathbb{Q}}_p / \mathbb{Q}_p)$ with fixed Hodge-Tate weights and large $\mathcal{L}$-invariant. Second, in dimension $d$, let $V_n$ be a sequence of crystalline representations converging in a certain geometric sense to a crystalline representation $V$. We show that for any refined version $(V, σ)$ of $V$ (or equivalently for any chosen triangulation of its attached $(φ, Γ)$-module $D_{\text{rig}} (V)$ over the Robba ring), there exists a sequence of refinement $σ_n$ of each of the $V_n$ such that the limit as refined representations $(V_n , σ_n )$ converges to the $(V, σ)$. This result does not hold under the weaker assumption that $V_n$ converges only uniformly $p$-adically to $V$ (in the sense of Chenevier, Khare and Larsen). |
| title | On the existence of analytic families of G-stable lattices and their reductions |
| topic | Number Theory 11F80, , 11F85, 14G20, 11F03 |
| url | https://arxiv.org/abs/2401.00462 |