Anabelian Geometry

by Florian Pop, Univ of Bonn

The aim of the talk was to give a sketch of the proof of a result as follows. First let k be the algebraic closure of some finite field or of the rationals ℚ. For any function field K|k and a prime ℓ ≠ chark we denote by K^{ℓ} a maximal pro-ℓ Galois extension of K, and set G_{K}^{ℓ} = Gal(K^{ℓ}|K). Then we have the following:

Main Result. There exists a group theoretic recipe by which we can recover the field (K, +, .) from G_{K}^{ℓ} in a functorial way, provided td(K|k) > 1. This recipe is invariant under isomorphisms of profinite groups. Thus given a further function field K′|k′ as above, we have:

G_{K}^{ℓ} ≅ G_{K′}^{ℓ} ⇔ K^{i} ≅ K′^{i},

where ( )^{i} means pure inseparable closure. More precisely, one has a canonical bijection:

Isom^{i}(K′^{i}, K^{i}) ≅ Out(G_{K}^{ℓ}, G_{K′}^{ℓ})

where Isom^{i} means up to Frobenius twists, and Out(G_{K}^{ℓ}, G_{K′}^{ℓ}) means outer isomorphisms of profinite groups.

Comments/Remarks.

1) First we remark that the above result completes the Program proposed by Bogomolov [Bo] over k as above. See also Bogomolov–Tschinkel [B–T].

2) “By taking limits” the above result implies a corresponding assertion for the full Galois groups.

Furthermore, the above result implies Grothendieck’s birational anabelian Conjecture for finitely generated infinite fields of transcendence degree > 1. This follows from the fact that for such fields K_{0}|k_{0}, where k_{0} is the constant field, the canonical projection G_{K_{0}} → G_{k_{0}} is encoded in G_{K_{0}}.

Finally, if td(K|k) = 1, then by a result of Harbater and Pop, it follows that G_{K}^{ℓ} is the pro-ℓ free group on countably many generators. Therefore it carries no information about the isomorphy type of K.

Because of the same reason one cannot expect a Hom-form of the Main Result above, as proved by Mochizuki in the case of function fields K_{1}|k_{1} over a sub-p-adic base field k_{1}. Indeed for any two function fields K|k and K′|k′ as above, one can find “many” surjective homomorphisms as follows: G_{K}^{ℓ} → G_{k(t)}^{ℓ} → G_{K′}^{ℓ}, and naturally, these cannot be of geometric nature.

Motivation/Consequence.

The above result is related in an intimate way to the so-called Ihara/Oda–Matsumoto Conjecture, which itself is related to the following idea of Grothendieck about looking for a (non-tautological) description of the absolute Galois group of the rationals G_{ℚ}, as follows:

Let V be a category of (geometrically connected) ℚ-varieties. For every X ∈ V one considers X(ℂ) the complex points of X, and π_{1}^{top}(X(ℂ)) its topological fundamental group (up to inner conjugation). We finally denote by π = π̂^{top} and π^{ℓ} = π̂^{top} the profinite and the pro-ℓ completion of the topological fundamental group, respectively. These are exactly the geometric, respectively the pro-ℓ quotient of the geometric fundamental group functor of V. Thus we get two functors

π_{V} : V → (proGroups), π_{V}^{ℓ} : V → (pro-ℓ-Groups).

Clearly, by the functoriality of the fundamental group, since all varieties from V are defined over ℚ, the absolute Galois group G_{ℚ} = Gal(\overline{ℚ}|ℚ) acts on the homotopy category of V, i.e., there exist group homomorphisms

ı_{V} : G_{ℚ} → Aut(π_{V}), ı_{V}^{ℓ} : G_{ℚ} → Aut(π_{V}^{ℓ}).

Now the problem / question is the following:

1) Give “interesting” categories V such that Aut(π_{V}) and/or Aut(π_{V}^{ℓ}) can be computed.

2) Find V as above such that the resulting ı_{V} and/or ı_{V}^{ℓ} is an isomorphism.

Specifically, Grothendieck proposed to work with categories V consisting of moduli spaces M_{g,n}, e.g. V_{0} = {M_{0,4} , M_{0,5}}. In this last case, Aut(π_{V_{0}}) =: ĜT is the celebrated Grothendieck–Teichmüller group... There is a lot of research in this area of mathematics by several people (Drinfeld, Ihara, Deligne, Belyi, Nakamura, Schneps, Lochak, Harbater, Matsumoto, etc.). In particular, if the category V contains M_{0,4} = ℙ^{1}\{0, 1, ∞}, then both ı_{V} and ı_{V}^{ℓ} are injective.

It is a question by Ihara/Oda–Matsumoto, whether there do exist categories V for which ı_{V} or ı_{V}^{ℓ} are isomorphisms. This question was answered in a positive sense by the present author in 1998 (unpublished).

The remark we want to make here is that actually the Main Result above implies a very strong form of the “positive answer” to the above question. Indeed, let k = \overline{ℚ}, and X some ℚ-variety as above. Set K = k(X). Then denoting by V_{X} a basis of Zariski open neighbourhoods of the generic point of X and inclusions, we have:

• Aut(π_{V_{X}}^{ℓ}) = Out(G_{K}^{ℓ}) and Aut(π_{V_{X}}) = Out(G_{K}). On the other hand, Out(G_{K}^{ℓ}) ≅ Aut(K) by the Main Result above. Therefore, since X is defined over ℚ, we have an exact sequence of the form

1 → Aut_{k}(K) → Aut(K) → Aut(\overline{ℚ}) = G_{ℚ} → 1 .

In particular, if K is rigid, i.e., Aut_{k}(K) = {1}, then Aut(π_{V_{X}}^{ℓ}) = G_{ℚ}.

References:

[Bo] Bogomolov, F. A.: On two conjectures in birational algebraic geometry, in Algebraic Geometry and Analytic Geometry, ICM-90 Satellite Conference Proceedings, ed A. Fujiki et al., Springer Verlag Tokyo 1991.

[B–T] Bogomolov, F. A., Tschinkel, Y.: Reconstruction of function fields, Manuscript,

March 2003.

[P1] Pop, F.: The birational anabelian conjecture revisited, Manuscript, Oct 2002.

[P2] Pop, F.: Pro-ℓ birational anabelian geometry over algebraically closed fields, Manuscript, June 2003.