Representations of the Kottwitz gerbes

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 Rep(Kt_F) and the functor G → B(F,G) defined by Kottwitz in [28]. In particular, we show that if F is the function field of a curve over F_q, then Rep(Kt_F) is equivalent to the category of Drinfeld isoshtukas. In the case of number fields, we establish the existence of various fibre functors on Rep(Kt_Q) and its subcategories and show that Scholze’s conjecture [41, Conjecture 9.5] follows from the full Tate conjecture over finite fields [47]