Representations of the Kottwitz gerbes

TL;DR

This paper explores the category of representations of Kottwitz gerbes over different fields, establishing equivalences with known categories and connecting to deep conjectures in number theory and algebraic geometry.

Contribution

It demonstrates the equivalence of Rep(Kt_F) with Drinfeld isoshtukas over function fields and links the existence of fiber functors to major conjectures in number theory.

Findings

Rep(Kt_F) is equivalent to Drinfeld isoshtukas over function fields.

Existence of fiber functors on Rep(Kt_Q) related to Tate conjecture.

Scholze's conjecture follows from the full Tate conjecture over finite fields.

Abstract

Let $F$ be a local or global field and let $G$ be a linear algebraic group over $F$. We study Tannakian categories of representations of the Kottwitz gerbes $\text{Rep}(\text{Kt}_{F})$ and the functor $G\mapsto B(F, G)$ defined by Kottwitz in [28]. In particular, we show that if $F$ is a function field of a curve over $\mathbb{F}_q$, then $\text{Rep}(\text{Kt}_F)$ is equivalent to the category of Drinfeld isoshtukas. In the case of number fields, we establish the existence of various fiber functors on $\text{Rep}(\text{Kt}_{\mathbb{Q}})$ and its subcategories and show that Scholze's conjecture [41, Conjecture 9.5] follows from the full Tate conjecture over finite fields [47].