Recovering Fields from Their Decomposition Graphs

Recall that the birational anabelian conjecture originating in ideas presented in Grothendieck’s Esquisse d’un Programme [G1] and Letter to Faltings [G2], asserts roughly the following: First, there should exist a group theoretical recipe by which one can recognize the absolute Galois groups GK of finitely generated infinite fields K among all the profinite groups. Second, if G = GK is such an absolute Galois group, then the group theoretical recipe recovers the field K from GK in a functorial way. Third, the recipe should be invariant under open homomorphisms of absolute Galois groups. In particular, the category of finitely generated infinite fields (up to Frobenius twist) should be equivalent to the category of their absolute Galois groups and open outer homomorphisms between these groups. A first instance of this situation is the celebrated Neukirch–Uchida Theorem, which says that global fields are characterized by their absolute Galois groups. I will not go into further details about the results concerning Grothendieck’s (birational) anabelian geometry, but the interested reader can find more about this in Szamuely’s Bourbaki Séminaire talk [Sz], and Faltings’ Séminare Bourbaki talk [Fa], and newer results by Stix [St], Mochizuki [Mz], Saidi–Tamagawa [S–T], and Koenigsmann [Ko], Minhyong Kim [Ki] concerning the (birational) section conjecture. The idea behind Grothendieck’s anabelian geometry is that the arithmetical Galois action on rich geometric fundamental groups (like the geometric absolute Galois group) makes objects very rigid, so that there is no room left for non-geometric morphisms between such rich fundamental groups endowed with arithmetical Galois action. On the other hand, Bogomolov [Bo] advanced at the beginning of the 1990’s the idea that one should have anabelian type results in a total absence of an arithmetical action as follows: Let ` be a fixed rational prime number. Consider function fieldsK|k over algebraically closed fields k of characteristic 6= `. For each such a function field K|k, let G ′′ K := Gal(K ′′|K) be the Galois group of a maximal pro-` abelian-by-central Galois extension K ′′|K. Note that if G := [G, GK ](G ) ∞ , where i ≥ 0 and G = GK , are the central `∞ terms of the absolute Galois group GK of K, then we have G ′′ K = G /G. Further, G ′ K = G /G is the Galois group of the maximal pro-` abelian sub-extension K ′|K of K ′′|K; and denoting by G(∞) the intersection of all the G, it follows that GK(`) := GK/G (∞) is the maximal pro-` quotient of GK . Now the program initiated by Bogomolov in loc.cit. has as ultimate goal to recover function fieldsK|k with tr. deg(K|k) > 1 as above from G ′′ K in a functorial way. (Note