FINITE TRIPOD VARIANTS OF I/OM ON IHARA S QUESTION / ODA MATSUMOTO CONJECTURE FLORIAN POP

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1 FINITE TRIPOD VARIANTS OF I/OM ON IHARA S QUESTION / ODA MATSUMOTO CONJECTURE FLORIAN POP Abstract. In this note we introduce and prove a wide generalization and sharpening of Ihara s question / Oda-Matsumoto conjecture, for short I/OM. That leads to a quite concrete topological/combinatorial description of absolute Galois groups, in particular of Gal Q = Aut(Q), as envisioned by Grothendieck in his Esquisse d un Programme. 1. Introduction/Motivation A consequence of the results of this paper is a positive answer to a question by Ihara from the 1980 s, which in the 1990 s became a conjecture by Oda Matsumoto, for short classical I/OM. In essence, the classical I/OM is about giving combinatorial/topological descriptions of the absolute Galois group of the rational numbers Gal Q = Aut(Q). Before giving the results in their full strength, let me briefly present the broader context in which the classical I/OM evolved as one of the main problems in Grothendieck s anabelian program, which itself grew out of [G1], [G2] (see [GGA]). To fix notation and context, let G out be the category of profinite groups and outer homomorphisms. For geometrically integral Q- varieties X, setting X := X Q Q, one has: First, viewing the algebraic fundamental group π 1 (X) := π1 et (X, ) of X as an object in G out renders keeping track of base points irrelevant. Further, X(C) endowed with the complex topology is a nice topological ( space, ) and π 1 (X) is the profinite completion of the topological fundamental group π top 1 X(C), ; hence π1 (X) is in a precise sense an object of combinatorial/topological nature. Second, the canonical exact sequence of étale fundamental groups 1 π 1 (X) π1 et (X) Gal Q 1 gives rise to a canonical representation ρ X : Gal Q Out ( π 1 (X) ) = Aut G out(π 1 (X)), which is compatible with the canonical projections π(x) π(y ) defined by morphisms X Y of Q-varieties. Hence if V is any category of geometrically integral Q-varieties, its algebraic fundamental group functor π V : V G out, X π 1 (X) is well defined, and one gets a canonical representation: ρ V : Gal Q Aut(π V ), σ ( ρ X (σ) ) X V Thus the question of giving topological/combinatorial descriptions of Gal Q would follow from giving categories V of geometrically integral Q-varieties for which ρ V is an isomorphism. Among other things, Grothendieck suggested to use subcategories V T of the Teichmüller modular tower T of all the moduli spaces M g,n, and try to answer the two questions: Date: Variant of January, ey words and phrases. Anabelian geometry, function fields, (generalized) [quasi] prime divisors, decomposition graphs, Hilbert decomposition theory, pro-l Galois theory, algebraic/étale fundamental group 2010 MSC. Primary 11G99, 12F10, 12G99, 14A99. Partially supported by the Simons Foundation Grant

2 First, for which categories V is the representation ρ V injective. Second, describe the image im(ρ V ) Aut(π V ), and particular decide whether ρ V is surjective. There was and is an intensive and extensive effort to answer the questions above and related ones, starting with work by Deligne [De], Ihara [I1], see also [I2], [I3], Drinfeld [Dr], and subsequently by many others, e.g. [An], [F1], [F2], [H-Ma], [H-Sch], [H-Mz], [HLS], [I-M], [LNS1], [LNS2], [L-Sch], [Ma], [M-T], [Na], [N-Sch], [Sch], to mention a few. In particular, there is a canonical embedding of Gal Q in the Grothendieck Teichmüller group Gal Q ĜT Aut( F 2 ), 1 as well as in its more sophisticated variants Gal Q ĜT. On the other hand, it turns out that all these more or less abstractly defined subgroups of Aut( F 2 ) are actually of the form ĜT = Aut(π V ) for properly chosen categories V of geometrically integral Q-varieties; e.g. ĜT = ĜT V 0, where V 0 := {M 0,4, M 0,5 } is the full subcategory of T with objects M 0,4, M 0.5, cf. Harbater Schneps [H-Sch]. On the other hand, the other categories V under discussion, are not necessarily subcategories of T. Concerning concrete general results, the nature of ρ V in the above cases and in general is only partially understood. First, concerning the injectivity of ρ V, Drinfel d remarked that using Belyi s Theorem [Be] it follows that ρ V is injective, provided U 0 := M 0,4 = P 1 \{0, 1, } lies in V, and Voevodsky showed that the same is true if X V, where X := E \{ } is the complement of a point in an elliptic curve E; Matsumoto [Ma] showed that the same holds if X V for some affine hyperbolic curve X, and finally, Hoshi Mochizuki [H Mz] proved that ρ V is injective as soon as V contains any hyperbolic curve (complete or not). On the other hand, the question about describing non-tautologically the image im(ρ V ), in particular the question about the surjectivity of the representation ρ V, was/is less understood. Ihara asked in the 1980 s whether ρ V is an isomorphism, provided V = Var Q ; and Oda Matsumoto conjectured (based on some motivic evidence) in the 1990 s that Ihara s question has a positive answer. Let classical I/OM stand for Ihara s question/oda Matsumoto conjecture: Classical I/OM. Prove that if V = Var Q, then ρ V : Gal Q Aut(π V ) is an isomorphism. The author gave a proof (end of 1990 s) of the above classical I/OM, and slightly later, André [An] showed that the p-adic tempered I/OM holds. [This variant of the I/OM is obtained by replacing Q by Q p and π 1 (X) by the tempered fundamental group π temp 1 (X), which carries more information than π 1 (X).] Author s original proof of the classical I/OM was never published, because shortly later, he started developing a completely new approach to tackle I/OM types questions. That approach allows among other things to formulate and prove (birational) pro-l abelian-by-central variants of I/OM, which are much stronger than and imply the classical I/OM, cf. [P5]. 2 In a nutshell, the basic idea is as follows: Let k 0 be an arbitrary field, k := k 0. For a category V of geometrically integral k 0 -varieties, let F be the category of the functions fields := k(x), X V having as morphisms the k-embeddings L := k(y ) k(x) =: defined by the dominant V-morphisms X Y. Let 1 Here, F 2 is the profinite free group on two generators. 2 Among other things, the present note renders the officially unpublished [P5] obsolete. 2

