Tannakian Fundamental Groups Associated to Galois Groups

The goal of this paper is to provide background, heuristics and motivation for several conjectures of Deligne [12, 8.2, p. 163], [12, 8.9.5, p. 168] and [23, p. 300] and Goncharov [15, Conj. 2.1], presumably along the lines used to arrive at them. A complete proof of the third and fourth of these conjectures and partial solutions of the first two are given in [19]. A second goal of this paper is to show that the weighted completion of a profinite group developed in [19] and a key ingredient in the proofs referred to above, can be defined as the tannakian fundamental group of certain categories of modules of the group. This should help clarify the role of weighted completion in [19]. Acknowledgements: We would like to thank Marc Levine for clarifying several points about motivic cohomology and Owen Patashnick for his helpful comments on the manuscript.