On the locally analytic $\text{Ext}^1$-conjecture in the $\text{GL}_2(L)$ case
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908562451595264 |
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| author | Su, Benchao |
| author_facet | Su, Benchao |
| contents | Let $L$ be a finite extension of $\mathbb{Q}_p$. We calculate the dimension of $\text{Ext}^1$-groups of certain locally analytic representations of $\text{GL}_2(L)$ defined using coherent cohomology of Drinfeld curves. Furthermore, let $ρ_p$ be a $2$-dimensional continuous representation of $\text{Gal}(\bar L/L)$, which is de Rham with parallel Hodge-Tate weights $0,1$ and whose underlying Weil-Deligne representation is irreducible. We prove Breuil's locally analytic $\text{Ext}^1$ conjecture for such $ρ_p$. As an application, we show that the isomorphism class of the multiplicity space $Π^{\text{an}}_{\text{geo}}(ρ_p)$ of $ρ_p$ in the pro-étale cohomology of Drinfeld curves uniquely determines the isomorphism class of $ρ_p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_17683 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the locally analytic $\text{Ext}^1$-conjecture in the $\text{GL}_2(L)$ case Su, Benchao Number Theory Let $L$ be a finite extension of $\mathbb{Q}_p$. We calculate the dimension of $\text{Ext}^1$-groups of certain locally analytic representations of $\text{GL}_2(L)$ defined using coherent cohomology of Drinfeld curves. Furthermore, let $ρ_p$ be a $2$-dimensional continuous representation of $\text{Gal}(\bar L/L)$, which is de Rham with parallel Hodge-Tate weights $0,1$ and whose underlying Weil-Deligne representation is irreducible. We prove Breuil's locally analytic $\text{Ext}^1$ conjecture for such $ρ_p$. As an application, we show that the isomorphism class of the multiplicity space $Π^{\text{an}}_{\text{geo}}(ρ_p)$ of $ρ_p$ in the pro-étale cohomology of Drinfeld curves uniquely determines the isomorphism class of $ρ_p$. |
| title | On the locally analytic $\text{Ext}^1$-conjecture in the $\text{GL}_2(L)$ case |
| topic | Number Theory |
| url | https://arxiv.org/abs/2504.17683 |