3 C be a Serre class of groups, e.g., finite (abelian-by-central) [l-groups], unipotent/linear, etc., and π C 1 (X) and Gal C be the corresponding completions of π 1 (X), respectively Gal. Then paralleling the discussion above, one can formulate the pro-c I/OM for V and the pro-c I/OM for F, where the latter should be rather called the birational pro-c I/OM for V. Moreover, if V contains a basis of open neighborhoods of the generic point η X for every X V, e.g., V = Var k0, then by taking limits one gets: Every automorphism Φ C V Aut(π C V ) gives rise to an automorphism Φ C F Aut(Gal C F), that is, to automorphisms Φ C Gal C, F, compatible with all F-morphisms L. And an easy verification shows that the pro-c I/OM for V follows from the birational pro-c I/OM for V. In particular, for C the class of all the finite abelian-by-central l-groups, l char(k 0 ), one gets the (birational) pro-l abelian-bycentral I/OM for V, as introduced [P5] and proved there for sufficiently rich categories V by using techniques developed to tackle the so called Bogomolov s Program; see [P3], [P4] for details about the latter. This also suggests that in the case of other classes C, like the ones mentioned above, the corresponding pro-c I/OM type results might lead to Galois group like objects of interest in arithmetic/algebraic geometry. See Remark 2.10 for such an instance. For the rest of the paper, we introduce notations as follows: Notations 1.1. Let l be a fixed prime number, and k 0 an arbitrary base field, char(k 0 ) l. For geometrically integral k 0 -varieties X, let π 1 (X) Π c X Π X be the pro-l abelian-bycentral, respectively pro-l abelian (quotients of the) algebraic fundamental group of X. We notice the following: 1) First, the canonical projections π 1 (X) Π c X Π X give rise canonically to projections Aut G out( π1 (X) ) Aut G out(π c X ) Aut(Π X), which usually are not injective or surjective. 2) One has Π X = H 1 et(x, Z l ), and Aut c (Π X ) := im ( Aut G out(π c X ) Aut(Π X )) /Z l consists of the automorphisms compatible with : H 1 et(x, Z l ) H 1 et(x, Z l ) H 2 et(x, Z l ). Finally, for categories V of geometrically integral k 0 -varieties, consider the corresponding quotients of π V and the resulting representation of Gal k0 below π V Π c V Π V, ρ c V : Gal k0 Aut(π V ) Aut c (Π V ). Notice that the classical I/OM is a rather theoretical question of foundational nature. On the other hand, by the discussion above, the (birational) pro-l abelian-by-central I/OM for V is quite concrete and relates in down-to-earth terms to the étale l-adic cohomology of the category V. Further we notice that, strictly speaking, the sufficiently rich hypothesis under which the (birational) pro-l abelian-by-central I/OM for V was proved in [P5] requires: a) Every Y V is dominated by some X V satisfying: dim(x) > 1 and V contains some basis B X of Zariski open neighborhoods U i of the generic point η X X. b) For X and U i B X as above, V contains (virtually) all dominant morphisms U i U 0, where U 0 = P 1 \{0, 1, } is the tripod. Especially condition b) is quite restrictive and moves away from and beyond the Teichmüller tower type situation. This being said, the aim of this note is to prove similar I/OM type results but under much weaker hypotheses, by weakening hypothesis b) to the extent that V contains the morphisms U i U 0 defined by only (finitely many) rational maps ϕ t : X U 0, given in advance, necessary to rigidify V. 3

4 This gives much more concrete descriptions of Gal k0 for k 0 global and/or local fields, that might be used in studying representations of Gal k0 and the (birational) Tate conjectures. Example. The birational Grothendieck Teichmüller groups ĜT bir and ĜT c bir Recall that V 0 := {M 0,4, M 0,5 }, and let ϕ i : M 0,5 M 0,4 be the morphisms of V 0 defined by forgetting the i th marked point for 1 i 5. Further, one has M 0,4 = P 1 \{0, 1, } = U 0, and M 0,5 = U 0 U 0 \ U0 with U0 the image of the diagonal morphism U 0 U 0 U 0. Hence M 0,4 = Spec Q[t 0, 1 t 0, 1 1 t 0 ], M 0,5 = Spec Q[t 1, t 2, 1 t 1, 1 t 2, 1 1 t 1, 1 1 t 2, 1 t 1 t 2 ], and the projections ϕ i : M 0,5 M 0,4 are defined by the field embeddings Q(t 0 ) Q(t 1, t 2 ), t 0 t Θ 0, where Θ 0 := {t 1, t 2, t 1 t 2, t, t } with t, t Q(t 1, t 2 ) explicitly computable. We set Θ := {t 1, t 2, t 2 t 1 }, and for an arbitrary but otherwise fixed basis B = {U i i I} (w.r.t. inclusion) for the complements of curves C i = V (f i ) M 0,5, consider the category: V 0,bir := V 0,Θ, B with objects B {U 0 }, and having as morphisms, first, the canonical inclusions U j U i for C i C j and id U0, and second, the projections ϕ t : U i U 0 defined by Q(t 0 ) Q(t 1, t 2 ), t 0 t Θ. Then in the above notation, one has the resulting canonical representations ( ) ρ V0,bir : Gal Q Aut(π V0,bir ) =: ĜT bir, ρ c V 0,bir : Gal Q Aut c (Π V0,bir ) =: ĜT c bir An easy verification shows that the category V 0,bir satisfies Hypothesis (H), from Definition/Remark 2.2 below. Hence in this concrete situation, by Theorem 2.6, 1) and Theorem 2.7, 1) below, one gets the following far reaching generalization of the results from [P5]: Theorem. The canonical representations ρ V0,bir and ρ c V 0,bir from ( ) above are isomorphisms. 2. Presentation of the results As already mentioned, the results of this note refine and generalize the ones from [P5]. Essential technical steps and tools for the proofs developed here are new and go beyond what was done in loc.cit. The results presented here hold and will be proved over arbitrary perfect base fields k 0. In particular, the classical I/OM over Q is a consequence of the tame I/OM, as given below in Theorem 2.7. We begin by introducing/recalling notation and terminology. Recall that G out is the category of profinite groups and outer (continuous) group homomorphisms, i.e., for given G, H G out, an element of Hom G out(g, H) is of the form Inn H f with f : G H a continuous and Inn H the inner automorphisms of H. We set k := k 0, and given a geometrically integral k 0 -variey X and its base change X := X k0 k, we view the algebraic fundamental group π 1 (X) := π1 et (X, ) of X as an object in G ouṭ Hence the ambiguity resulting from base points vanishes, and by mere definitions one has Out ( π 1 (X) ) = Aut G out(π 1 (X)). Further, via the canonical exact sequence 1 π 1 (X) π et 1 (X) Gal k0 1 one gets a representation ρ X : Gal k0 Out ( π 1 (X) ) = Aut G out(π 1 (X)), and by the functoriality of the étale fundamental group, the collection of all the representations (ρ X ) X, is compatible with the base changes of k 0 -morphisms f : X Y of geometrically integral k 0 -varieties. In particular, for every category V of geometrically integral varieties over k 0, its algebraic fundamental group functor π V :V G out gives rise to a representation 4

