Rigid inner forms over global function fields
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866911106343108608 |
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| author | Dillery, Peter |
| author_facet | Dillery, Peter |
| contents | We construct an fpqc gerbe $\mathcal{E}_{\dot{V}}$ over a global function field $F$ such that for a connected reductive group $G$ over $F$ with finite central subgroup $Z$, the set of $G_{\mathcal{E}_{\dot{V}}}$-torsors contains a subset $H^{1}(\mathcal{E}_{\dot{V}}, Z \to G)$ which allows one to define a global notion of ($Z$-)rigid inner forms. There is a localization map $H^{1}(\mathcal{E}_{\dot{V}}, Z \to G) \to H^{1}(\mathcal{E}_{v}, Z \to G)$, where the latter parametrizes local rigid inner forms (cf. [Kal16, Dil23]) which allows us to organize local rigid inner forms across all places $v$ into coherent families. Doing so enables a construction of (conjectural) global $L$-packets and a conjectural formula for the multiplicity of an automorphic representation $π$ in the discrete spectrum of $G$ in terms of these $L$-packets. We also show that, for a connected reductive group $G$ over a global function field $F$, the adelic transfer factor $Δ_{\mathbb{A}}$ for the ring of adeles $\mathbb{A}$ of $F$ serving an endoscopic datum for $G$ decomposes as the product of the normalized local transfer factors from [Dil20]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2110_10820 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Rigid inner forms over global function fields Dillery, Peter Representation Theory Number Theory 11R39, 11F70, 11R58, 11E72, 11S37, 14F20 18F20 We construct an fpqc gerbe $\mathcal{E}_{\dot{V}}$ over a global function field $F$ such that for a connected reductive group $G$ over $F$ with finite central subgroup $Z$, the set of $G_{\mathcal{E}_{\dot{V}}}$-torsors contains a subset $H^{1}(\mathcal{E}_{\dot{V}}, Z \to G)$ which allows one to define a global notion of ($Z$-)rigid inner forms. There is a localization map $H^{1}(\mathcal{E}_{\dot{V}}, Z \to G) \to H^{1}(\mathcal{E}_{v}, Z \to G)$, where the latter parametrizes local rigid inner forms (cf. [Kal16, Dil23]) which allows us to organize local rigid inner forms across all places $v$ into coherent families. Doing so enables a construction of (conjectural) global $L$-packets and a conjectural formula for the multiplicity of an automorphic representation $π$ in the discrete spectrum of $G$ in terms of these $L$-packets. We also show that, for a connected reductive group $G$ over a global function field $F$, the adelic transfer factor $Δ_{\mathbb{A}}$ for the ring of adeles $\mathbb{A}$ of $F$ serving an endoscopic datum for $G$ decomposes as the product of the normalized local transfer factors from [Dil20]. |
| title | Rigid inner forms over global function fields |
| topic | Representation Theory Number Theory 11R39, 11F70, 11R58, 11E72, 11S37, 14F20 18F20 |
| url | https://arxiv.org/abs/2110.10820 |