On some mod $p$ representations of quaternion algebra over $\mathbb{Q}_p$
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914814368940032 |
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| author | Hu, Yongquan Wang, Haoran |
| author_facet | Hu, Yongquan Wang, Haoran |
| contents | Let $F$ be a totally real field in which $p$ is unramified and $B$ be a quaternion algebra which splits at at most one infinite place. Let $\overline{r}:\mathrm{Gal}(\overline{F}/F)\to \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ be a modular Galois representation which satisfies the Taylor-Wiles hypotheses. Assume that for some fixed place $v|p$, $B$ ramifies at $v$ and $F_v$ is isomorphic to $ \mathbb{Q}_p$ and $\overline{r}$ is generic at $v$. We prove that the admissible smooth representations of the quaternion algebra over $\mathbb{Q}_p$ coming from mod $p$ cohomology of Shimura varieties associated to $B$ have Gelfand-Kirillov dimension $1$. As an application we prove that the degree two Scholze's functor vanishes on supersingular representations of $\mathrm{GL}_2(\mathbb{Q}_p)$. We also prove some finer structure theorem about the image of Scholze's functor in the reducible case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_01464 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On some mod $p$ representations of quaternion algebra over $\mathbb{Q}_p$ Hu, Yongquan Wang, Haoran Number Theory Representation Theory Let $F$ be a totally real field in which $p$ is unramified and $B$ be a quaternion algebra which splits at at most one infinite place. Let $\overline{r}:\mathrm{Gal}(\overline{F}/F)\to \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ be a modular Galois representation which satisfies the Taylor-Wiles hypotheses. Assume that for some fixed place $v|p$, $B$ ramifies at $v$ and $F_v$ is isomorphic to $ \mathbb{Q}_p$ and $\overline{r}$ is generic at $v$. We prove that the admissible smooth representations of the quaternion algebra over $\mathbb{Q}_p$ coming from mod $p$ cohomology of Shimura varieties associated to $B$ have Gelfand-Kirillov dimension $1$. As an application we prove that the degree two Scholze's functor vanishes on supersingular representations of $\mathrm{GL}_2(\mathbb{Q}_p)$. We also prove some finer structure theorem about the image of Scholze's functor in the reducible case. |
| title | On some mod $p$ representations of quaternion algebra over $\mathbb{Q}_p$ |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2201.01464 |