Manuscript, variant of June, 2003; 1–40

Pro-ℓ birational anabelian geometry over algebraically closed fields I

By Florian Pop*

Introduction

Let ℓ be a fixed rational prime number. For every field K of positive characteristic ≠ ℓ containing the ℓ^{th} roots of unity, let K^{ℓ}|K be a maximal pro-ℓ Galois extension, and G_{K}^{ℓ} = Gal(K^{ℓ}|K) its Galois group.

The aim of this paper is to prove a geometric pro-ℓ version of Grothendieck’s birational anabelian conjecture over algebraic closures of finite fields as follows:

Theorem. Let F be the category of all function fields K|k, with td(K|k) > 1 and k an algebraic closure of some finite field. Then there exists a group theoretic recipe by which we can recover every field K ∈ F from G_{K}^{ℓ}, up to a pure inseparable extensions. This recipe is invariant under profinite group isomorphisms. In particular, if K and L are in F, there exists a canonical bijection

Isom^{i} (L^{i}, K^{i}) ⟶ Out (G_{K}^{ℓ}, G_{L}^{ℓ}) ,

where ( )^{i} denotes pure inseparable closure, and Isom^{i} means up to Frobenius twists, and Out denotes outer isomorphisms of profinite groups.

Equivalently, if Φ : G_{K}^{ℓ} → G_{L}^{ℓ} is an isomorphism of profinite groups, then up to a Frobenius twist, there exists a unique field isomorphism ϕ : L^{i}^{ℓ} → K^{i}^{ℓ} such that Φ(g) = ϕ^{−1} g ϕ for all g ∈ G_{K}^{ℓ}. In particular, ϕ(L^{i}) = K^{i}.

Remarks.

I first want to mention that this manuscript is motivated by the program initiated by Bogomolov [Bo]. And the result above completes that program in the case of function fields over algebraic closures of finite fields. At least at the level of rough ideas, the paper [Bo] was quite inspiring for me... and at a first glance there is a lot of similarity of what we are doing here with loc.cit.. Without going into details, we hope nevertheless that the advised reader will realize the essential differences. Compare also with Bogomolov–

* Ideas used in this work originate in part from my visit to the IAS Princeton in 1994 and 1996. I would like to thank Pierre Deligne for several fruitful discussions and suggestions, see also [P2], [P3]. The main technical details of the present manuscript were solved during my visit to the IAS Princeton in 2002. During this last visit I was partially supported by the NSF grant DMS 9729992.

Tschinkel [B–T2].

1) The Theorem above implies corresponding “full Galois assertions”, i.e., the corresponding assertions for the full Galois group G_{K} in stead of the quotient G_{K}^{ℓ}.

Second, the corresponding questions concerning the Galois characterisation of finitely generated fields K with td(K) > 1 in positive characteristic, see [P2], [P3], [P4] can be reduced to the Main Theorem above. Thus the above result generalises in a non-trivial way the celebrated results by Neukirch, Ikeda, Iwasawa, Uchida concerning the Galois characterisation global fields.

2) The above result cannot be true in the case td(K|k) = 1, as in this case G_{K}^{ℓ} is a pro-ℓ free group on countably many generators, thus its structure does not depend on K at all.

Further, because of the reason just mentioned above, one cannot expect a Hom-form type result in the above context, as proved by Mochizuki over sub-p-adic fields. Indeed, for any given function field K|k there exist “many” surjective homomorphisms of profinite groups G_{K} → G_{k(t)}, given say by field k-embeddings k(t) ↪ K such that k(t) is relatively closed in K; and further, since G_{k(t)} is profinite free, see Harbater [Ha] and Pop [Pop], there are “many” surjective projections G_{k(t)} → G_{K}. Thus finally, there are many surjective group homomorphisms G_{K} → G_{K} which do not arise in a geometric way...

3) Finally, we remark that in stead of working with the full pro-ℓ Galois group G_{K}^{ℓ}, one could work as well with truncations of it, e.g., with ones coming from the central series. This is doable, but the resulting assertions are quite technical, and at the moment maybe too complicated in order to be interesting...

Rough idea of Proof (Comp. with Bogomolov [Bo])

Let 𝔎|𝔨 be an extension of fields of characteristic ≠ ℓ, and suppose that 𝔨 is algebraically closed. We consider algebraic extensions 𝔎′|𝔎 with the following properties:

i) 𝔎′|𝔎 is a pro-ℓ Galois extension.

ii) There exists a Galois sub-extension 𝔎_{1} of 𝔎′|𝔎 such that 𝔎′|𝔎_{1} is a maximal pro-ℓ Abelian extension of 𝔎_{1}.

We remark that the extension K^{ℓ}|K as introduced before the Main Theorem as well as K^{ℓ,ab}|K satisfies the conditions i), ii) above.

To fix notations, let G𝔎^{′} = Gal(𝔎′|𝔎), and 𝕋_{ℓ} ⟶^{ı} ℤ_{ℓ} a fixed identification as G𝔎^{′} -modules. By Kummer Theory there is a functorial isomorphism

(∗)  δ̂_{𝔎} : 𝔎̂ ⟶ H^{1}(𝔎, 𝕋_{ℓ}) ⟶^{ı} Hom(G𝔎^{′} , ℤ_{ℓ}) ,

where 𝔎̂ denotes the ℓ-adic completion of the multiplicative group 𝔎^{×} of 𝔎. Since 𝔨^{×} is divisible, and 𝔎^{×}/𝔨^{×} is a free Abelian group, the ℓ-adic completion homomorphism defines an exact sequence

1 → 𝔨^{×} ↪ 𝔎^{×} ⟶^{ȷ_{𝔎}} 𝔎̂ .

Denote P(𝔎) = 𝔎^{×}/𝔨^{×} inside 𝔎̂, and view P(𝔎) as the projectivization of the (infinite) dimensional 𝔨-vector space (𝔎, +). Suppose that we have a Galois theoretic recipe in order to detect: First the image of P(𝔎) inside the Galois theoretically “known” 𝔎̂. Second, the projective lines in P(𝔎). Remark that the multiplication l_{x} by any non-zero element x ∈ 𝔎 defines an “automorphisms” of P(𝔎) which respects “co-lineations”. This automorphism is noting but the translation l_{ȷ_{𝔎} (x)} in P(𝔎), this time viewed again as the multiplicative group P(𝔎) = 𝔎^{×}/𝔨^{×}. Now using the Fundamental Theorem of projective geometry, see e.g. Artin [A], it follows that the additive structure of 𝔎 can be deduced from the knowledge of all the projective lines in P(𝔎). And finally, since the “multiplications” l_{ȷ_{𝔎} (x)} do respect this structure, we finally deduce from this the field structure of 𝔎, as well the field extension 𝔎|𝔨.

Now coming back to the case of a function fields K|k as in the Theorem above, the problems we have to tackle are the following: Give Galois theoretic recipes in order to detect:

I) K_{(ℓ)} := K^{×} ⊗ ℤ_{(ℓ)} inside K̂.

This almost answers the question about detecting P(K): We know namely its ℤ_{(ℓ)}-version, but not P(K) itself.

II) P(K) and its projective lines.

III) Using the functoriality of the construction, show that the recipe for detecting (K, +, ·) is invariant under isomorphisms.

Organization of the paper

In the first part we put together the necessary tools for the proof as follows:

a) First, a “pro-ℓ Local theory” similar to the Local Theory form [P1], etc.. It relies on results by Ware [W], Koenigsmann [Ko], see also Bogomolov–

Tschinkel [B–T1], Efrat [Ef], Engler–Koenigsmann [E–K]. The aim of this

local theory is to recover in a functorial way from G_{K}^{ℓ} the set 𝔇_{K} of all the Zariski prime divisors of K|k, where K is any function field over an algebraic closure k of a finite field such that td(K|k) > 1.

b) Part of the local theory is to show that in the context from above, the whole inertia structure which is significant for us is encoded in G_{K}^{ℓ}.

c) Third, a more technical result by which we recover the “geometric sets of prime divisors” of function fields K|k as above. By definition, a geometric set of prime divisors of K is the set of Zariski prime divisors 𝔇_{X} ⊂ 𝔇_{K} defined by the Weil prime divisors of some quasi-projective normal model X of K. This result itself relies on de Jong’s theory of alterations [J].

In the second part of the manuscript, we give a simplified version of a part of the “abstract non-sense” from Part II of [P4] (which follows a suggestion by Deligne [D2]), reminding one in some sense of the abstract class field theory. We define so called pre-divisorial Galois formations. The aim of this theory is to lay an axiomatic strategy for the proof of the main result.

The interesting example of pre-divisorial Galois formations are the geometric Galois formations, which arise from geometry (and arithmetic).

A very basic results here is Proposition 3.18, which shows that in the case K|k is a function field with td(K|k) > 1 and k an algebraic closure of a finite field, the geometric Galois formations on G = G_{K}^{ℓ,ab} are group theoretically encoded in G_{K}^{ℓ}.