5 ρ V : Gal k0 Aut(π V ), where Aut(π V ) is the automorphism group of π V. In down to earth terms, the elements Φ Aut(π V ) are the families Φ = (Φ X ) X V, Φ X Out ( π 1 (X) ) = Aut G out(π 1 (X)), which are compatible with π 1 (f) : π 1 (X) π 1 (Y ) for all V-morphisms f : X Y. Next we recall the pro-l abelian-by-central I/OM from [P5] in detail. Let π 1 Π c Π be the pro-l abelian-by-central and the pro-l abelian quotients of π 1 as introduced in Notations 1.1. Then by mere definitions, Π c X Π X are the maximal pro-l quotients of π 1(X) with Π X abelian, and ker(π c X Π X ) in the center of Πc X. Since the kernels in π 1(X) Π c X Π X are characteristic subgroups, there are canonical projections: Aut G out(π 1 (X)) Aut G out(π c X) Aut(Π X ). Hence for every category V of geometrically integral k 0 -varieties, the canonical morphisms of functors π V Π c V Π V give rise to homomorphisms Aut(π V) Aut(Π c V ) Aut(Π V ). Further, Z l acts by multiplication on Π X, and by general group theoretical non-sense, that action lifts to a Z l -action on Πc X. Hence we get naturally a representation: ρ c V : Gal k 0 Aut c (Π V ) := im ( Aut(Π c V ) Aut(Π V )) /Z l. Conjecture (pro-l abelian-by-central I/OM over k 0 ). Let V = Var k0 be the category of geometrically integral k 0 -varieties. Then ρ c V : Gal k 0 Aut c (Π V ) is an isomorphism. We will prove more precise and much stronger assertions than the above pro-l abelian-bycentral I/OM over k 0. In order to present the results, we need some preparation as follows: First, concerning general terminology, let p X : X S, p Y : Y S be given S-schemes. When speaking about a morphism f : X Y, we mean a pair (f, f S ), where f S : S S is a scheme isomorphism such that f S p X = p Y f. In particular, if f S = id S, then f is actually an S-morphism. We denote by Hom S (X, Y ) Hom(X, Y ) the corresponding spaces of S-morphisms, respectively morphisms from X to Y. Second, if char(s) = p > 0, we tacitly assume that the schemes are perfect, i.e., the absolute Frobenius is an isomorphism of schemes, and identify two morphisms which differ by a Frobenius twist. To indicate this, we will use the notation Hom i (X, Y ), and to reduce the amount of explanation, we will use this notation in the case char(s) = 0 as well, where actually Hom i (X, Y ) = Hom(X, Y ). Let k 0 be a fixed perfect field with l char(k 0 ). By the convention above, when speaking about a k 0 -variety X, we will actually mean its perfect closure X i, and in particular, the function field k 0 (X) will be replaced by k 0 (X) i = k 0 (X i ). Finally, up to Frobenius twists, a morphism between k 0 -varieties X, Y is a morphism of schemes f : X Y which induces a field isomorphism f k0 : k 0 k 0 on k 0. Finally, up to Frobenius twists, the dominant rational maps ϕ : X Y are in bijection with the field embeddings k 0 (Y ) i k 0 (X) i which map k 0 onto itself. In particular, up to Frobenius twists, the automorphisms φ : k 0 (X) i k 0 (X) i with φ(k 0 ) = k 0 are in canonical bijection with the birational maps ϕ : X X, say φ ϕ. We also mention that in the case k 0 is replaced by its algebraic closure k, all the above facts hold, but something new specific to the situation happens. Namely, since k k(x) i is the unique maximal algebraically closed subfield, one has: Every field isomorphism φ of k(x) maps k isomorphically onto itself, and therefore originates from a unique birational map ϕ : X X up to Frobenius twists. But since φ does not necessarily map k 0 onto/into itself, ϕ is not necessarily induced by base change from a birational map X X. 5

6 Recall that U 0 := Spec k 0 [t 0, 1/t 0, 1/(1 t 0 )] = P 1 k 0 \{0, 1, } is the k 0 -tripod (terminology by Hoshi Mochizuki) with canonical parameter t 0 on P 1, and that Aut i (U 0 ) = Aut i (k 0 ) S 3, and Aut i k 0 (U 0 ) = S 3. Actually, setting U t0 := {t 0, 1 t 0, 1/t 0, 1/(1 t 0 ), t 0 /(t 0 1), (t 0 1)/t 0 }, the representatives φ of elements in Aut i k 0 (U 0 ) are defined by t 0 t pe φ for some e Z, t φ U t0. Up to Frobenius twists, the rational dominant maps ϕ t : X U 0 are in bijection with the field embeddings φ t : k 0 (t 0 ) k 0 (X), t 0 t k 0 (X) and φ t (k 0 ) = k 0, and ϕ t is defined on all sufficiently small open subsets U X. Definition/Remark 2.1. In the above notations, for every open subset U X there exists a unique maximal open subset U max X such that U U max and the canonical projection Π U Π Umax is an isomorphism, or equivalently, ker(gal Π U ) = ker(gal Π Umax ); in particular, U max is uniquely determined by ker(π Π U ). We say that U is maximal, if U max = U, and notice that U X is maximal if and only if U X is maximal. For a set of Zariski open subsets B of X, we denote B max := {U max U B}, and notice that the base change B max of B max under k 0 k is precisely {V max V B}. Further, if B is a basis of open neighborhoods of the generic point η X, then so is B max, thus so is B max for η X. Finally, let ϕ : X X be a birational map. Then ϕ is defined on U max for all sufficiently small open subsets U X, and if ϕ is defined on some U max, then ϕ(u max ) = ϕ(u) max. Definition/Remark 2.2. In the above notations, let Θ k 0 (X) be a subset of non-constant functions, and ϕ t : X U 0, t 0 t Θ, be the corresponding dominant rational k 0 -maps. 1) For a basis of neighborhoods B of the generic point η X X, consider the small category V X := V X,Θ, B with objects B {U 0 }, and morphisms as follows: First, id U0 and the canonical inclusions U j U i, and second, the restrictions ϕ t,i := ϕ t Ui, t Θ, provided ϕ t is defined on U i. 2) Let ϕ : X i X i be a birational map. We say that ϕ is V X -compatible, if ϕ satisfies: i) There exists ϕ 0 Aut i (U 0 ) such that ϕ 0 ϕ t = ϕ t ϕ, t Θ, as rational maps. ii) If U B and ϕ is defined on U max, then ϕ(u max ) = U max. The set of all the V X -compatible birational maps is a subgroup Aut i V X () Aut i (), and the image of the canonical embedding Gal k0 Aut( i ) lies actually in Aut i V X (). 3) We say that V X is rigid, if it satisfies the following equivalent conditions: i) The restriction of every φ Aut i V X () to k 0 (X) i is a power of Frobenius. ii) The canonical embedding Gal k0 Aut i V X () is surjective, thus Aut i V X ()= Gal k0. Hypothesis (H): The following are satisfied: dim(x) > 1, k(x) = k(θ), V X is rigid. Remark 2.3. The following hold (the proofs being straightforward verifications): 1) The fact that V X satisfies Hypothesis (H) above is somehow the generic case. Indeed: a) V X is rigid, provided i k 0 is geometrically rigid, i.e., Aut k ( i ) = 1 = Aut i () k0. In general, if ϕ : X X is not a Frobenius twist, there are arbitrarily small open subsets U X with ϕ(u max ) U max. Hence if B is chosen randomly, then V X is rigid. b) = k(θ), provided Θ contains a basis of a linear space L for X which defines the birational class of X, i.e, the canonical rational map X P( L ) is a birational map. 6

