Rigid inner forms over global function fields

TL;DR

This paper constructs a global gerbe framework over function fields to define and organize rigid inner forms and local-global relations, facilitating the study of automorphic representations and transfer factors.

Contribution

It introduces a novel fpqc gerbe over global function fields that coherently organizes local rigid inner forms into a global structure, advancing the understanding of automorphic forms and transfer factors.

Findings

Construction of an fpqc gerbe over global function fields.

Organization of local rigid inner forms into global families.

Decomposition of the adelic transfer factor into local factors.

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].