Introduction to birational anabelian geometry

Summary

We survey recent developments in the Birational Anabelian Geometry program aimed at the reconstruction of function fields of algebraic varieties over algebraically closed fields from pieces of their absolute Galois groups.

Introduction

The essence of Galois theory is to lose information, by passing from a field k, an algebraic structure with two compatible operations, to a (profinite) group, its absolute Galois group G_k or some of its quotients. The original goal of testing solvability in radicals of polynomial equations in one variable over the rationals was superseded by the study of deeper connections between the arithmetic in k, its ring of integers, and its completions with respect to various valuations on the one hand, and (continuous) representations of G_k on the other hand. The discovered structures turned out to be extremely rich, and the effort led to the development of deep and fruitful theories: class field theory (the study of abelian extensions of k) and its nonabelian generalizations, the Langlands program. In fact, techniques from class field theory (Brauer groups) allowed one to deduce that Galois groups of global fields encode the field:

Theorem 1 [Neukirch 1969; Uchida 1977]. Let K and L be number fields or function fields of curves over finite fields with isomorphic Galois groups

of their maximal solvable extensions. Then

In another, more geometric direction, Galois theory was subsumed in the theory of the etale fundamental group. Let X be an algebraic variety over a field k. Fix an algebraic closure and let K = k(X) be the function field of X. We have an associated exact sequence

of étale fundamental groups, exhibiting an action of the Galois group of the ground field k on the geometric fundamental group.