7 c) Finally, if Θ is a linear space such that the canonical rational map X P( Θ ) is a birational map, then = k(θ) and V X is rigid for all B. 2) If V X is rigid, then Aut c (Π VX ) consists of all the systems of automorphisms ( (Φ i ) i, Φ 0 ) with Φ i Aut c (Π Ui ) and Φ 0 Aut c (Π U0 ) such that for all U j U i and ϕ t,i : U i U 0, t Θ, the diagrams below are commutative: ( ) Φ j Φ Π Uj ΠUj Π i Ui ΠUi can can Π(ϕ t,i ) Π(ϕ t,i ) Φ i Φ ΠUi Π 0 U0 ΠU0 Π Ui Definition/Remark 2.4. In the above context, let ( (Φ i ) i, Φ 0 ) Aut c (Π VX ) be given. 1) Since Π c Π is the projective limit of the projective system (Πc U i Π Ui ) i I, it follows from ( ) above that (Φ i : Π Ui Π Ui ) i defines a unique Φ Aut c (Π ) satisfying: i) Let π t : Π Π U0 be the canonical projections defined by ı t : k 0 (t 0 ) k 0 (X), t 0 t Θ. Then π t Φ = Φ 0 π t, and in particular, Φ ( ker(π t ) ) = ker(π t ). ii) Let p Ui : Π Π Ui be the canonical projection. Then p Ui Φ = Φ i p Ui, and in particular, Φ ( ker(p Ui ) ) = ker(p Ui ). 2) Given an automorphism Φ Aut c (Π ), we say that Φ is V X -compatible, if it satisfies conditions i), ii) above, and let Aut c V X (Π ) be the set of all such automorphisms of Π. An easy verification shows that one has canonical group embeddings: Aut c (Π VX ) Aut c V X (Π ) Aut c (Π ). 3) Recalling the group of V X -compatible automorphisms Aut i V X () as introduced in Definition/Remark 2.2, 2), one has that the canonical map Aut i () Aut c (Π ) arising from Galois Theory is injective, and gives rise by restriction to a canonical embedding: Aut i V X () Aut c V X (Π ). The stronger/more precise form of the pro-l abelian-by-central I/OM for V X is as follows: Conjecture [(Birational) pro-l abelian-by-central I/OM for V X ]. If V X satisfies Hypothesis (H), then Gal k0 Aut c (Π VX ) Aut c V X (Π ) are isomorphisms. Definition 2.5. Let V be a category of geometrically integral k 0 -varieties. 1) For X, Y V, we say that X dominates Y, denoted Y X, if there exists a dominant morphism X Y which is a V-morphism. 2) We say that V is connected, if for every X, Y in V there exist X 0,..., X 2m in V such that X 0 = X, X 2m = Y, and for 0 i < m one has X 2i, X 2i+2 X 2i+1. 3) We say that V satisfies Hypothesis (H), if for every X V there exists some X V such that V contains a subcategory V X satisfying Hypothesis (H), and there is some U V X with X U and Π U Π X surjective. Theorem 2.6. Let k 0 be a perfect field. In the above notations the following hold: 1) Suppose that the category V X satisfies Hypothesis (H). Then the resulting canonical representations Gal k0 Aut c (Π VX ) Aut c V X (Π ) are isomorphisms. 2) Let V be a connected category satisfying Hypothesis (H). Then the canonical representation ρ c V : Gal k 0 Aut c (Π V ) is an isomorphism. 7

8 An application of Theorem 2.6 is the following strengthening of the classical I/OM: In the general context above, replace π 1, Gal by their valuation tame quotients π t 1, Gal ṭ Then for every category of geometrically integral k 0 -varieties V one gets a representation ρ t V : Gal k0 Aut(π t V). Further, in the context of V X above, every Φ t Aut(πV t X ), say given by Φ 0, (Φ t i) i, defines an automorphism Φ t Out(Gal), t which is V X -compatible, i.e., maps ker ( Gal t π1(u t i ) ) onto itself, thus induces isomorphisms Φ t i : π1(u t i ) π1(u t i ), U i V X, and p t t Φ t = Φ 0 p t t for p t t : Gal t Π U0, t Θ. Hence if Aut VX (Gal) t Out(Gal) t denotes the subgroup of V X -compatible automorphisms, then one has a canonical embedding Aut(π t V X ) Aut VX (Gal t ). Theorem 2.7. Let k 0 be a perfect field. In the above notations the following hold: 1) Suppose that the category V X satisfies Hypothesis (H). Then the canonical representations Gal k0 Aut(π t V X ) Aut VX (Gal t ) are isomorphisms. 2) Let V be a connected category satisfying Hypothesis (H). Then the canonical representation ρ t V : Gal k 0 Aut(π t V) is an isomorphism. The essential technical tool in the proof of the above results is Theorem 2.9 below, which is related to Bogomolov s Program as initiated in [Bo], see rather [P3], Introduction. Definition/Remark 2.8. In the above notations, let Θ \k be a non-empty set, endowed with a bijection θ : Θ Θ, t u, and for t Θ, recall ϕ t : X U 0 and the corresponding π t : Π Π U0. We say that ϕ : X X, respectively Φ Aut c (Π ), are weakly Θ-compatible, if for every t u, there is ϕ 0 Aut(U 0 ), respectively Φ 0 Aut(Π U0 ), depending on t, u, such that ϕ t ϕ 0 = ϕ u ϕ, respectively Φ 0 π t = π u Φ. Notice that ϕ and/or Φ being weakly Θ-compatible is in general much weaker than conditions i) from Definition/Remark 2.2, 2), respectively 2.4, 1) above. Further, the corresponding subsets Aut Θ ( i ) Aut( i ), Aut c Θ (Π ) Aut(Π ) are actually subgroups, and the canonical embedding Aut( i ) Aut c (Π ) defines an embedding Aut Θ ( i ) Aut c Θ (Π ). Theorem 2.9. Let k be a function field with td( k) > 1, and Θ satisfy = k(θ). Then the canonical embedding Aut Θ ( i ) Aut c Θ (Π ) is an isomorphism. Equivalently, for every Φ Aut c Θ (Π ) there exists φ Aut(i ), unique up to Frobenius twists and satisfying: i) φ defines Φ, i.e., letting φ be the prolongation of φ to the maximal pro-l abelian extension, there exists ε Z l such that ε Φ(g) = φ 1 g φ for all g Π. ii) For t u under θ : Θ Θ, there exists t φ {t, 1 t, 1/t, 1/(1 t), t/(t 1), (t 1)/t} and a power p e, e Z, of the characteristic exponent p of k, such that φ(u) = t pe φ. It turns out that Theorem 2.6 follows relatively easily from Theorem 2.9, whereas Theorem 2.7 follows from Theorem 2.6 and some extra (partially quite technical) work. The techniques for the proof of Theorem 2.9 are the ones developed to tackle Bogomolov s Program, supplemented by some new ideas. Namely using Π c endowed with π t : Π Π U0, t Θ, one has the following: First, Proposition 3.10 gives a recipe to recover the divisorial subgroups of Π, and based on that, Proposition 3.11 gives a group theoretical recipe to recover the total decomposition graph G D tot of k, as introduced/defined in [P3], see section 3. 8