Finally in the last Section we prove the main Theorem announced above. The main tool here is Proposition 4.1, which gives a Galois characterization of the “rational projections”.

Thanks: I would like to thank several people who showed interest in this work, both for criticism, suggestions, and careful reading. My special thanks go to D. Harbater, N. Katz, J. Koenigsmann, P. Lochak, H. Nakamura, F. Oort, T. Szamuely, J. Stix, A. Tamagawa, etc..

1. Local theory and inertia elements

In this section we first recall the main facts concerning the Local theory from [P1], but in a pro-ℓ setting. Our first aim is to give to give a group theoretic recipe for finding information about the space of Zariski prime divisors in a functorial way.

Let 𝔎|𝔨 be a function field over some base field 𝔨, and td(𝔎|𝔨) = d > 0. We consider the family of all models X_{i} → 𝔨 of 𝔎|𝔨 such that the structure sheaf of X_{i} is a sheaf of sub-rings of 𝔎 with inclusions as structure morphisms. On the family of all the X_{i}’s there exists a naturally defined domination relation as follows: X_{j} ≥ X_{i} if there exists a surjective 𝔨-morphism φ_{ji} : X_{j} → X_{i} which at the structure sheaf level is defined by inclusions. Let 𝔓𝔯𝔬𝔧_{𝔎} be the subfamily of all projective, normal models of the function field 𝔎. The following is well known, see e.g. Zariski–Samuel [Z–S], Ch.VI, especially §17:

• Every complete model is dominated by some X_{i} ∈ 𝔓𝔯𝔬𝔧_{𝔎} (Chow Lemma).

• The set 𝔓𝔯𝔬𝔧_{𝔎} is increasingly filtered with respect to ≥, hence it is a surjective projective system.

• Denote ℜ_{𝔎} = lim_{⟵}_{i} X_{i} as topological spaces. We will call ℜ_{𝔎} the Riemann space of 𝔎. The points of ℜ_{𝔎} are in bijection with the space of all valuation 𝔨-rings of 𝔎. For v = (x_{i})_{i} in ℜ_{𝔎} one has: x_{i} is the centre of v on X_{i} in the usual sense, and O_{v} = ∪_{i}O_{X_{i},x_{i}}.

Fact/Definition 1.1. Using e.g. [BOU], Ch.IV, §3, one shows that for a point v = (x_{i})_{i} in ℜ_{𝔎} the following conditions are equivalent:

i) For i sufficiently large, x_{i} has co-dimension 1, or equivalently, x_{i} is the generic point of a prime Weil divisor of X_{i}. Hence v is the discrete 𝔨-valuation of 𝔎 with valuation ring O_{X_{i},x_{i}}.

ii) td(𝔎v | 𝔨) = td(𝔎|𝔨) − 1.

We will say that a point v = (x_{i})_{i} in ℜ_{𝔎} satisfying the above equivalent conditions is a Zariski prime divisor of 𝔎|𝔨. We denote the space of all Zariski prime divisors of 𝔎|𝔨 by 𝔇_{𝔎} ⊂ ℜ_{𝔎}. One has: The space 𝔇_{𝔎} ⊂ ℜ_{𝔎} of all Zariski prime divisors of 𝔎 is the union of the spaces 𝔇_{i} of prime Weil divisors of all the models X_{i} of 𝔎|𝔨 in 𝔓𝔯𝔬𝔧_{𝔎} (if we identify every prime Weil divisor with the discrete valuation on 𝔎 it defines).

More generally, let 𝔎′|𝔎 be an arbitrary algebraic extension, and v′ a valuation on 𝔎′. Then we will say that v′ is a Zariski prime divisor (of 𝔎 or of 𝔎′), if its restriction v to 𝔎 is a Zariski prime divisor of 𝔎|𝔨.

A) On the decomposition group

Let 𝔎′|𝔎 be some Galois field extension as in the Introduction, Rough idea of proof, and v is a valuation on 𝔎′ which is trivial on 𝔨. Let Z_{v}, T_{v}, and V_{v} be respectively the decomposition group, the inertia group, and the ramification group of v in G𝔎^{′} := Gal(𝔎′|𝔎). Since ℓ ≠ char(𝔨), and v is trivial on 𝔨, V_{v} is trivial. We denote by 𝔎^{T}, and 𝔎^{Z} the corresponding fixed fields.

Fact 1.2. The following are well known facts from Hilbert decomposition, and/or ramification theory for general valuations: 𝔎′v | 𝔎v is a Galois field extension, which also satisfies the properties i), ii), from loc.cit.. Let G𝔎v^{′} = Gal (𝔎′v | 𝔎v) be its Galois group. One has a canonical exact sequence

1 → T_{v} → Z_{v} ⟶^{π_{v}} G_{v} := G𝔎v^{′} → 1 .

Moreover, v(𝔎^{T}) = v(𝔎^{Z}) = v𝔎, and 𝔎v = 𝔎^{Z}v; and further v𝔎′ is an ℓdivisible hull of v𝔎.

Finally, there exists a pairing Ψ_{𝔎}′ : T_{v} × v𝔎′ → μ_{𝔎′v}, (g, vx) ↦ (gx/x)v, and the following hold: The left kernel of Ψ_{𝔎}′ is exactly V_{v} = {1}, and the right kernel of Ψ_{𝔎}′ is v𝔎. In particular, T_{v} is Abelian. Further, Ψ is compatible with the action of G_{v} on T_{v} via π_{v}. In particular, this action is via the cyclotomic character, thus trivial, as 𝔎v contains by hypothesis the algebraically closed field 𝔨. Therefore, we finally have: Z_{v} ≅ T_{v} × G_{v}.

1) Let B = (vx_{i})_{i} be an 𝔽_{ℓ}-basis of v𝔎 / ℓ. Then T_{v} ≅ ℤ_{ℓ}^{B}, and finally

Z_{v} ≅ T_{v} × G𝔎v^{′} ≅ ℤ_{ℓ}^{B} × G𝔎v^{′}

as profinite groups. In particular, cd(Z_{v}) = cd(G𝔎v^{′}) + |B|, where cd denotes the cohomological dimension.

2) Thus finally there exist canonical isomorphisms θ^{v} : v̂𝔎 → Hom(T_{v}, ℤ_{ℓ}) and θ_{v} : T_{v} → Hom(v𝔎, ℤ_{ℓ}).

3) Recalling the notations and remarks from Introduction, one gets a commutative diagram of the form:

𝔎^{×}  ⟶^{v}  v𝔎
↓^{ȷ_{𝔎}}  ↓_{θ^{v}}
Hom(G𝔎^{′} , ℤ_{ℓ}) ⟶^{ȷ^{v}} Hom(T_{v}, ℤ_{ℓ})

4) Let U_{v} ⊂ 𝔎^{×} denote the v-units, and U_{v}^{1} = 1 + 𝔪_{v} ⊂ U_{v} the principal v-units. Then U_{v}^{1} becomes ℓ-divisible in 𝔎^{Z}, and (𝔎v)^{×} = U_{v}/U_{v}^{1}. We have a commutative diagram of the form:

U_{v}  ⟶  𝔎v
↓^{ȷ_{𝔎}}  ↓^{ȷ_{𝔎 v}}
Hom(G𝔎^{′} , ℤ_{ℓ}) ⟶^{ȷ_{v}} Hom(G𝔎v^{′}, ℤ_{ℓ})

B) Recovering 𝔇_{K} from G_{K}^{ℓ}

Definition/Remark 1.3. In the notations from Introduction and the above ones, let K|k be a function field with td(K|k) > 0 and k an algebraic closure of a finite field.

1) The decomposition group Z_{v} ⊂ G_{K}^{ℓ} of some divisorial valuation v ∈ 𝔇_{K^{ℓ}} is called a divisorial subgroup.

2) A subgroup Z ⊂ G_{K}^{ℓ} which is isomorphic to a divisorial subgroup of some function field of transcendence degree equal to td(K|k) is called a divisorial like subgroup.

Proposition 1.4. Let K|k be a function field with td(K|k) > 1 and k an algebraic closure of a finite field. Let Z_{v} ⊂ G_{K}^{ℓ} be a divisorial subgroup, say defined by a divisorial valuation v on K^{ℓ}. Let T_{v} be the inertia group of v. Then the following hold:

(1) Z_{v} is self-normalising in G_{K}^{ℓ}. Further, if Z_{v′} ≠ Z_{v} is another divisorial subgroup, then one has: Z_{v′} ∩ Z_{v} = 1.

(2) T_{v} ≅ ℤ_{ℓ} as a G_{Kv}^{ℓ}-module. Further T_{v} is the unique maximal pro-ℓ Abelian normal subgroup of Z_{v}. And finally Z_{v} ≅ T_{v} × G_{Kv}^{ℓ} ≅ ℤ_{ℓ} × G_{Kv}^{ℓ}.

