On the Birational Anabelian Program Initiated by Bogomolov (i)

Let ` be a fixed rational prime number. Consider function fields K|k over algebraically closed fields k of characteristic 6= `. For each such a function field K|k, let Π K := Gal(K ′′|K) be the Galois group of a maximal pro-` abelian-by-central Galois extension K ′′|K, and ΠK = Gal(K ′|K) be the Galois group of the maximal pro-` abelian sub-extension K ′|K of K ′′|K. At the beginning of the 1990’s, Bogomolov [1] initiated a program which has as ultimate goal to recover function fields K|k with td(K|k) > 1 as above from Π K . (Note that Bogomolov denotes Π c K by PGal c K .) If successful, this program would go far beyond Grothendieck’s birational anabelian conjectures, as k being algebraically closed, there is no arithmetical action in the game. The program is far from being completed, and the present manuscript settles that program in the case k is an algebraic closure of a finite field.