Anabelian Geometry

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In this last chapter we want to give some idea of the “anabelian” program of A. Grothendieck. The term anabelian should be read as “far from being abelian” and as we understand the matter, a group is far enough away from being abelian if all of its subgroups of finite index have a trivial center. The principal idea is the following: in topology, a space X of type K(π,1) is determined by its fundamental group π up to weak homotopy equivalence. If we require that X is a CW-complex, then X is already determined up to strong homotopy equivalence. The “anabelian” idea is that something similar should also be true for schemes, i.e. a scheme X which is an étale K(π,1) should be essentially reconstructible from its étale fundamental group. This is obviously not correct in general, but it should be true under certain conditions; for example, X should be absolutely finitely generated and $\pi^{\mathit{et}}_{1}(X)$ is supposed to be “anabelian”. The smallest constituents of this anabelian world are points, i.e. spectra of fields which are finitely generated over their prime fields. Here the étale fundamental group is just the absolute Galois group. Finite fields have an abelian absolute Galois group (in fact all these fields have the same Galois group $\hat{\mathbb{Z}}$ ), and so the first objects of interest are global fields. In §1,2 we will present some results on “anabelian properties” of global fields, which already existed before Grothendieck formulated his program. We will explain the general conjectures in §3.