Proof. To (1): Both assertions follow using a result of F. K. Schmidt, see e.g., Pop [P1], Proposition 1.3; see also the proof of Proposition 1.14 from loc.cit.. Concerning (2), the only non-obvious part is the fact that given any function field K′ = Kv over k, the Galois group G_{K′}^{ℓ} has no non-trivial Abelian normal subgroups. This follows from the Hilbertianity of Kv.

A first content of the local theory is that “morally” the converse of the above Proposition is also true, i.e., if Z ⊂ G_{K}^{ℓ} is a divisorial like subgroup, then it comes from a Zariski prime divisor.

Proposition 1.5. Let K|k be a function field with td(K|k) = d > 1 and k an algebraic closure of a finite field. Then one has:

(1) For every divisorial like subgroup Z of G_{K}^{ℓ} there exists a unique divisorial valuation v of K^{ℓ} such that Z ⊂ Z_{v} and char(Kv) ≠ ℓ.

(2) Moreover, if T ⊂ Z is the unique maximal Abelian normal subgroup of Z, then T = Z ∩ T_{v}, where T_{v} is the inertia subgroup of Z_{v}.

Therefore, the space 𝔇_{K^{ℓ}} of all Zariski prime divisors of K^{ℓ} is in bijection with the divisorial subgroups of G_{K}^{ℓ}. This bijection is given by v ↦ Z_{v}.

Proof. The main step in the proof is the following ℓ-Lemma below, which replaces the q-Lemma from [P1], Local theory. After having the ℓ-Lemma, the remaining steps in the proof are similar to (but easier than) the ones from the proof of Theorem 1.16 from [P1], Local theory. We will skip the remaining details.

The ℓ-Lemma (revisited). In the context of Proposition 1.5, let Z_{0} ≅ ℤ_{ℓ}^{d} be a closed subgroup of G_{K}^{ℓ}. Then there exists a valuation v_{0} of K^{ℓ} such that Z_{0} ⊂ Z_{v_{0}}, and char(Kv) ≠ ℓ.

There are several ways to prove the ℓ-Lemma above: First, one could develop the corresponding model theoretic machinery, and proceed as in [P1], Local theory. Second, one can apply the results from Ware [W], and Koenigsmann [Ko], more precisely Engler–Koenigsmann [E–K]; see also Efrat [Ef]. Or third, apply Bogomolov–Tschinkel [B–T1].

C) Inertia elements of G_{K}^{ℓ}

Let K|k be a function field with td(K|k) > 1 and k an algebraic closure of a finite field. In this subsection we give a description of an important class of inertia elements in G_{K}^{ℓ} via divisorial inertia elements.

Definition. 1.6. Let K′|K be a Galois extension, and let denote G′ := Gal(K′|K). For a valuation v of K and a prolongation v′ of v to K′, let T_{v′} ⊂ Z_{v′} be its inertia, respectively decomposition group in G′.

1) An element g of G′ is called a v-inertia element, if g ∈ T_{v′}, for some prolongation v′|v. In general, an element g of G′ is called an inertia element, if there exists v such that g is a v-inertia element.

Denote by 𝔦𝔫𝔯(K′) is the set of all the inertia elements in G′.

2) An inertia element g of G′ is called divisorial inertia element, if g is a v-inertia element to some divisorial valuation v of 𝔎.

Denote by 𝔡𝔦𝔳.𝔦𝔫𝔯(K′) is the set of all divisorial inertia elements of G′.

Remarks 1.7. Let K_{0} be a finitely generated infinite field, e.g. a global field. If char(K_{0}) > 0, then denote by K = K_{0}k a maximal constant extension of K_{0}. (Thus k is an algebraic extension of the constants of K_{0}, and K|k is a function field as above.) As above, let K′_{0}|K_{0} be some Galois extension, and G′_{0} its Galois group. Thus if K_{0} has positive characteristic, then K′ := K′_{0}k is a Galois extension of K, and let G′ ⊂ G′_{0} is a closed subgroup of G′_{0}.

a) Let X → ℤ be a model of K_{0}. Recall, that an element σ of Gal(K′_{0}|K_{0}) is called a Frobenius element of K′_{0} (over X), if there exits a regular closed point x ∈ X, and a decomposition groups D_{x} ⊂ Gal(K′_{0}|K_{0}) over x, such that σ ∈ D_{x} is in the pre-image of the Frobenius at x in D_{x}. Let 𝔉𝔯𝔬𝔟(K′_{0}) be the set of all the Frobenius elements in Gal(K′_{0}|K_{0}) (for the several models of K_{0} as above). By the Chebotarev Density Theorem, 𝔉𝔯𝔬𝔟(K′_{0}) is dense in Gal(K′_{0}|K_{0}).

b) In contrast to the situation above, the set 𝔡𝔦𝔳.𝔦𝔫𝔯(K′_{0}) is in general not dense in Gal(K′_{0}|K_{0}). Namely, if T′ is the closed subgroup of Gal(K′_{0}|K_{0}) generated by all the divisorial inertia elements, and if X is any complete regular model of K_{0} (if any such models do exist), then Gal(K′_{0}|K_{0}) / T′ ≅ π_{1}(X).

This is a totally different situation than that of the Frobenius elements of global fields. But a much more stronger assertion holds, as follows. First, recall that for every valuation v on K one has the following fundamental inequality, see e.g. [BOU], Ch.IV, §10, 3:

td(K|k) ≥ rr(vK) + td(Kv|k)

where rr(vK) the rational rank of the group vK. A valuation of K is called defectless, if the above inequality is an equality.

Fact. The most prominent defectless valuations are the “iterations” of Zariski prime divisors as follows: Let δ ≤ d = td(K|k) be a fixed positive integer. Define inductively a sequence of function fields K_{k} endowed with Zariski prime divisors v_{k} as follows: v_{1} is a Zariski prime divisor of K_{1} := K, and set K_{2} := K_{1}v_{1}, etc.. Thus inductively: v_{k} is a Zariski prime divisor of K_{k} := K_{k−1}v_{k−1} for all k ≤ δ. Finally set v = v_{δ} ∘ … ∘ v_{1} as a valuation on K. Then v is a “generalised” discrete valuation of K with vK ≅ ℤ^{δ} lexicographically ordered, and td(Kv|k) = td(K_{δ}v_{δ}|k) = td(K|k) − δ. Therefore v is defectless.

Theorem 1.8. In the notations from the Definition above, let K|k be a function field with k an algebraically closed base field, K′|K be a Galois sub-extension of K^{ℓ} | K. Then the following hold:

(1) 𝔦𝔫𝔯(K′) is closed in Gal(K′|K).

(2) The closure of 𝔡𝔦𝔳.𝔦𝔫𝔯(K′) in Gal(K′|K) contains all the inertia elements at defectless valuations v which are trivial on k.

Proof. In both cases we can suppose that K′ = K^{ℓ}, as both closed subsets and inertia elements are closed under the projection G_{K}^{ℓ} → Gal(K′|K).

To (1): Let g ≠ 1 lie in the closure of 𝔦𝔫𝔯(K^{ℓ}). Equivalently, for every finite Galois sub-extension K_{i}|K of G_{K}^{ℓ} | K, there do exist:

i) A quasi divisorial valuation v_{i} on K^{ℓ}.

ii) Some g_{i} in the inertia group T_{i} := T_{v_{i}} of v_{i} in G_{K}^{ℓ}, such that g and g_{i} have the same restriction to K_{i}.

We will show that g is a v-inertia element of G_{K}^{ℓ}. First, in the notations from above, let ℓ^{n_{i}} be the common order of (the restriction of) g and g_{i} on K_{i}. Further, let K_{0} be the fixed field of g in K^{ℓ}, thus K^{ℓ} | K_{0} has Galois group generated by g. By Kummer Theory, there exits some x ∈ K_{0} which is not an ℓ^{th} power in K_{0}^{×}; and for every α_{n} ∈ K^{ℓ} such that α^{ℓ^{n}} = x, and K_{n} := K_{0}[α_{n}] is the unique extension of K_{0} of degree ℓ^{n} inside K^{ℓ}. Moreover, we can suppose that α_{n}^{ℓ} = α_{n−1}, where α_{0} := x.

In a first approximation, we consider only those K_{i} which contain x, as this is a co-final set of finite Galois sub-extensions of K^{ℓ} | K. In a second approximation, we consider the subset of those K_{i} for which α_{n_{i}} ∈ K_{i}. We claim that this is also co-final in the set of all finite Galois sub-extensions of K^{ℓ} | K. Indeed, it is sufficient to show that the restriction of g to K_{i}[α_{n_{i}}] has order ℓ^{n_{i}}. But this follows from the fact that the restriction of g to both K_{i} and K_{n_{i}} := K[x, α_{n_{i}}] is ℓ^{n_{i}}.

