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Le groupe fondamental de la droite projective moins trois points. (Fundamental group of the straight line minus three points). (French) Zbl 0742.14022
Galois groups over $$\mathbb{Q}$$, Proc. Workshop, Berkeley/CA (USA) 1987, Publ., Math. Sci. Res. Inst. 16, 79-297 (1989).
[For the entire collection see Zbl 0684.00005.]
Let $$X$$ be an algebraic variety over a number field $$k$$ and $$x\in X(k)$$. A general and very important problem is the following: Let $$\hat\pi_ 1$$ be the profinite completion of the fundamental group $$\pi_ 1(X\otimes_ k\mathbb{C},x)$$. Then using the Grothendieck theory of motifs try to understand the structure of $$\hat\pi_ 1$$ together with the action of $$\hbox{Gal}(\bar k/k)$$ on $$\hat\pi_ 1$$. Grothendieck stressed that this object is particularly interesting in the case when $$X$$ is the complex projective line minus the three points $$\{0,1,\infty\}$$. This problem is very difficult because one does not dispose of an adequate language to tackle it. That is why the author considers (as a first step toward a solution of the problem) the case of “$$\hat\pi_ 1$$ rendu nilpotent”, i.e. the quotients $$\hat\pi_ 1^{(N)}$$ of $$\hat\pi_ 1$$ by the subgroups of its descending central series. This latter object is particularly interesting because it turns out that it is quite close to the cohomology. To this end the author makes precise the meaning of a “ realisation systems” and constructs such a “realisation system” Lie$$(\hat\pi^{(N)}_{1 mot})$$ in such a way that $$X$$ is the projective line minus three points the gradation with respect to the weights filtration of Lie$$(\hat\pi^{(N)}_{1 mot})$$ is just the Lie algebra over $$H_ 1(X)_{mot}\bmod\mathbb{Z}^{N+1}$$ (the central series). Here $$H_ 1(X)_{mot}$$ is the sum of two copies of the Tate motif $$\mathbb{Q}(1)$$. It follows that $$\hbox{Lie}(\hat\pi_{1 mot}^{(N)})$$ is an iterated extension of the Tate motif $$\mathbb{Q}(n)$$.
To pursue this program the author develops a whole theory in detail. Several open questions and conjectures are also discussed.

##### MSC:
 14H30 Coverings of curves, fundamental group 14A20 Generalizations (algebraic spaces, stacks)
##### MathOverflow Questions:
$$p$$-adic realisation of Kummer motive and Frobenius matrix