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arXiv · 2205.06510

Representations of the Kottwitz gerbes

Sergei Iakovenko
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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].

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Sergei Iakovenko. 2022-05-13. Representations of the Kottwitz gerbes. https://arxiv.org/abs/2205.06510

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