Now for a co-final set K_{i}|K as above, let v_{i} and g_{i} with the properties i), ii), above. We claim that v_{i}(x) is not divisible by ℓ in v_{i}K. Indeed, otherwise let u ∈ K such that ℓ·vu = vx. Then y = x/u^{ℓ} is a v_{i}-unit. Thus in the inertia field K^{T_{i}} we have: y is an ℓ^{th} power, and hence x is an ℓ^{th} power in K^{T_{i}}. Therefore, α_{1} lies in K^{T_{i}}. Hence g_{i} acts trivially on α_{1}. This contradicts the fact that g acts non-trivially on α_{1} ! Therefore, we finally have: x is not a v_{i}-unit, and K^{ℓ} = K^{g_{i}}[(α_{n})_{n}].

Now consider and ultrafilter D on the (index set I = {i} of the) family K_{i} such that for every n, D contains all the K_{i} with n_{i} ≥ n. By the “general non-standard non-sense” we have the following facts:

a) ^{∗}K := K^{ℓ}^{I}/ D carries the valuation^{∗}v = ∏ v_{i} / D such that the residual field ^{∗}K^{∗}v has characteristic ≠ ℓ.

b) 𝔏 := ∏ K^{g_{i}}/ D has a canonical embedding into ^{∗}K. Furthermore, one has 𝔏^{ℓ} = 𝔏[(α_{n})_{n}], and ^{∗}v x is not divisible by ℓ in ^{∗}v 𝔏.

c) Since each v_{i} is pro-ℓ Henselian on K^{g_{i}}, it follows that the restriction 𝔳 of ^{∗}v to 𝔏 is pro-ℓ Henselian on 𝔏.

From this we deduce: 𝔏 ∩ K^{ℓ} = K_{0} is the fixed field of g. Therefore, if v is the restriction of 𝔳 to K^{ℓ} ⊂ 𝔏^{ℓ}, then v is pro-ℓ Henselian on K_{0}. And since vx is not divisible by ℓ in vK_{0}, it follows that T_{v} = G_{K_{0}}^{ℓ}. Thus g lies in 𝔦𝔫𝔯(K^{ℓ}) .

To (2): Let g ≠ 1 be an inertia element at some v. Generally, in order to show that g is an accumulation point of divisorial inertia elements, w.l.o.g. we can suppose that g is not an ℓ^{th}-power in T_{v}.

As in the proof of (1), let K_{0} be the fixed field of g in K^{ℓ}, and (α_{n})_{n} the compatible system of roots α_{n}^{ℓ^{n}} = x of x such that K_{n} := K_{0}[α_{n}] is the unique extension of degree ℓ^{n} of K_{0}. By replacing x by its inverse (if necessary), we can suppose that vx > 0. Finally consider a co-final set of finite Galois subextensions K_{i}|K of G_{K}^{ℓ} | K as in the proof of assertion (1) above, i.e., such that: x ∈ K_{i}, and the restriction of g to K_{i} has order ℓ^{n_{i}}, and α_{n_{i}} ∈ K_{i}, where α_{n_{i}} are as above. Hypothesis/Remark. For every i, there exist projective models X_{i} → k of K_{i}^{g}|k such that the centre 𝔵_{i} of v is a regular point.

We remark that by Abhyankar’s desingularization Theorem, Abhyankar [A], this hypothesis is satisfied if td(K|k) ≤ 2. Further, if v is a defectless valuation, the above hypothesis is satisfied by the main result of Knaf–

Kuhlmann, [K–K].

For every i consider a projective model X_{i} → k of K_{i}^{g}|k, such that the center 𝔵_{i} of v on X_{i} is a regular point. Equivalently, O_{v} dominates the regular local ring O_{i} := O_{X_{i},𝔵_{i}} of 𝔵_{i}. In particular, if M_{i} = O_{i} ∩ M_{v} is the maximal ideal of O_{i}, then vx > 0 implies x ∈ M_{i}. On the other hand, since O_{i} is regular, it is a factorial ring. Let x = ϵπ_{1}^{ν_{1}} … π_{r}^{ν_{r}} be its representation as product of powers of non-associate prime elements in O_{i}.

Claim. There is an exponent ν_{m} which is not divisible by ℓ.

Indeed, by its choice, x is not an ℓ^{th} power in O_{v}^{g}. Further, O_{v}^{g} is pro-ℓ Henselian. As it dominates O_{i}, it contains a pro-ℓ Henselisation O_{i}^{h} of O_{i}. Further, since ϵ is unit in O_{i}, and the residue field κ_{i} = O_{i}/M_{i} = k is algebraically closed, it follows that ϵ is an ℓ^{th} powers in O_{i}^{h}. Therefore, if all the exponents ν_{m} were divisible by ℓ, then x would be an ℓ^{th} power in O_{i}^{h} ⊂ O_{v}^{g}, contradiction!

Now let ν_{m_{i}} be an exponent which is not divisible by ℓ. Let v_{i}^{g} be the divisorial valuation on K_{i}^{g} defined by the prime element π_{m_{i}}, and let v_{i} be some prolongation of v_{i}^{g} to K^{ℓ}. Then we have:

a) The value v_{i}(x) = v_{i}^{g}(x) = ν_{m_{i}} is not divisible by ℓ.

b) Since α_{i} ∈ K_{i} satisfies α_{i}^{ℓ^{n_{i}}} = x, we have ℓ^{n_{i}} · v_{i}(α_{i}) = v_{i}(x). Thus v_{i} is totally (tamely) ramified in K_{i}|K_{i}^{g}.

In particular, by general decomposition theory, it follows that the restriction of T_{v_{i}} to K_{i} contains Gal (K|K_{i}^{g}). Thus, there exists g_{i} ∈ T_{v_{i}} such that g_{i} and g coincide on K_{i}.

This concludes the proof of assertion (2) of Theorem 2.10.

2. First consequences of the local theory

Let K|k be a function field over with td(K|k) > 1 and k an algebraic closure of a finite field. Using the Theorem above one can recover from G_{K}^{ℓ} certain geometric invariants as follows.

A) Recovering td(K|k).

We first remark that from the generalized Milnor Conjectures it immediately follows that cd(G_{K}^{ℓ}) = td(K|k). But this fact can be deduced from the local theory as follows:

Fact. 2.1. In the above context we have:

1) The following conditions are equivalent, and they are satisfied if an only if td(K|k) = 2:

(i) G_{Kv}^{ℓ} is pro-ℓ free for some v ∈ 𝔇_{Kv}. (ii) G_{Kv}^{ℓ} is pro-ℓ free for all v ∈ 𝔇_{Kv}.

2) By induction, we can characterize d = td(K|k) as follows: The following conditions are equivalent, and they are satisfied if an only if td(K|k) = d:

(j) For some v ∈ 𝔇_{K}, the pro-ℓ group G_{Kv}^{ℓ} has the property characterizing the fact that td(Kv|k) = d − 1.

(jj) G_{Kv}^{ℓ} has the property characterizing the fact that td(Kv) = d − 1 for all v ∈ 𝔇_{v}.

Proof. It is sufficient to prove 1). First assume td(K|k) = 2. Then Kv is a function field in one variable over k. Thus the result follows by the fact that the absolute Galois group of Kv is profinite free, see e.g. Harbater [Ha], or Pop [Po]. Conversely, suppose that G_{Kv}^{ℓ} is pro-ℓ free for some v ∈ 𝔇_{K}. By contradiction, suppose that td(K|k > 2. Since td(Kv|k) = td(K|k) − 1 > 1, G_{Kv}^{ℓ} contains divisorial subgroups of the form Z_{w} ≅ ℤ_{ℓ} × G_{(Kv)w}^{ℓ}, which have cd > 1. Thus G_{Kv}^{ℓ} is not pro-ℓ free, contradiction!

B) Recovering the residually divisorial inertia

In the usual hypothesis from above, let v be a Zariski prime divisor, and G_{Kv}^{ℓ} the residual Galois extension. In order to have a name, we will say that the divisorial inertia elements 𝔡𝔦𝔳.𝔦𝔫𝔯(Kv^{ℓ}) are v-residually divisorial inertia elements of G_{K}^{ℓ}.

By the local theory, if td(K|k) > 2, then td(Kv|k) > 1. Thus the Zariski prime divisors of Kv are encoded in G_{Kv}^{ℓ} as shown in Proposition 1.5 of the Local Theory. In particular, the v-residually divisorial inertia elements of G_{Kv}^{ℓ} are encoded in G_{Kv}^{ℓ} (and indirectly, in G_{K}^{ℓ} too).

Nevertheless, if td(K|k) = 2, a fact which by the subsection above is encoded in G_{K}^{ℓ}, then the residue Galois group G_{Kv}^{ℓ} is pro-ℓ free. Thus there is no group theoretic way to recognise the inertia elements of G_{Kv}^{ℓ} from G_{Kv}^{ℓ} itself, as this group does not depend on Kv at all...

