Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Peter Dillery

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.

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

Abstract

We geometrize the basic cohomology set for a global function field . We do this by constructing a v-stack which has localization maps to Fargues' analogous stack for all places of and whose semistable locus is the disjoint union of for all . We also prove a version of Tate-Nakayama duality for , which lets us state a conjectural multiplicity formula for discrete automorphic representations of adapted to this new cohomology set.
Paper Structure (22 sections, 50 theorems, 102 equations)

This paper contains 22 sections, 50 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Motivation and overview
  3. Structure of the paper
  4. Notation and conventions
  5. Acknowledgements
  6. Gerbes and their cohomology
  7. A review of global gerbes
  8. The Kottwitz gerbe
  9. The Kaletha gerbe
  10. Unramified complexes of tori
  11. A Kaletha gerbe with restricted ramification
  12. A new gerbe
  13. The cohomology of $\textnormal{Kott}_{F}^{1/\infty}$
  14. A Tannakian description
  15. The global extended Kottwitz set
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $G$ be any connected reductive group over $F$. Define $B_{e}(G)$ to be all elements of $H^{1}_{\textnormal{\'{e}t}}(\textnormal{Kott}_{F} \times_{F} \textnormal{Kal}_{F}, G)$ whose restriction to the band of $\textnormal{Kal}_{F}$ is central. Then:

Theorems & Definitions (99)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • ...and 89 more