Moduli of $G$-bundles on rigid gerbes over affine curves

TL;DR

This paper constructs a geometric stack to understand basic cohomology sets of G-bundles over global function fields, linking local and global structures and proposing a conjectural automorphic multiplicity formula.

Contribution

It introduces a v-stack framework for basic cohomology sets of G-bundles, connecting local and global moduli spaces and establishing a duality principle.

Findings

Construction of the v-stack $ ext{Bun}_{G,F}^{e}$ with localization maps.

Identification of the semistable locus with a union of moduli stacks.

A Tate-Nakayama duality for the basic cohomology set.

Abstract

We geometrize the basic cohomology set $H^{1}(\text{Kal}_{F}, G)_{\text{basic}}$ for a global function field $F$. We do this by constructing a v-stack $\text{Bun}_{G,F}^{e}$ which has localization maps to Fargues' analogous stack $\text{Bun}_{G,F_{v}}^{e}$ for all places $v$ of $F$ and whose semistable locus is the disjoint union of $\text{Bun}_{G_{b},F}$ for all $b \in H^{1}(\text{Kott}_{F} \times_{F} \text{Kal}_{F},G)_{\text{basic}}$. We also prove a version of Tate-Nakayama duality for $H^{1}(\text{Kott}_{F} \times_{F} \text{Kal}_{F},G)_{\text{basic}}$, which lets us state a conjectural multiplicity formula for discrete automorphic representations of $G(\mathbb{A}_{F})$ adapted to this new cohomology set.