On modular rigidity for ${\rm GL}_n$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910617558843392 |
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| author | Matringe, Nadir Mínguez, Alberto Sécherre, Vincent |
| author_facet | Matringe, Nadir Mínguez, Alberto Sécherre, Vincent |
| contents | Let $k$ be a global field and $\mathbb{A}_k$ be its ring of adeles. Let $\ell$ be a prime number and fix a field isomorphism from $\mathbb{C}$ to $\overline{\mathbb{Q}}_{\ell}$. Let $Π_1$ and $Π_2$ be cuspidal automorphic representations of ${\rm GL}_n(\mathbb{A}_k)$ for some integer $n\geq1$. In this paper, we study the following question: assuming that there is a finite set $S$ of places of $k$ containing all Archimedean places and all finite places above $\ell$ such that, for all $v\notin S$, the local components $Π_{1,v} \otimes_{\mathbb{C}} \overline{\mathbb{Q}}_{\ell}$ and $Π_{2,v} \otimes_{\mathbb{C}} \overline{\mathbb{Q}}_{\ell}$ are unramified and their Satake parameters are congruent mod $\ell$, are the local components $Π_{1,w} \otimes_{\mathbb{C}} \overline{\mathbb{Q}}_{\ell}$ and $Π_{2,w} \otimes_{\mathbb{C}} \overline{\mathbb{Q}}_{\ell}$ integral, and do their reductions mod $\ell$ share an irreducible factor for all non-Archimedean places $w$ not dividing $\ell$? We show that, under certain conditions on $Π_1$ and $Π_2$, the answer is yes. We also give a simple proof when $k$ is a function field. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_15209 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On modular rigidity for ${\rm GL}_n$ Matringe, Nadir Mínguez, Alberto Sécherre, Vincent Representation Theory 11F70 Let $k$ be a global field and $\mathbb{A}_k$ be its ring of adeles. Let $\ell$ be a prime number and fix a field isomorphism from $\mathbb{C}$ to $\overline{\mathbb{Q}}_{\ell}$. Let $Π_1$ and $Π_2$ be cuspidal automorphic representations of ${\rm GL}_n(\mathbb{A}_k)$ for some integer $n\geq1$. In this paper, we study the following question: assuming that there is a finite set $S$ of places of $k$ containing all Archimedean places and all finite places above $\ell$ such that, for all $v\notin S$, the local components $Π_{1,v} \otimes_{\mathbb{C}} \overline{\mathbb{Q}}_{\ell}$ and $Π_{2,v} \otimes_{\mathbb{C}} \overline{\mathbb{Q}}_{\ell}$ are unramified and their Satake parameters are congruent mod $\ell$, are the local components $Π_{1,w} \otimes_{\mathbb{C}} \overline{\mathbb{Q}}_{\ell}$ and $Π_{2,w} \otimes_{\mathbb{C}} \overline{\mathbb{Q}}_{\ell}$ integral, and do their reductions mod $\ell$ share an irreducible factor for all non-Archimedean places $w$ not dividing $\ell$? We show that, under certain conditions on $Π_1$ and $Π_2$, the answer is yes. We also give a simple proof when $k$ is a function field. |
| title | On modular rigidity for ${\rm GL}_n$ |
| topic | Representation Theory 11F70 |
| url | https://arxiv.org/abs/2409.15209 |