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. [8, 6]) 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 [6].