The solution to recovering the divisorial inertia elements in the residual Galois group G_{Kv}^{ℓ} in general in the following: Let v_{0} be a Zariski divisor of Kv, and T_{v_{0}} ⊂ Z_{v_{0}} its inertia, respectively decomposition group in G_{Kv}^{ℓ}. Further let w = v_{0} ∘ v the composition of v_{0} with v. Thus one has Kw = (Kv)v_{0} and 0 → vK → wK → v_{0}(Kv) → 0 in a canonical way. Further, via the canonical exact sequence 1 → T_{v} → Z_{v} ⟶^{π_{v}} G_{Kv}^{ℓ} → 1 we have: T_{w} ⊂ Z_{w} are exactly the pre-images of T_{v_{0}} ⊂ Z_{v_{0}} via π_{v}.

On the other hand, T_{w} consists of inertia elements of G_{K}^{ℓ}, thus by Theorem 1.8 it follows that T_{w} is contained in the topological closure of 𝔡𝔦𝔳.𝔦𝔫𝔯(K^{ℓ}). And this last set is known, as all the divisorial subgroups Z_{v} together with their inertia groups T_{v} are encoded in G_{K}^{ℓ}, as shown in Proposition 1.5. Finally, in the “trouble case” td(K|k) = 2, we deduce the following: The image of 𝔡𝔦𝔳.𝔦𝔫𝔯(K^{ℓ}) ∩ Z_{v} in G_{Kv}^{ℓ} via π_{v} consists of exactly all the v-residually inertia elements of G_{Kv}^{ℓ}. And finally, by general decomposition theory, a v-residually inertia element g_{v} generates an inertia group of G_{Kv}^{ℓ} if and only if the pro-cyclic closed subgroup T_{g_{v}} generated by g_{v} in G_{Kv}^{ℓ} is maximal among the pro-cyclic subgroups of G_{Kv}^{ℓ}. Summarising we the following:

Fact 2.2. In the above context, let g_{v} ∈ G_{Kv}^{ℓ} be a given element, and T_{g_{v}} the pro-cyclic subgroup generated by g_{v}. Then T_{g_{v}} is an inertia subgroup of G_{Kv}^{ℓ} if and only if the following holds:

1) If td(K|k) = 2, then g_{v} lies in the image of 𝔡𝔦𝔳.𝔦𝔫𝔯(K^{ℓ}) ∩ Z_{v} in G_{Kv}^{ℓ}, and T_{g_{v}} is maximal among the pro-cyclic subgroups of G_{Kv}^{ℓ}.

2) If td(K|k) > 2, then T_{g_{v}} is the inertia group of a divisorial subgroup of G_{Kv}^{ℓ} as resulting from Proposition 1.5 from the Local Theory for G_{Kv}^{ℓ}.

C) More about the case td(K|k) = 2

We first recall some basic facts concerning Galois theory over curves and their function fields over algebraically closed fields as follows. Let 𝔎|𝔨 be a function field in one variable over the algebraically closed field 𝔨. Let X → 𝔨 be the unique complete normal model of 𝔎|𝔨, thus X → 𝔨 is projective and smooth. The closed points a ∈ X are in one-to-one correspondence with the divisorial valuations v_{a} of 𝔎|𝔨.

Let ℓ ≠ char be some prime number, and G𝔎^{′} = Gal(𝔎^{ℓ,ab}|𝔎) be the Galois group of a maximal abelian pro-ℓ Galois extension of 𝔎. Let T_{a} = Z_{a} be the inertia/decomposition groups over each a ∈ X(𝔨) in G. (N.B., these groups depend only on a, and not on the specific Zariski prime divisor of 𝔎^{ℓ,ab}|𝔨 used to define each particular T_{a} = Z_{a}.)

a) Canonical inertia generators Let g ≥ 0 be genus of X. It is well known (e.g., using for instance generalized Jacobians, or the specialization theorem of Grothendieck) that there exist elements ρ_{1}, σ_{1}, …, ρ_{g}, σ_{g} in G𝔎^{′}, and “Abelian inertia elements” τ_{a} ∈ T_{a} (all a ∈ X(𝔨)) such that one has:

G𝔎^{′} = G =< ρ_{1}, σ_{1}, …, ρ_{g}, σ_{g}, (τ_{a})_{a} | ∏_{a}τ_{a} = 1 > .

In particular, if τ′_{a} ∈ T_{a} are such that ∏_{a} τ′_{a} = 1 inside G, then there exists a unique ϵ ∈ ℤ_{ℓ} such that τ′_{a} = τ_{a}^{ϵ} for all a ∈ X.

We will say that (τ_{a})_{a∈X} is a canonical system of Abelian inertia generators in G𝔎^{′}. Such a system is unique up to simultaneous multiplication by an element of ℤ_{ℓ}^{×}.

b) π_{1}^{ℓ,ab}(X) and the genus

In the context above, let T ′_{𝔎} be the closed subgroup of G𝔎^{′} generated by the inertia subgroups T_{a} (all a). Then we have a canonical exact sequence of the form:

1 → T ′_{𝔎} → G𝔎^{′} → π′_{1}(X) → 1 .

In particular, π′_{1}(X) is the free Abelian pro-ℓ of rank 2g, where g is the genus of X. Thus the genus of X is encoded in Gℓ,ab_{𝔎} endowed via the inertia groups (T_{a})_{a∈X}.

c) Detecting Div^{0}(X) For a ∈ X, recall the commutative diagram form section 1, A), 4):

(a)

𝔎^{×} ⟶^{can} Hom(G𝔎^{′} , ℤ_{ℓ})
↓^{v_{a}}  ↓^{res}
v_{a}𝔎 ⟶^{ȷ^{a}} Hom(T_{a}, ℤ_{ℓ})

For every a, let γ_{a} ∈ v_{a}𝔎 be the unique positive generator, thus γ_{a} = v_{v}π_{a} for every uniformizing parameter π_{a} at a. Then T_{a} has a unique generator τ_{a}^{0} such that ȷ^{a} (γ_{a})(τ_{a}^{0}) = 1 ∈ ℤ_{ℓ}. In particular, if x ∈ 𝔎^{×} is arbitrary, and v_{a}x = γ_{x} = m_{a,x} γ_{a} for some integer m_{a,x}, then ȷ_{v_{a}}(γ_{x})(τ′_{a}) = m_{a,x} inside ℤ_{ℓ}. We will call τ_{v}^{0} the canonical inertia generator of T_{a} (all a).

It is a well known fact, that 𝔗^{0} = (τ_{a}^{0})_{a} is also a canonical system of Abelian inertia generators in G𝔎^{′}, and Div^{0}(X) is canonically isomorphic to

𝔇0_{𝔗^{0}} = {ψ ∈ Hom(T ′_{𝔎} , ℤ_{ℓ}) | ψ(τ_{a}^{0}) ∈ ℤ (all a), ψ(τ_{a}^{0}) = 0 (almost all a) }.

Fact 2.3. Let 𝔗 = (τ_{a})_{a} be an arbitrary canonical system of Abelian inertia generators, thus τ_{a}^{0} = τ_{a}^{ϵ} (all a) for some ℓ-adic unit ϵ. Further denote

𝔇0_{𝔗} = {ψ ∈ Hom(T ′_{𝔎} , ℤ_{ℓ}) | ψ(τ_{a}) ∈ ℤ (all a), ψ(τ_{a}) = 0 (almost all a) }.

Then ϵ · Div^{0}(X) = ϵ · 𝔇0_{𝔗^{0}} = 𝔇0_{𝔗} inside Hom(T ℓ,ab_{𝔎}, ℤ_{ℓ}).

Prof. Clear from the discussion above.

Fact 2.4. Now let K|k be a function field with td(K|k) = 2, and k the algebraic closure of a finite field. Let v be a Zariski prime divisor of K, and G_{Kv}^{ℓ} the corresponding residual Galois group. Then by Fact 2.2 above, the inertia groups in G_{Kv}^{ℓ} are known. Thus applying the above discussion in the case 𝔎 = Kv and X = X_{v}, we get the following:

1) The inertia groups (T_{a})_{a∈X_{v}} and the canonical generating systems of inertia 𝔗_{v} = (τ_{a})_{a∈X_{v}} can be deduced in a group theoretic way from G_{K}^{ℓ}.

2) The exact sequence 1 → T_{Kv}^{ℓ,ab} → G_{Kv}^{ℓ,ab} → π_{1}^{ℓ,ab}(X_{v}) → 1 , can be deduced in a group theoretic way from G_{K}^{ℓ}. Thus also the genus g_{v} of X_{v}, as π_{1}^{ℓ,ab}(X_{v}) has ℤ_{ℓ}-rank equal to 2g_{v}.

3) Each canonical generating systems of inertia 𝔗_{v} = (τ_{a})_{a∈X_{v}} defines in a canonical way the subgroup

𝔇0_{𝔗_{v}} = {ψ ∈ Hom(T_{Kv}^{ℓ,ab}, ℤ_{ℓ}) | ψ(τ_{a}) ∈ ℤ (all a), ψ(τ_{a}) = 0 (almost all a) }

which up to multiplication by an ℓ-adic unit equals the image of Div^{0}(X_{v}) in Hom(T_{Kv}^{ℓ,ab}, ℤ_{ℓ}) defined canonically via Kummer Theory.

