Lifting $G$-Valued Galois Representations when $\ell \neq p$
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866916424813903872 |
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| author | Booher, Jeremy Cotner, Sean Tang, Shiang |
| author_facet | Booher, Jeremy Cotner, Sean Tang, Shiang |
| contents | In this paper we study the universal lifting spaces of local Galois representations valued in arbitrary reductive group schemes when $\ell \neq p$. In particular, under certain technical conditions applicable to any root datum we construct a canonical smooth component in such spaces, generalizing the minimally ramified deformation condition previously studied for classical groups. Our methods involve extending the notion of isotypic decomposition for a $\textrm{GL}_n$-valued representation to general reductive group schemes. To deal with certain scheme-theoretic issues coming from this notion, we are led to a detailed study of certain families of disconnected reductive groups, which we call weakly reductive group schemes. Our work can be used to produce geometric lifts for global Galois representations, and we illustrate this for $\mathrm{G}_2$-valued representations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_03768 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Lifting $G$-Valued Galois Representations when $\ell \neq p$ Booher, Jeremy Cotner, Sean Tang, Shiang Number Theory Algebraic Geometry 11F80 In this paper we study the universal lifting spaces of local Galois representations valued in arbitrary reductive group schemes when $\ell \neq p$. In particular, under certain technical conditions applicable to any root datum we construct a canonical smooth component in such spaces, generalizing the minimally ramified deformation condition previously studied for classical groups. Our methods involve extending the notion of isotypic decomposition for a $\textrm{GL}_n$-valued representation to general reductive group schemes. To deal with certain scheme-theoretic issues coming from this notion, we are led to a detailed study of certain families of disconnected reductive groups, which we call weakly reductive group schemes. Our work can be used to produce geometric lifts for global Galois representations, and we illustrate this for $\mathrm{G}_2$-valued representations. |
| title | Lifting $G$-Valued Galois Representations when $\ell \neq p$ |
| topic | Number Theory Algebraic Geometry 11F80 |
| url | https://arxiv.org/abs/2211.03768 |