Alterations and Birational Anabelian Geometry

Abstract

The basic idea of Grothendieck’s anabelian geometry is that under certain “anabelian hypotheses” the étale fundamental group of a scheme contains all the geometric and arithmetic information about the scheme in discussion, that is to say, the scheme isfunctorially encodedin its étale fundamental group. Such ideas are not completely new, the first assertion of this type being the celebrated result of Artin-Schreier from the middle of the Twenties, which asserts that if the absolute Galois group of some field is non-trivial and finite, then the field in discussion is real closed. This is nevertheless not an assertion about the structure/isomorphy type of the field in discussion, but rather about its elementary theory. It was the attempt to give ap-adic analogueof the Artin-Schreier Theorem which lead Neukirch to the question whether the isomorphy type of a number field (as a field) isgroup theoreticallyencoded in the isomorphy type of the absolute Galois group (as a profinite group) of the number field in discussion.