D) Geometric families of Zariski prime divisors

First we recall the basic definitions and facts. Let 𝔎|𝔨 be a function field over an algebraically closed base field 𝔨.

Fact/Definition 2.5. In the above context, for a quasi-projective normal model X → 𝔨 of 𝔎|𝔨, let 𝔇_{X} be the Zariski prime divisors of 𝔎|𝔨 it defines. We will say that a subset 𝔇 ⊂ 𝔇_{𝔎| 𝔨} is a geometric set of Zariski prime divisors, if it satisfies the following equivalent conditions:

(i) For all projective normal models X → 𝔨 of 𝔎|𝔨 one has: 𝔇 and 𝔇_{X} are almost equal.^{2}

(ii) There exists quasi-projective normal models X′ → 𝔨 of 𝔎|𝔨 such that the equality 𝔇 = 𝔇_{X′} holds.

Our aim in this subsection is to recall a criterion for describing the geometric sets 𝔇 of divisorial valuations of 𝔎|𝔨, and to show that the criterion is encoded group theoretically in G_{K}^{ℓ} in the case K|k is a function field with td(K|k) > 1 and k the algebraic closure of a finite field. See [P4], Section 3, for more details for general function fields.

Recall that a line on a 𝔨-variety is by definition an integral 𝔨-subvariety 𝔩 ⊆ X, which is a curve of geometric genus equal to 0. We denote by X^{line} the union of all the lines on X.

We will say that a variety X → 𝔨 is very unruly if the set X^{line} is not dense in X. In particular, a curve X is very unruly if and only if its geometric genus g_{X} is positive.

Further recall that being very unruly is a birational notion. Thus we will say that a function field 𝔎|𝔨 with td(𝔎|𝔨 = d > 0 is very unruly, if 𝔎|𝔨 has models X → 𝔨 which are very unruly.

Suppose that d > 1. We call a Zariski prime divisor of 𝔎|𝔨 very unruly, if 𝔎v | 𝔨 is very unruly. A Zariski prime divisor v of 𝔎|𝔨 is very unruly if and only if there exists a normal model X → 𝔨 of 𝔎|𝔨, and a very unruly prime Weil divisor X_{1} of X such that v = v_{X_{1}}.

Now let 𝔇 be a set of Zariski prime divisors of 𝔎|𝔨. For every finite extension 𝔎_{i}|𝔎, let 𝔇_{i} be the prolongation of 𝔇 to 𝔎_{i}. Thus in particular, 𝔇_{i} is a set of Zariski prime divisors of 𝔎_{i}|𝔨. As shown in loc.cit., one has the following:

Proposition 2.6. In the context above, set d = td(𝔎|𝔨). A set 𝔇 of Zariski prime divisors of 𝔎|𝔨 is geometric if and only if the following conditions are satisfied:

2 We say that two sets are almost equal, if their symmetric difference is finite.

(i) There exists a finite ℓ-elementary extension 𝔎_{0}|𝔎 of degree ≤ ℓ^{d} such that 𝔇_{0} is almost equal to the set of all very unruly prime divisors of 𝔎_{0}.

(ii) If 𝔎_{2}|𝔎 is any ℓ-elementary extension of degree ≤ ℓ^{d}, and 𝔎_{1} = 𝔎_{2}𝔎_{0} is the compositum, then 𝔇_{1} is almost equal to the set of all very unruly prime divisors of 𝔎_{1}.

From this Proposition one deduces the following inductive procedure on d = td(𝔎|𝔨) for deciding whether a given set 𝔇 of Zariski prime divisors is geometric, respectively whether 𝔎|𝔨 is very unruly.

Criterion 2.7. In the above context one has:

1) Case d = 1:

P_{geom}^{(1)}(𝔇): A set of Zariski prime divisors 𝔇 of 𝔎|𝔨 is geometric if and only if it is almost equal to the set of all Zariski prime divisors 𝔇_{𝔎| 𝔨}.

P_{v.u.}^{(1)}(𝔎|𝔨): 𝔎|𝔨 is very unruly if and only if the geometric genus g_{X} of the complete normal model X → 𝔨 of 𝔎|𝔨 satisfies g_{X} > 0.

2) Case d > 1: Then by induction on d, we already have criteria P_{geom}^{(δ)} and P_{v.u.}^{(δ)} which assure that sets of Zariski prime divisors are geometric, respectively that function fields are very unruly (all 1 ≤ δ < d). We now make the induction step d = td(𝔎|𝔨) as follows:

P_{geom}^{(d)}(𝔇): The criterion given by Proposition 2.6 is satisfied for the set of Zariski prime divisors 𝔇 of 𝔎|𝔨.

Here we remark that given a finite extension 𝔎_{i}|𝔎, the assertion “ v_{i} is a very unruly Zariski prime divisor of 𝔎_{i}|𝔨 ” which is essentially used in loc.cit., is actually equivalent to “ 𝔎_{i}v_{i}|𝔨 satisfies P_{v.u.}^{(d−1)}(𝔎_{i}v_{i}|𝔨) ”.

P_{v.u.}^{(d)}(𝔎|𝔨): There exists a subset 𝔇 ⊂ 𝔇_{𝔎| 𝔨} such that P_{geom}^{(d)}(𝔇) holds; and further, P_{v.u.}^{(d−1)}(𝔎v |𝔨) is true for almost all 𝔎v |𝔨 (v ∈ 𝔇).

We finally come to case K|k with td(K|k) > 1 and k the algebraic closure of a finite field. We show that Criterion 2.7 can be interpreted in G_{K}^{ℓ}.

Indeed, by Proposition 1.5, the Zariski prime divisors of K^{ℓ} are in bijection with the divisorial subgroups of G_{K}^{ℓ} via v ↦ Z_{v}; thus a bijection 𝔇^{ℓ} ↦ Z_{𝔇^{ℓ}} from sets of Zariski prime divisors 𝔇^{ℓ} ⊂ 𝔇_{K^{ℓ}} to sets of divisorial subgroups Z of G_{K}^{ℓ}. Moreover, a set 𝔇^{ℓ} is the prolongation to K^{ℓ} of a set of Zariski prime divisors 𝔇 ⊂ 𝔇_{K} if and only if Z_{𝔇^{ℓ}} is invariant under conjugation in G_{K}^{ℓ}. If this is the case, then every conjugacy class represents an element of 𝔇.

For Z^{ℓ} a set of divisorial subgroups of G_{K}^{ℓ}, which is closed under conjugation in G_{K}^{ℓ}, let 𝔇_{Z^{ℓ}} be the corresponding set of Zariski prime divisors of K^{ℓ}. For every sub-extension K_{i}|K of K^{ℓ}|K, let 𝔇_{Z}^{i} be the restriction of 𝔇_{Z}^{ℓ} to K_{i}. Thus 𝔇_{Z}^{i} is exactly the prolongation of 𝔇_{Z} to K_{i}.

Now suppose that d = 2. Let v be some Zariski prime divisor of K|k. Let further X_{v} → k be the complete smooth model of Kv|k. Then by Fact 2.4, the inertia groups T_{a} ⊂ G_{Kv}^{ℓ} (all a ∈ X_{v}), thus π_{1}(X_{v}) and the genus g_{X_{v}} of X_{v} can be recovered from G_{K}^{ℓ}. The same is true if we replace K|k by some finite sub-extension K_{i}|K of K^{ℓ}|K.

Therefore, by induction on d = td(K|k) > 0, we have the following Galois translation of the Criterion above.

Gal-Criterion 2.8. In the above context one has:

1) Case d = 2:

Gal P_{geom}^{(2)}(𝔇_{Z}) is said to be satisfied, if the Galois theoretic translation of P_{geom}^{(2)}(𝔇_{Z}) given above holds.

Gal P_{v.u.}^{(2)}(K|k) is said to be satisfied, if for some 𝔇_{Z} as above one has: Gal P_{geom}^{(2)}(𝔇_{Z}) holds, and g_{X_{v}} > 0 for almost all v ∈ 𝔇_{Z}.

2) Case d > 1:

By induction on d, we already have the Galois translation Gal P_{geom}^{(δ)} and Gal P_{v.u.}^{(δ)} of the criteria P_{geom}^{(δ)} and P_{v.u.}^{(δ)} (all 1 ≤ δ < d). We now make the induction step d = td(K|k) as follows:

Gal P_{geom}^{(d)}(𝔇_{Z}) is said to be satisfied, if (the Galois translation of) the criterion given by Proposition 2.6 is satisfied.

