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| Format: | Preprint |
| Veröffentlicht: |
2026
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2604.01846 |
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| _version_ | 1866917380172546048 |
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| author | He, Yiqin |
| author_facet | He, Yiqin |
| contents | Let $ρ_p$ be an $n$-dimensional non-critical semistable $p$-adic Galois representation of the absolute Galois group of $\mathrm{Q}_p$ with regular Hodge--Tate weights. Let $\mathrm{D}$ be the associated $(φ,Γ)$-module over the Robba ring. By combining Ding's and Breuil--Ding's methods for the crystalline case with Qian's computation of higher extension groups of locally analytic generalized Steinberg representations, we capture the full information of the $p$-adic Hodge parameters of $ρ_p$ on the automorphic side by considering several Steinberg subquotients of $\mathrm{D}$ and the "crystalline" Hodge parameters between them. These results also admit geometric and Lie-algebraic reformulations on flag varieties related to the moduli space of Hodge parameters. We then construct an explicit locally analytic representation $π_{1}(ρ_p)$ and explicitly describe which Hodge-parameters information of $ρ_p$ it determines. In particular, if the monodromy rank of $ρ_p$ is at most $1$, $π_{1}(ρ_p)$ determines $ρ_p$. When $ρ_p$ comes from a $p$-adic automorphic representation, we show that $π_{1}(ρ_p)$ is a subrepresentation of the $\mathrm{GL}_n(\mathrm{Q}_p)$-representation globally associated to $ρ_p$, under mild hypotheses. Although it is still difficult to construct an explicit representation $π_{1}(ρ_p)$ that determines $ρ_p$, our results provide new evidence for the $p$-adic Langlands program in general semistable cases and demonstrate the broad applicability of Ding's, Breuil--Ding's, and Qian's methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01846 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Towards the $p$-adic Hodge parameters in semistable representations of $\mathrm{GL}_n(\mathrm{Q}_p)$ He, Yiqin Number Theory Let $ρ_p$ be an $n$-dimensional non-critical semistable $p$-adic Galois representation of the absolute Galois group of $\mathrm{Q}_p$ with regular Hodge--Tate weights. Let $\mathrm{D}$ be the associated $(φ,Γ)$-module over the Robba ring. By combining Ding's and Breuil--Ding's methods for the crystalline case with Qian's computation of higher extension groups of locally analytic generalized Steinberg representations, we capture the full information of the $p$-adic Hodge parameters of $ρ_p$ on the automorphic side by considering several Steinberg subquotients of $\mathrm{D}$ and the "crystalline" Hodge parameters between them. These results also admit geometric and Lie-algebraic reformulations on flag varieties related to the moduli space of Hodge parameters. We then construct an explicit locally analytic representation $π_{1}(ρ_p)$ and explicitly describe which Hodge-parameters information of $ρ_p$ it determines. In particular, if the monodromy rank of $ρ_p$ is at most $1$, $π_{1}(ρ_p)$ determines $ρ_p$. When $ρ_p$ comes from a $p$-adic automorphic representation, we show that $π_{1}(ρ_p)$ is a subrepresentation of the $\mathrm{GL}_n(\mathrm{Q}_p)$-representation globally associated to $ρ_p$, under mild hypotheses. Although it is still difficult to construct an explicit representation $π_{1}(ρ_p)$ that determines $ρ_p$, our results provide new evidence for the $p$-adic Langlands program in general semistable cases and demonstrate the broad applicability of Ding's, Breuil--Ding's, and Qian's methods. |
| title | Towards the $p$-adic Hodge parameters in semistable representations of $\mathrm{GL}_n(\mathrm{Q}_p)$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2604.01846 |