Uniformization of \mathcal{G}-bundles

TL;DR

This paper proves conjectures related to the structure of moduli stacks of G-torsors over curves, including a uniformization theorem, and computes their connected components and Picard groups for certain cases.

Contribution

It establishes the uniformization theorem for G-bundles in this setting and computes key invariants of the moduli stacks, advancing understanding of their geometric structure.

Findings

Proved the uniformization theorem for G-bundles on curves.

Computed the connected components of the moduli stacks.

Calculated the Picard group for simply connected G.

Abstract

We show some of the conjectures of Pappas and Rapoport concerning the moduli stack of $\mathcal{G}$-torsors on a curve C, where $\mathcal{G}$ is a semisimple Bruhat-Tits group scheme on C. In particular we prove the analog of the uniformization theorem of Drinfeld-Simpson in this setting. Furthermore we apply this to compute the connected components of these moduli stacks and to calculate the Picard group of the stack of torsors in case $\mathcal{G}$ is simply connected.