Birational anabelian geometry
by
Florian Pop
Univ of Bonn

A celebrated Theorem by Neukirch, Ikeda, Iwasawa, Uchida asserts that the isomorphy type of a number field (and more general, of a global field) is encoded in its absolute Galois group. This theorem is the first approximation of a much broader picture, called Grothendieck's birational anabelian geometry. Roughly speaking, it asserts that every infinite finitely generated field K should be encoded in (an explicite, group theoretic way in) its absolute Galois group G_K, and (iso)morphisms between such fields should correspond to topologically open (iso)morphisms between the corresponding Galois groups.

We plan to give an overview of the known results, and to indicate some of the main ideas of the proofs. We will stress on the so called ``local theory'' and show how to recover information of geometric nature from the absolute Galois group. Valuation theory and regularisation, resp. alterations à la de Jong, play here an essential role.

Date received: July 14, 1999