Compatibility of canonical $\ell$-adic local systems on Shimura varieties
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917794378940416 |
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| author | Klevdal, Christian Patrikis, Stefan |
| author_facet | Klevdal, Christian Patrikis, Stefan |
| contents | For a Shimura variety $(G, X)$ in the superrigid regime and neat level subgroup $K_0$, we show that the canonical family of $\ell$-adic representations associated to a number field point $y \in \mathrm{Sh}_{K_0}(G, X)(F)$,
\[
\left\{ ρ_{y, \ell} \colon \mathrm{Gal}(\overline{\mathbb{Q}}/F) \to G^{\mathrm{ad}}(\mathbb{Q}_{\ell}) \right\}_{\ell},
\]
form a compatible system of $G^{\mathrm{ad}}(\mathbb{Q}_{\ell})$-representations: there is an integer $N(y)$ such that for all $\ell$, $ρ_{y, \ell}$ is unramified away from $N(y) \ell$, and for all $\ell \neq \ell'$ and $v \nmid N(y)\ell \ell'$, the semisimple parts of the conjugacy classes of $ρ_{y, \ell}(\mathrm{Frob}_v)$ and $ρ_{y, \ell'}(\mathrm{Frob}_v)$ are ($\mathbb{Q}$-rational and) equal. We deduce this from a stronger compatibility result for the canonical $G(\mathbb{Q}_{\ell})$-valued local systems on connected Shimura varieties inside $\mathrm{Sh}_{K_0}(G, X)$. Our theorems apply in particular to Shimura varieties of non-abelian type and represent the first such independence-of-$\ell$ results in non-abelian type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_03863 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Compatibility of canonical $\ell$-adic local systems on Shimura varieties Klevdal, Christian Patrikis, Stefan Number Theory Algebraic Geometry 11G18, 14G35 For a Shimura variety $(G, X)$ in the superrigid regime and neat level subgroup $K_0$, we show that the canonical family of $\ell$-adic representations associated to a number field point $y \in \mathrm{Sh}_{K_0}(G, X)(F)$, \[ \left\{ ρ_{y, \ell} \colon \mathrm{Gal}(\overline{\mathbb{Q}}/F) \to G^{\mathrm{ad}}(\mathbb{Q}_{\ell}) \right\}_{\ell}, \] form a compatible system of $G^{\mathrm{ad}}(\mathbb{Q}_{\ell})$-representations: there is an integer $N(y)$ such that for all $\ell$, $ρ_{y, \ell}$ is unramified away from $N(y) \ell$, and for all $\ell \neq \ell'$ and $v \nmid N(y)\ell \ell'$, the semisimple parts of the conjugacy classes of $ρ_{y, \ell}(\mathrm{Frob}_v)$ and $ρ_{y, \ell'}(\mathrm{Frob}_v)$ are ($\mathbb{Q}$-rational and) equal. We deduce this from a stronger compatibility result for the canonical $G(\mathbb{Q}_{\ell})$-valued local systems on connected Shimura varieties inside $\mathrm{Sh}_{K_0}(G, X)$. Our theorems apply in particular to Shimura varieties of non-abelian type and represent the first such independence-of-$\ell$ results in non-abelian type. |
| title | Compatibility of canonical $\ell$-adic local systems on Shimura varieties |
| topic | Number Theory Algebraic Geometry 11G18, 14G35 |
| url | https://arxiv.org/abs/2303.03863 |