Rigid inner forms over global function fields

Abstract

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 $\pi$ 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 $\Delta_{\mathbb{A}}$ 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].