9 Second, using the Construction 4.6, one gives in Proposition 4.7 a recipe to recover the geometric rational quotients of G D tot. Moreover, the group theoretical recipes to recover these objects are preserved under all the automorphisms Φ Aut c Θ (Π ) and/or Φ Aut Θ(Gal), t see Propositions 4.7 and Lemma 7.6 below. Thus by the main result of [P3], Introduction, it follows that every Φ Aut c Θ (Π ) originates from geometry, i.e., there exists ε Z l such that ε Φ is defined by some automorphism φ : i k i k, etc. Remark In very recent work, Topaz [To3] gives yet another refinement of the (birational) pro-l abelian-by-central I/OM from [P5], in the spirit of the comments in the middle of the Introduction/Motivation above. He introduces, namely, and proves mod-l abelian-bycentral variants of I/OM as follows: First, consider the mod-l abelian-by-central and mod-l abelian, quotients π 1 π1 c π1 a of π 1. Second, for U a U 0 open, consider categories U a similar to the categories V X above, but satisfying extra conditions, e.g., Θ = k 0 (X)\k 0 consists of all the non-constant functions (as done in [P5] as well), and the dimension restriction dim(x) > 4. Then Aut c (π a U a ) = Gal k0, thus giving a purely combinatorial description of Gal k0. Nevertheless, for the time being, it is unclear whether any of the mod-l abelian-by-central variants of I/OM from [To3] holds for the coarser categories V X and/or V which satisfy Hypothesis (H) as introduced above, e.g., Θ finite, and/or 1< dim(x)<5. Thanks: I would like to thank all who showed interest in this work, among whom: Yves André, Ching-Li Chai, Pierre Deligne, Jakob Stix, Alexander Schmidt and Tamás Szamuely for technical discussions and help, Pierre Lochak for insisting that these facts should be thoroughly investigated, and V. Abrashkin, R. Hain, Y. Ihara, Minhyong im, M. Matsumoto, N. Nakamura, M. Saidi, A. Tamagawa for discussions on several occasions. Last but not least, many thanks to the referee, for the careful reading of the manuscript and suggestions. Special thanks are due to the University of Heidelberg and the University of Bonn for the excellent working conditions during my visits there (academic year ) as visiting scientist supported by the Research Prize of the Humboldt Foundation. 3. Recovering the total decomposition graph A) Recalling basics about (quasi) divisorial subgroups We begin by recalling a few basic definitions/notations from valuation theory, including Hilbert decomposition/ramification theory in pro-l abelian field extensions, l char. First, for an arbitrary field Ω containing µ l, and a valuation v of Ω, let T v Z v be the inertia/decomposition groups of v in Π Ω, and Ω Z Ω T be the corresponding fixed fields in the maximal pro-l abelian extension Ω Ω. (Note that because Π Ω is abelian, T v Z v depend on v only, and not on the prolongation of v to Ω used to define them.) Further, let Uv 1 := 1 + m v O v =: U v be the principal v-units, respectively the v-units in Ω. Then by [P1], Fact 2.1, see also Topaz [To1], [To2], one has that Ω Z Ω Z1 := Ω[ l Uv 1 ], and Ω T Ω T 1 := Ω[ l U v ]. We denote Tv 1 := Gal ( ) Ω Ω T 1 Tv, Zv 1 := Gal ( ) Ω Ω Z1 Zv and call Tv 1 Zv 1 the (minimized) inertia/decomposition groups of v. Since vω = Ω /U v and Ωv = U v /Uv 1, by ummer theory and Pontrjagin duality, setting δ := dim(vω/l), one has: ( ) T 1 v = Hom cont ( vω, Zl (1) ) = Z δ l, Π 1 Ωv := Z 1 v /T 1 v = Hom cont ( Ωv, Z l (1) ). We notice the following: First, if char(ωv) l, then by [P1], Fact 2.1, one has that Zv 1 = Z v and Π 1 Ωv = Π Ωv. Second, if char(ωv) = l, one has: Since l char(ω), one must have 9

10 char(ω) = 0. Further, Tv 1 Zv 1 T v, thus Π 1 Ωv T v/tv 1 has trivial image in Π Ωv = Z v /T v, and the residue field of Ω Z1 v contains l Ωv. Second, let Ω κ be a function field, say Ω = κ(z) is the function field of some (geometrically) integral κ-variety Z. A defectless valuation, or a valuation without defect, of Ω κ is any valuation v of Ω which satisfies the Abhyankar equality td(ω κ) = td(ωv κv) + rr(vω/vκ), where we denote rr(a) := dim Q A Q the rational rank of any abelian group A. Suppose that κ = κ. Then given a defectless valuation v of Ω κ, the following hold, see e.g., [h]: a) vω/vκ is a finitely generated free abelian group, and Ωv κv is a function field. b) The restriction v 1 := v Ω1 of v to any function subfield Ω 1 κ Ω κ is defectless. Coming back to the context from Introduction, recall that a prime divisor of k is a discrete valuation v of which is trivial on k and has a function field v as residue field satisfying td(v k) = td( k) 1. We call T v Z v a divisorial subgroup of Π. It turns out that knowing the divisorial subgroups T v Z v of Π is one of the key technical ingredients in reconstructing the function field k from its pro-l abelian-by-central Galois group Π c. Unfortunately, at the moment there is no group theoretical recipe to recover the divisorial groups T v Z v from the group theoretical information encoded in Π c in the case of an arbitrary algebraically closed base field k. The best one can do so far in general is to recover the larger class of all the (minimized) quasi divisorial subgroups Tv 1 Zv 1 of Π from the group theoretical information encoded in Π c. The precise definitions and result are as follows: First, a valuation v of k, which is not necessarily trivial on k, is called a quasi prime divisor of k provided it satisfies the following: i) v vk and td(v kv) = td( k) 1. ii) No proper coarsening of v satisfies these properties. Condition i) implies that v is defectless on k, hence v/vk = Z, and v kv is a function field. Second, a quasi prime divisor v of k is a prime divisor of k iff v is trivial on k. Let L k k be a function subfield of k, and w := v L for some quasi prime divisor v of k. Since both the residual transcendence degree and the rational rank are additive in towers of function field extension, one has the following: Remark 3.1. In the above notations, there are only two possibilities for v and w, namely: a) rr(wl/wk) = 1, or equivalently, w is a quasi prime divisor of L k. b) td(l k) = td(lw kw), thus w is by definition a constant reduction (à la Deuring) of L k. The first point we want to make is that Galois theory encodes the nature of the above restriction w = v L of v to L l as follows: Fact 3.2. Let p L : Π Π L be the canonical projection. Then the following hold: 1) p L maps T 1 v Z 1 v into T 1 w Z 1 w, and p L (Z 1 v ) Z 1 w, p L (T 1 v ) T 1 w are open subgroups. 2) Therefore, w is a quasi prime divisor of L k if p L (T 1 v ) 1, respectively a constant reduction of L k if p L (T 1 v ) = 1. The second point we make is the result about recovering the quasi divisorial subgroups of Π from Π c is as follows, see [P1] and Topaz [To1], [To2]: 10