Here we remark that given a finite extension K_{i}|K, the assertion “ v_{i} is a very unruly Zariski prime divisor of K_{i}|k,” which is essentially used in loc.cit., is actually equivalent to Gal P_{v.u.}^{(δ)}(K_{i}v_{i} | k)

Gal P_{v.u.}^{(d)}(K|k) is said to be satisfied if and only if for some set 𝔇_{Z} as above satisfying Gal P_{geom}^{(d)}(𝔇_{Z}) one has: For almost all conjugacy classes of decomposition groups Z_{v}, the condition Gal P_{v.u.}^{(d−1)}(Kv|k) is satisfied.

3. Abstract pro-ℓ Galois theory

In this section we develop a “pro-ℓ Galois theory”, which in some sense has a flavor similar to one of the abstract class field theory. The final aim of this theory is to provide a machinery for recovering function fields (over algebraically closed base fields) from their pro-ℓ Galois theory in an axiomatic way. The material here is a simplified version of the corresponding part from Pop [P4].

A) Axioms and definitions

The context is the following: Let ℓ be a fixed prime number, and Z a quotient of ℤ_{ℓ}.

Definition. A pre-divisorial level δ ≥ 0 (pro-ℓ) Galois formation G = (G, ℨ) is defined by induction on δ to be an Abelian pro-ℓ group G endowed with extra structure as follows:

Axiom I) The level δ = 0:

A level δ = 0 pre-divisorial Galois formation is simply G = (G, / O).

We will say that G has level δ > 0, if the following Axioms II, III, are inductively satisfied.

Axiom II) Decomposition structure:

ℨ = (Z_{v})_{v} is a family of closed subgroups of G. We call Z_{v} the decomposition group at v, and suppose that each Z_{v} is endowed with a subgroup T_{v} such that:

i) T_{v} ≅ Z as an abstract pro-ℓ groups, and T_{v} ∩ T_{w} = {1} for all v ≠ w. For every co-finite subset 𝔘_{i} of valuations v, let T_{𝔘_{i}} be the closed subgroup

of G generated by the T_{v} (v ∈ 𝔘_{i}). A system (𝔘_{i})_{i} of such 𝔘_{i} is called co-final, if every finite set of valuations is contained in the complement of some 𝔘_{i}.

ii) There exist co-final systems (𝔘_{i})_{i} with T_{v} ∩T_{𝔘_{i}} = {1} (all i, and v ∉ 𝔘_{i}).

Axiom III) Induction:

Every G_{v} := Z_{v}/T_{v} caries itself the structure of a pre-divisorial Galois formation of level (δ − 1).

Convention. Let G = (G, ℨ) be a level δ ≥ 0 pre-divisorial Galois formation. In order to have a uniform notation, we enlarge the index set v (which in the case δ = 0 is empty) by a new symbol, which we denote v_{∗}, by setting Z_{v}_{∗} = G and T_{v}_{∗} = {1}. We will say that v_{∗} is the “trivial valuation”. Thus G is the decomposition group of the trivial valuation, and its “inertia group” is the trivial group. In particular, the residue Galois group at v_{∗} is G_{v}_{∗} = G.

Definition/Remark 3.1. Let G = (G, ℨ) be a pre-divisorial Galois formation of some level δ ≥ 0. Consider any α such that 0 ≤ α ≤ δ.

1) By induction on δ it is easy to see that one can view G = (G, ℨ) canonically as a pre-divisorial Galois formation of level α.

2) We define inductively the system G^{(α)} = (G_{α,i})_{i} of the α-residual predivisorial Galois formations of G as follows: First, if α = 0, then this system consists of G only. In general, if 0 < α ≤ δ, we recall that every residue group G_{v} carries the structure of a pre-divisorial Galois formation of level (δ − 1); thus of level (α − 1) by remark 1) above. Let G_{v} be this pre-divisorial Galois formation. Then by induction, the system of the (α − 1)-residual pre-divisorial Galois formations G_{v}^{(α−1)} of each G_{v} are defined. We then set G^{(α)} = (G_{v}^{(α−1)})_{v}.

We remark that the “correct” notation for the system of the α-residual pre-divisorial Galois formations (G_{α,i})_{i} is to index it by multi-indices of length α of the form v = (v_{i_{1}} … v_{i_{α}}), where v_{i_{1}} is a valuation of G, v_{i_{2}} is a valuation of G_{v}_{1}, etc..

3) For every multi-index v as above, let G_{v} be the profinite group on which the v-residual pre-divisorial Galois formation G_{v} is based. Then we will call G_{v} a v-residual group of G. One can further elaborate here as follows: Given a multi-index v = (v_{i_{1}} … v_{i_{α}}), we can define inductively the following:

a) The v-decomposition group Z_{v} of G, as being —inductively on α— the pre-image of Z_{(v_{i_{2}}...v_{iα})} ⊆ G_{v_{i_{1}}} in Z_{v}_{i_{1}} via Z_{v_{i_{1}}} → G_{v_{i_{1}}} .

b) The inertia group T_{v} of Z_{v}, as being the kernel of Z_{v} → G_{v}. From the definition it follows that T_{v} ≅ Z^{α}.

Definition/Remark 3.2. Let G = (G, (Z_{v})_{v}) be a pre-divisorial Galois formation of some level δ′ ≥ 0.

1) We denote L̂_{G} = Hom(G, Z) and call it the (ℓ-adic completion of the) pre-divisorial pre-field formation defining G.

From now on suppose that δ′ > 0.

2) Let T ⊆ G be the closed subgroup generated by all the inertia groups T_{v} (all v). We set π_{1,G} := G/T and call it the abstract fundamental group of G. One has a canonical exact sequence

1 → T → G → π_{1,G} := G/T → 1

Taking Z-Homs, we get an exact sequence of the form

0 → Û_{G} := Hom(π_{1,G}, Z) ⟶^{can} L̂_{G} := Hom(G, Z) ⟶^{ȷ^{G}} Hom(T, Z).

We will call Û_{G} = Hom(π_{1,G}, Z) the unramified part of L̂_{G}. And if no confusion is possible, we will identify Û_{G} with its image in L̂_{G}.

3) We now have a closer look at the structure of L̂_{G}. For an arbitrary v we have inclusions T_{v} ↪ Z_{v} ↪ G. Thus we can/will consider/denote restriction maps as follows:

ȷ^{v} : L̂_{G} = Hom(G, Z)⟶^{res_{Zv}}Hom(Z_{v}, Z)⟶^{res_{v}}Hom(T_{v}, Z).

We set Û_{v} = ker(ȷ^{v}) and call it the v-units in L̂_{G}. Thus the unramified part of L̂_{G} is exactly Û_{G} = ∩_{v}ker(ȷ^{v}).

We further denote L_{G,fin} = { x ∈ L̂_{G} | ȷ^{v}(x) = 0 for almost all v }. We remark that by Axiom II, ii), L_{G,fin} is dense in L̂_{G}. Indeed, for a co-final system (𝔘_{i})_{i} as at loc.cit., denote G_{i} = G/T_{𝔘_{i}} and T_{i} = T/T_{𝔘_{i}}. We have a canonical exact sequence

1 → T_{i} → G_{i} → π_{1,G} → 1 ,

and T_{i} is generated by the images T_{v,i} of T_{v} in G_{i} (v ∉ 𝔘_{i}). Clearly, the image of the inflation map

inf_{i} : Hom(G_{i}, Z) → Hom(G, Z)

is exactly Δ_{i} := {x ∈ L̂_{G} | ȷ^{v}(x) = 0 for all v ∈ 𝔘_{i} }. Finally, taking inductive limits over the co-final system (𝔘_{i})_{i}, the density assertion follows.

A closed submodule Δ ⊂ L̂_{G} is said to have finite co-rank, if Δ ⊂ L_{G,fin}, and Δ/ Û_{G} is a finite Z-module (or equivalently, Δ is contained in ker(ȷ^{v}) for almost all v). Clearly, the sum of two finite co-rank submodules of L̂_{G} is again of finite co-rank. Thus the set of such submodules is inductive. And one has:

L_{G,fin} = ∪_{Δ} (all finite co-rank Δ)

4) By the discussion above, the family (ȷ^{v})_{v} gives rise canonically to a continuous homomorphism ˆ⊕_{v} ȷ^{v} of ℓ-adically complete ℤ_{ℓ} -modules

ˆ⊕_{v} ȷ^{v} : L̂_{G} = Hom(G, Z) → Hom(T, Z) ↪ ˆ⊕_{v} Hom(T_{v}, Z) .

We will identify Hom(T, Z) with its image inside ˆ⊕_{v} Hom(T_{v}, Z). Therefore, ȷ^{G} = ˆ⊕_{v} ȷ^{v} on L̂_{G}.

We will denote D̂iv_{G} := ˆ⊕_{v} Hom(T_{v}, Z) and call it the (ℓ-adic completion of the) abstract divisor group of G. We will say that the image of L̂_{G} in D̂iv_{G} is the divisorial quotient (or the divisorial part) of L̂_{G}.

