Category Of C-Motives Over Finite Fields

TL;DR

This paper introduces a new semi-simple, non-neutral Tannakian category of motives over algebraic closures of finite fields, advancing the understanding of motives in the context of function fields and their applications to $G$-Shtukas.

Contribution

It generalizes previous constructions of motivic categories, providing a framework more suitable for applications like the Langlands-Rapoport conjecture over function fields.

Findings

The category is semi-simple and non-neutral Tannakian.

It possesses all expected fiber functors.

Applications to $G$-Shtukas and Langlands-Rapoport conjecture are discussed.

Abstract

In this article we introduce and study a motivic category in the arithmetic of function fields, namely the category of motives over an algebraic closure $L$ of a finite field with coefficients in a global function field over this finite field. It is semi-simple, non-neutral Tannakian and possesses all the expected fiber functors. This category generalizes the previous construction due to Anderson and is more relevant for applications to the theory of $G$-Shtukas, such as formulating the analog of the Langlands-Rapoport conjecture over function fields. We further develop the analogy with the category of motives over $L$ with coefficients in $\mathbb{Q}$ for which the existence of the expected fiber functors depends on famous unproven conjectures.