11 Fact 3.3. Let Π c Π be the canonical projection, and for subgroups G Π, let G Π c be their preimages in Π c. Then the following hold: 1) Let d be the maximal positive integer such that Π contains subgroups = Z d l with abelian preimage Π c. Then d = td( k). 2) The minimized quasi divisorial subgroups of Π are precisely the pairs T Z which are maximal satisfying the following: i) Z contains subgroups = Z d l having an abelian preimage Π c. ii) T = Z l, and its preimage T Π c is the center of Z Π c. B) Recovering the projection p κt : Π Π κt from π t : Π Π U0 In the context and notations of Theorem 2.6, let t \k be any non-constant function, and κ t be the relative algebraic closure of k(t) in. Then κ t k(t) is a finite field extension, hence the projection Π κt Π U0 defined by t 0 t has an open image. Therefore, since the canonical projection p κt : Π Π κt is (by mere definitions) surjective, it follows that the canonical projection π t : Π Π U0 defined by k(t 0 ), t 0 t, has an open image in Π U0. Our aim is to show that there exist group theoretical recipes to recover the projection p κt : Π Π κt from the given group theoretical projection π t : Π Π U0. Notations 3.4. In the above context, consider/define: a) The set Q 0 t of all the quasi prime divisors v of k with π t (Zv 1 ) Π U0 open. b) The closed subgroup Tt 0 := Tv 1 v Q 0 t Π generated by the minimized inertia groups Tv 1, v Q 0 t, and the resulting canonical projection p 0 t : Π Π 0 t := Π /Tt 0. c) The set Q t of all the quasi prime divisors v of k such that image p 0 t (Zv 1 ) Π 0 t of Zv 1 under p 0 t : Π Π 0 t := Π /Tt 0 is not a topologically finitely generated group. d) The closed subgroup T t := Tv 1 v Q t Π generated by the minimized inertia groups Tv 1, v Q t, and the resulting canonical projection p t : Π Π t := Π /T t. We proceed by shedding some light on the objects defined above. Lemma 3.5. For every quasi prime divisor v of k the following are equivalent: i) v Q 0 t. ii) t is a v-unit and residually transcendental, that is, t v is non-constant. iii) The restriction of v to k(t) is the Gauss valuation defined by v k := v k and t. In particular, v κt is a constant reduction of κ t k. Proof. First, the equivalence of ii), iii) follows by mere definitions. For the reverse implication ii), iii) i), we notice that Π U0 = Z 2 l is noting but the Galois group of the maximal extension 0 k(t 0 ) unramified outside t 0 = 0, 1,. On the other hand, since t, t 1 v generate a Z-submodule of rank two in (v) /(kv), it follows by mere definitions, that the image of Z 1 v in Π U0 under t 0 t, is isomorphic to Z 2 l as well, thus open in Π U 0 = Z 2 l. Finally, for the implication i) ii), suppose that π t (Z 1 v ) is open in Π U0. Then by mere definitions, it follows that there exist v-units θ tk, η (t 1)k, such that their images θ, η in v generate a Z-module of rank two in (v) /(kv). Hence setting θ = t/a, η = b(t 1) with a, b k, we must have t = aθ, and η = baθ b. We claim that a, b are v k -units. Indeed, one has the following case-by-case analysis: 11

12 - v k (b) < 0. Since η = b(aθ 1) is a v-unit, we must have v(aθ 1) > 0, hence aθ 1+m v. Since θ is v-unit, so must be a, and a θ = 1 in v. Hence θ = 1/a kv, contradiction! - v k (b) > 0. Then v(aθ 1) < 0, hence v(aθ) < 0. Thus η = (ba)θ b implies that ab is a v k -unit, and η, θ have equal images in (v) /(kv), contradiction! - v k (b) = 0, i.e., b is a v k -unit. Then aθ = b 1 η + 1 is a v-unit (because the RHS is so), and since θ is a v-unit, so is a. Thus conclude that a, b both are v k -units. To proceed, notice that we have t = θ/a. Since a is a v k -unit, and θ is v-residually transcendental, it follows that t is a v-unit, and t is v-residually transcendental. For the next Lemma, we recall the following basic facts about the pro-l abelian birational fundamental group Π 1, k of k (and correspondingly, for its subfield κ t k, etc.), see [P3], Appendix for further details. First, for every set of quasi prime divisors Q of k, let T Q Π be the closed subgroup generated by (Tv 1 ) v Q. We set Π 1,Q := Π /T Q and call Π 1,Q the pro-l abelian fundament group of Q. The pro-l abelian fundamental group of the set of all the prime divisors D k of k is called the (pro-l abelian) birational fundamental group of k, and is denoted by Π 1, k. Notice/recall that Π 1, k equals the abelian pro-l (quotient of the) fundamental group Π X of any complete regular model X of k if such models exist. In any case, Π X is a quotient of Π 1, k for every complete normal model X, and there always exist normal projective models X of k such that Π 1, k = Π X. In particular Π 1, k is topologically finitely generated, or equivalently, it is a finite Z l -module. Lemma 3.6. In the above notations, set k := κ t, and := k. The following hold: 1) p κt : Π Π κt factors through p 0 t : Π Π 0 t, say p κt = q 0 t p 0 t with q 0 t : Π 0 t Π κt. 2) 0 t := ker(q 0 t ) = ker(π 0 t Π κt ) is a quotient of Π 1, k hence a finite Z l -module. Proof. To 1): First, recall that Π κt = Gal ( ) κ t κ t is the Galois group of the maximal pro-l abelian extension κ t κ t of κ t. Hence one has ker(p κt ) = Gal ( κ t). Second, by mere definitions, for every valuation v of one has that Tv 1 = Gal ( ) Uv, where Uv is the group of v-units, and Uv := [ l Uv ]. Third, the fact that p κt : Π Π κt factors through p 0 t : Π Π 0 t is equivalent to ker(p 0 t ) ker(p κt ). On the other hand, since ker(p 0 t ) is generated by Tv 1, v Q 0 t, one has: ker(p 0 t ) ker(p κt ) iff Tv 1 ker(p κt ), v Q 0 t. Switching to field extensions via the Galois correspondence, the inclusion Tv 1 ker(p κt ) is equivalent to κ t Uv, v Q 0 t, hence equivalent to κ t Uv for all v Q 0 t. On the other hand, since k is algebraically closed, k is l-divisible, hence l Uv = l k U v. Hence by ummer theory and mere definition, the inclusion κ t Uv is equivalent to ( ) κ t k U v. To conclude, we notice that the above inclusion follows from the fact that v κt is a constant reduction of κ t k. Indeed, since k is algebraically closed, hence vk is divisible, it follows that v(κ t ) = v(k). Hence every u κ t is of the form u = au 1 for some a k with v(a) = v(u), and u 1 U v κ t. This concludes the proof of assertion 1) of the Lemma. To 2): First, we notice that a prime divisor v of k lies in Q 0 t if and only if v is trivial on k(t) if and only if v lies in Q t. Hence the set D t of all such prime divisors v of k is nothing but the set of prime divisors D κ t of the function field κ t. Let t be the maximal subextension in which all v D t are unramified. Equivalently, if T Dt Π is the closed subgroup generated by Tv 1 = T v, v D t, it follows that Gal ( t ) = Π /T Dt. Clearly, 12

