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 Q for which the existence of the expected fiber functors depends on famous unproven conjectures.
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