Tannakian Fundamental Groups Associated to Galois Groups

Summary

The goal of this paper is to give background and motivation for several conjectures of Deligne and Goncharov concerning the action of the absolute Galois group on the fundamental group of the thrice punctured line, and to sketch solutions, complete and partial, of several of them. A major ingredient in these is the theory of weighted completion of profinite groups. An exposition of weighted completion from the point of view of tannakian categories is included.

1. Introduction

The goal of this paper is to provide background, heuristics and motivation for several conjectures of Deligne [14, 8.2, p. 163], [14, 8.9.5, p. 168] and [27, p. 300] and Goncharov [19, Conj. 2.1], presumably along the lines used to arrive at them. A complete proof of the third of these conjectures, and partial solutions of the remaining three are given in [23]. A second goal of this paper is to show that the weighted completion of a profinite group, developed in [23], 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 [23].

2. Motivic Cohomology

It is believed that there is a universal cohomology theory, called motivic cohomology. It should be defined for all schemes X. It is indexed by two integers m and n. The coefficient ring A is typically ℤ, ℤ/N, ℤ_ℓ, ℚ or ℚ_ℓ; the corresponding motivic cohomology group is denoted.