13 κ t, and recalling that k := κ t, one has that κ t = k, and therefore, k and are linearly disjoint over κ t. Hence one has an exact sequence of the form ( ) 1 Gal ( t κ t κ t) Gal ( t ) = Π 0 t q 0 t Gal ( κ t κ t ) = Πκt 1, in which 0 t := Gal ( t κ t) = ker(π 0 t Π κt ) is the κ t -geometric part of Π 0 t = Gal ( t ). Next recalling that := k, we set t := t k k. Then since k and are linearly disjoint over κ t, it follows that the canonical projection below is an isomorphism: t := Gal ( t ) Gal ( t κ t κ t) = 0 t. Let D t be the set of all the prolongations ṽ v of all the valuations v D t to. Since k is the base change of κ t under the algebraic extension(s) κ t κ t k, the following hold: First, since D t is the set of all the prime divisors of κ t, it follows that D t equals the set D k of all the prime divisors of k. Second, since each v D t is unramified in t, it follows that each prolongation ṽ of v to = k is unramified in t (because the latter is the base change of t under κ t k). Hence since D t = D k, we conclude that all the prime divisors of k are unramified in t. Therefore, t = Gal ( t ) is a quotient of Π 1, k, and so is its isomorphic quotient t 0 t = ker(π 0 t Π κt ). Lemma 3.7. In the Notations 3.4, the following hold: 1) The set Q t consists of all the quasi prime divisors v of k such that v κt is a constant reduction of κ t k. Hence Q 0 t Q t, thus by mere definitions Tt 0 T t, and p t : Π Π t factors through p 0 t : Π Π 0 t, say p t = q t p 0 t for a unique q t : Π 0 t Π t. 2) p κt : Π Π κt factors through p t : Π Π t, say p κt = q t p t with q t : Π t Π κt. Thus since 0 t = ker(qt 0 ) is a quotient of Π 1, k, so is t := ker(q t ) = ker(qt 0 )/ ker(q t ). Proof. To 1): First, if w := v κt is a constant reduction of κ t k, then the minimized decomposition group Zw 1 Π κt is not topologically finitely generated (by mere definitions). On the other hand, p κt (Zv 1 ) Zw 1 is an open subgroup, hence p κt (Zv 1 ) Π κt is not topologically finitely generated. Thus finally, p 0 t (Zv 1 ) is not finitely topologically generated either. Conversely, let v be a quasi prime divisor such that p 0 t (Zv 1 ) is not finitely topologically generated. By contradiction, suppose that w := v κt is not a constant reduction. Then w is a quasi prime divisor of κ t k, and therefore Zw 1 = Tw, 1 because td(κ t k) = 1. Since Tw 1 = Z l, one finally has Zw 1 = Z l. Finally, recalling that ker(π 0 t Π κt ) is finitely generated, and that p κt (Zv 1 ) Zw 1 has finite index, it follows that p 0 t (Zv 1 ) Π 0 t is topologically finitely generated, contradiction! The remaining assertions from assertion 1) of the Lemma are clear. To 2): First, as in the proof of assertion 1) of the Lemma 3.6 above, especially the proof of the inclusion ( ), it follows that Tv 1 ker(p κt ) for all v Q t. Hence p κt : Π Π κt factors through p t : Π Π t, i.e., there exists q t : Π t Π κt such that p κt = q t p t. Second, qt 0 : Π 0 t Π κt factors through q t : Π t Π κt, precisely, qt 0 = q t q t, with q t : Π 0 t Π t as introduced at 1). Therefore, ker(q t ) = ker(qt 0 )/ ker(q t ) is a quotient of ker(qt 0 ), as claimed. We next announce the group theoretical recipe to recover p κt from π t. Proposition 3.8. In the above notations, in order to simplify notations, for quasi divisorial subgroups T 1 v Z 1 v of Π, we set T 1 v := p t (T 1 v ), Z 1 v := p t (Z 1 v ) Π t. Then the following hold: 13

14 1) t is the unique Z l -submodule Π t satisfying the following: i) For all v Q t one has that Z 1 v T 1 v and T 1 v = 1. ii) There exist v Q t such that Z 1 v, hence Z 1 v = T 1 v. ( ) Therefore, the discussion above gives a group theoretical recipe to recover/reconstruct p κt : Π Π κt from Π c endowed with π t : Π Π U0. 2) The above recipe to recover p κt is invariant under Π U0 -isomorphisms as follows: Let L l be a function field with l algebraically closed field, and for u L\l, let κ u L and p κu : Π L Π κu, π u : Π L Π U0 defined by t 0 u L, be correspondingly defined. Let Φ : Π Π L be the abelianization of an isomorphism Φ c : Π c Πc L satisfying ker(π u ) = Φ ( ker(π t ) ). Then one has: a) Φ maps (Tv 1 ) v Q 0 t, (Tv 1 Zv 1 ) v Qt isomorphically onto (Tw) 1 w Q 0 u, (Tw 1 Zw) 1 w Qu, respectively, thus gives rise to isomorphisms Φ 0 t : Π 0 t Π 0 u, Φ t : Π t Π u which map 0 t, t isomorphically onto the corresponding 0 u, u. b) Hence one has that ker(p κu ) = Φ ( ker(p κt ) ). Moreover, the induced canonical isomorphism Φ t,u : Π κt Π κu defined by Φ maps the quasi divisorial subgroups of Π κt isomorphically onto the ones of Π κu. Proof. To 1): We begin by showing that t satisfies the requirement 1), i) from Proposition 3.8. First, let v be a quasi prime divisor of k whose restriction to κ t is not a constant reduction of κ t k. We claim that Z 1 v t T 1 v. Indeed, by Remark 3.1, it follows that v(κ t )/v(k ) = Z. Further, by Fact 3.2 combined with the fact that td(κ t k) = 1, it follows that p κt (Tv 1 ) = p κt (Zv 1 ). Hence Zv 1 Tv 1 ker(p κt ), thus Z 1 v T 1 ( v p t ker(pκt ) ). Hence taking into account that p t ( ker(pκt ) ) = ker(q t ) = t, we get Z 1 v T 1 v t. Next, since T 1 v = Z l = pκt (T 1 v ), it follows that p κt maps T 1 v isomorphically onto p κt (T 1 v ). Hence p κt = q t p t implies that p t : T 1 v T 1 v and q t : T 1 v p κt (T 1 v ) are isomorphisms as well. Finally, since t = ker(p t ), it follows that t T 1 v = 1, as claimed. We next prove that there exists a family (v i ) i I of prime divisors v i Q t satisfying: - p κt (T vi ) p κt (T vj ) = 1 for all i j. - p κt (T vi ), i I, consists of almost all the divisorial subgroups of Π κt. - Z vi = t T vi, i I. In particular, t and Z vi satisfy condition 1), ii) for all i. The proof of this is not difficult, but a little bit involved, and we will do it a few steps: a) First, recall that qt 0 = q t q t, and in the notations from (the proof of) Lemma 3.6, let := ker(q t ) t be the corresponding fixed field in t. Then Gal ( κ t) = t, and one has an exact sequence of abelian groups ( ) 1 Gal ( ) κ t = t Gal ( ) pκ = Π t t Gal ( κ t κ t ) = Πκt 1, which is a quotient of the exact sequence ( ) from the proof of Lemma 3.6, 2). Since Π κt a pro-l abelian free group (being the l-adic dual of κ t /k ), the exact sequence ( ) above is split. Hence there exists a Z/l-elementary abelian extension 1 with Gal ( 1 ) = t /l, and satisfying: 1 and κ t are linearly disjoint over, thus 1 and = k are linearly 14 is

15 disjoint over as well. 3 Hence 1 := 1 k is an abelian extension 1 with Galois group t /l, and recall that t /l is finite, because t was a finite Z l -module. b) Let X t be the projective smooth k-curve with k(x t ) = κ t, and consider a t /l cover of (proper/normal) geometrically integral X t -schemes X 1 X with generic geometric fiber the field extension 1. Then there exists an open subset U t X t such that for all s U t, the fiber X 1,s X s at s is a t /l-cover of (proper/normal) integral k-varieties. In particular, if X 1 η 1,s η s X are the generic points of X 1,s X s, the corresponding extension of local rings O ηx O η1,s is an étale and totally inert extension of local rings, i.e., one has [ 1 : ] = [κ(η 1,s ) : κ(η s )]. On the other hand, X 1,s X 1 and X s X are Weil prime divisors. Thus the corresponding valuations v 1 of 1, respectively v of are prime divisors of 1 k, respectively k, which satisfy: v 1 is the unique prolongation of v to 1, and the residue field extension of v 1 v is nothing but κ(η 1,s ) κ(η s ). Further, v κt = v s is the valuation of κ t defined by s X t. Hence if w is a prolongation of v to κ t, then w is totally inert in 1 κ t κ t, and in particular, the decomposition group Z 1,w of w in Gal ( 1 κ t κ t) is nothing but Z 1,w = Gal ( 1 κ t κ t). c) Recall that is the subextension of with Galois group Π t, or equivalently, the fixed field of T t in, one has that t = Gal ( κ t κ t), and 1 κ t κ t is the Galois subextension of κ t κ t with Galois group Gal ( 1 κ t κ t) = t /l. Hence the decomposition group Z w of w in κ t κ t satisfies: Z w t and Z w Z 1,w = Gal ( 1 κ t κ t) = t /l. Since t is a finite Z l -module by Lemma 3.7, 2), Nakayama Lemma implies Z w = t. Finally, by general decomposition theory, one has that: First, Z w = Z t,v t, where Z t,v is the decomposition group of v in Π t. Second, Z t,v = p t (Z v ) = Z v is the image of Z v Π under p t : Π Π t. Thus one has that t = Z w Z v. We thus conclude that for almost all closed points s i X t, the local ring O X,ηi of the generic point η i of X si is a DVR whose valuation v i satisfies the following: a) v i () = v si (κ t ), because the special fiber X si is reduced. Hence p κt (T vi ) = T vsi. b) t Z vi, hence Z vi = t T vi. Finally, to complete the proof of assertion 1) of the Proposition, we have to prove that t is the only closed subgroup of Π t satisfying the conditions i), ii) from assertion 1). The proof of this assertion is easily to axiomatize as follows: Let Π t be a further subgroup satisfying the conditions i), ii) from assertion 1). Since satisfies ii), there exists v Q t such that T 1 v = Z 1 v, and since t satisfies i), it follows that Z 1 v t T 1 v, thus finally, t T 1 v. Similarly, t T vi for all v i. Finally, since p κt (T vi ) p κt (T vj ) = 1 for i j, we can choose v i such that p κt (T v ) p κt (T vj ) = 1. Equivalently, we have ( t T 1 v) ( t T vi ) = t. Thus the equality t = will follows from the following quite general assertion: Fact 3.9. Let G be an arbitrary group, T, T 1 G be subgroups, and, 1 G be normal subgroups satisfying: First, 1 T, 1 T 1, and second, 1 T 1 = 1 = T, ( 1 T 1 ) ( 1 T ) = 1. Then = 1. 3 Recall that k := κ t. 15

16 Proof. First, since 1 T 1, every δ 1 1 is of the form δ 1 = δτ 1 with δ, τ 1 T 1. Second, 1 T, implies that δ = δ 1τ with δ 1 1, τ T. Therefore, the following holds: δ 1 = δτ 1 = δ 1ττ 1, hence δ 1 τ 1 1 =: g := δ 1τ. Since g = δ 1 τ1 1 1 T 1, g = δ 1τ 1 T, and by hypothesis ( 1 T 1 ) ( 1 T ) = 1, it follows that g 1, thus concluding that τ, τ 1 1. Since 1 T 1 = 1, we get τ 1 = 1, hence concluding that δ 1 = δτ 1 = δ. And since δ 1 1 was arbitrary, we finally get 1. For the converse inclusion, let δ be arbitrary. Since 1 T, one has δ = δ 1 τ with δ 1 1, τ T. Hence τ = δ1 1 δ, and δ 1 1 implies τ, and therefore, τ = 1 (because T = 1). Hence finally δ 1 = δ, and since δ 1 1 was arbitrary, we get 1. Thus finally = 1, as claimed. To 2): The proof is an easy exercise of sorting through the proof of assertion 1), using the Φ maps the quasi divisorial subgroups of Π onto those of Π L. C) Recovering the divisorial groups in Π from Π c endowed with π t, π t for κ t κ t In this subsection we give a group theoretical recipe which recovers the divisorial subgroups T v Z v of Π from Π c endowed with two projections π t, π t : Π Π U0 for t, t such that κ t κ t (that is, t, t are algebraically independent over k). First, by the discussion in the previous subsection, the projection p κt : Π Π κt can be recovered/reconstructed by a group theoretical recipe from Π c endowed with the projection π t : Π Π U0 defined by t 0 t. Further, for every quasi divisororial subgroup Tv 1 Zv 1 of Π, one has the following: p κt (Tv 1 ) 1 iff w := v κt is a quasi prime divisor of κ t k, and if so, then by Fact 3.2, one has that p κt (Tv 1 ) = p κt (Zv 1 ) Tw 1 = Zw 1 is open (and these groups are isomorphic to Z l ). And p κt (Tv 1 ) = 1 if and only if p κt (Zv 1 ) has infinite Z l -rank, and if so, then w := v κt is a constant reduction of κ t k and p κt (Zv 1 ) Zw 1 is an open subgroup. Clearly, the same holds, correspondingly, about p κt. Proposition In the above notations the following hold: 1) A quasi divisorial group Tv 1 Zv 1 of Π is divisorial, i.e., v is a prime divisor of k if and only if one of the following conditions is satisfied: i) p κt (Zv 1 ) Π κt is an open subgroup. ii) p κt (Zv 1 ) = p κt (Tv 1 ) and there exists a quasi divisorial group Tv 1 Z1 v of Π satisfying: First, p κt (Zv 1 ) Π κ t is an open subgroup, and second, p κt (Tv 1 ) p κt (Tv 1 ) 1. 2) The above recipe to recover the divisorial subgroups T v Z v of Π from Π c endowed with π t, π t is invariant under Π U0 -isomorphisms as follows: Let L l be a function field with l algebraically closed field, and π u, π u : Π L Π U0 be the projections defined by t 0 u, respectively t 0 u for some u, u L\l. Let Φ : Π Π L be the abelianization of an isomorphism Φ c : Π c Πc L satisfying Φ( ker(π t ) ) = ker(π u ), Φ ( ker(π t ) ) = ker(π u ). Then Φ maps the divisorial groups T v Z v of Π isomorphically onto the divisorial groups T w Z w of Π L. Proof. To 1): For the implication let T v Z v be a divisorial group of Π, and set w := v κt. First, if w is trivial, then Z w = Π κt. On the other hand, by Fact 3.2, one has that p κt (Z v ) is open in Z w = Π κt is open. Hence the first condition from assertion 1) is satisfied. Second, if w is non-trivial, then w k = v k being trivial, implies that w is a prime 16