We further set ℭ̂𝔩_{G} = coker(ȷ^{G}), and call it the (ℓ-adic completion of the) abstract divisor class group of G. Therefore, we finally have a canonical exact sequence

0 → Û_{G} ↪ L̂_{G} ⟶^{ȷ^{G}} D̂iv_{G} ⟶^{can} ℭ̂𝔩_{G} → 0 .

5) We say that G is complete curve like if the following holds: There exist generators τ_{v} of T_{v} such that ∏_{v} τ_{v} = 1, and this is the only pro-relation satisfied by the system of elements 𝔗 = (τ_{v})_{v}.

Further consider 0 ≤ δ < δ′. We say that G is δ-residually complete curve like if all the δ-residual pre-divisorial Galois formations G_{v} are residually complete curve like. In particular, “0-residually complete curve like” is the same as “residually complete curve like”.

6) Consider the exact sequence 1 → T_{v} → Z_{v} → G_{v} := Z_{v}/T_{v} → 1 given by Axiom II, i), (all v). Let inf_{v} : Hom(G_{v}, Z) → Hom(Z_{v}, Z) be the resulting inflation homomorphism.

Since T_{v} = ker(Z_{v} → G_{v}), it follows that res_{Z_{v}}(Û_{v}) is the image of the inflation map infl_{v}. Therefore there exists a canonical continuous homomorphism, which we call the v-reduction homomorphism:

ȷ_{v} : Û_{v} → Hom(G_{v}, Z) = L̂_{G_{v}}

where L̂_{G_{v}} is the ℓ-adic completion of the abstract v-residual field, i.e., the one attached to the v-residual Galois formation G_{v}.

7) Next let v be arbitrary, and G_{v} be the corresponding residual Galois formation. To G_{v} we have the corresponding exact sequence as defined for G at point 3) above:

0 → Û_{G_{v}} ↪ L̂_{G_{v}} ⟶^{ȷ^{Gv}} D̂iv_{G_{v}}

We will say that G is ample, if in the notations from point 3) above the following hold:

i) The canonical projection L̂_{G} ⟶^{(ȷ^{v})_{v}} ∏_{v∉𝔘_{i}} Hom(T_{v}, Z) is surjective.

ii) There exists v such that:

a) Δ_{i} ⊆ Û_{v} and ȷ_{v}(Δ_{i}) ⊆ L_{G_{v},fin}.

b) ker( Δ_{i} ⟶^{ȷ_{v}} L̂_{G_{v}} ⟶^{ȷ^{Gv}} D̂iv_{G_{v}} ) ⊆ Û_{G}.

Next consider 0 ≤ δ < δ′. We say that G is ample up to level δ, if either δ = 0, or by induction on δ the following hold: First, if δ = 1, then G is ample in the sense defined above. Second, if δ > 1, then G_{v} is ample up to level (δ−1) for all v.

Before going to the next Subsection, let us remark that Condition a) and b) for a co-final system (𝔘_{i})_{i} implies the corresponding assertions for every co-finite subset of valuations U, respectively for every finite co-rank module Δ (and this fact is obvious).

B) Abstract divisor groups

Convention. In order to avoid too technical formulations, we will suppose from now —if not explicitly otherwise stated— that Z = ℤ_{ℓ}. In particular, ℤ ⊂ ℤ_{(ℓ)} are subgroups/subrings of Z.

Definition 3.3.

1) Let M be an arbitrary Z-module. We say that subsets M_{1}, M_{2} of M are ℓ-adically equivalent, if there exists an ℓ-adic unit ϵ ∈ Z such that M_{2} = ϵ · M_{1} inside M. Correspondingly, given systems S_{1} = (x_{i})_{i} and S_{2} = (y_{i})_{i} of elements of M, we will say that S_{1} and S_{2} are ℓ-adically equivalent, if there exists an ℓ-adic unit ϵ ∈ Z such that x_{i} = ϵ y_{i} (all i).

2) Let M be an arbitrary ℓ-adically complete module. We will say that a ℤ_{(ℓ)}-submodule M_{(ℓ)} ⊆ M of M is a ℤ_{(ℓ)}-lattice in M, (for short, a lattice) if M_{(ℓ)} is a free ℤ_{(ℓ)}-module, and it is ℓ-adically dense in M, and it satisfies the following equivalent conditions:

a) M/ℓ = M_{(ℓ)}/ℓ

b) M_{(ℓ)} has a ℤ_{(ℓ)}-basis 𝔅 which is ℓ-adically independent in M. c) The condition b) above is satisfied for every ℤ_{(ℓ)}-basis of M_{(ℓ)}. More general, let N ⊆ M_{(ℓ)} ⊆ M be ℤ_{(ℓ)}-submodules of M such that

M/N is again ℓ-adically complete. We will say that M_{(ℓ)} is an N-lattice in M, if N ⊆ M_{(ℓ)}, and M_{(ℓ)}/N is a lattice in M/N.

4) Finally, in the context from 3) above, a true lattice in M is a free Abelian subgroup M of M such that M_{(ℓ)} := M ⊗ ℤ_{(ℓ)} is a lattice in M in the sense of 3) above. And we will say that a ℤ-submodule M ⊆ M containing N is a true N-lattice in M, if M/N is a true lattice in M/N.

Construction 3.4. Let G = (G, (Z_{v})_{v}) be a pre-divisorial Galois formation which is both ample up to level δ and δ-residually complete curve like (δ ≥ 0 given). Recall the last exact sequence from point 3) from Definition/Remark 3.2:

0 → Û_{G} ↪ L̂_{G} ⟶^{ȷ^{G}} D̂iv_{G} ⟶^{can} ℭ̂𝔩_{G} → 0 .

The aim of this subsection is to describe the ℓ-adic equivalence class of a lattice Div_{G} in D̂iv_{G}, which —in the case it exists— will be called the abstract divisor group of G. By construction, this will be equivalent to giving the equivalence class of a Û_{G}-lattice in L̂_{G}, which will then be exactly the preimage of Div_{G} in L̂_{G}.

The case δ = 0, i.e., G complete curve like.

In the notations from Definition/Remark 3.2, 5) above, let 𝔗 = (τ_{v})_{v} be the system of generators of the groups T_{v} as there. Let as call such a system a distinguished system of inertia generators. We remark that any two distinguished system of inertia generators are strictly ℓ-adically equivalent. Indeed, if τ′_{v} ∈ T_{v} is another generator of T_{v}, then τ′_{v} = τ_{v}^{−ϵ_{v}} for some ℓ-adic units ϵ_{v} ∈ Z. Therefore, if 𝔗′ = (τ′_{v})_{v} does also satisfy condition ii) from Definition/Remark 3.2, 5), then we have also ∏_{v} τ_{v}^{−ϵ_{v}} = 1. By the uniqueness of the relation ∏_{v} τ_{v} = 1, it follows that ϵ_{v} = ϵ for some fixed ℓ-adic unit ϵ ∈ Z, as claimed.

Now let 𝔗 = (τ_{v})_{v} be a distinguished system generators of T. Further let F_{𝔗} be the Abelian pro-ℓ free group on the system 𝔗 (written multiplicatively). Then one has a canonical exact sequence of pro-ℓ groups

1 → τ^{Z} → F_{𝔗} → T → 1 ,

where τ = ∏_{v} τ_{v} in F_{𝔗} is the pro-ℓ sum of the generators τ_{v} (all v). Taking ℓ-adically continuous Homs we get an exact sequence

0 → Hom(T, Z) → Hom(F_{𝔗}, Z) → Hom(τ^{Z}, Z) → 0.

Remark that Hom(F_{𝔗}, Z) ≅ D̂iv_{G} in a canonical way, and the last homomorphism is simply the summation: φ ↦ (τ ↦ ∑_{v} φ(τ_{v})). Thus Hom(T^{ab}, Z) consists of all the homomorphisms φ ∈ Hom(F, Z) with trivial “trace”.

Consider the system 𝔅 = (φ_{v})_{v} of all the functionals φ_{v} ∈ Hom(F_{𝔗}, Z) defined by φ_{v}(τ_{w}) = 1 if v = w, and 0, otherwise (all v, w). We denote by

Div_{𝔗,(ℓ)} = < 𝔅 >_{(ℓ)} ⊂ Hom(F, Z)

the ℤ_{(ℓ)}-submodule of Hom(F, Z) generated by 𝔅. Then 𝔅 is an ℓ-adic basis of Hom(F, Z), i.e., Div_{𝔗,(ℓ)} is ℓ-adically dense in Hom(F, Z), and there are no non-trivial ℓ-adic relations between the elements of 𝔅. We will say that 𝔅 = (φ_{v})_{v} is the “dual basis” to 𝔗. We next set

Div_{𝔗,ℓ}^{0} := { ∑_{v} a_{v} φ_{v} ∈ Div_{𝔗,(ℓ)} | ∑_{v} a_{v} = 0 } = Div_{𝔗,(ℓ)} ∩ Hom(T, Z). Clearly, if φ_{v_{0}} is fixed, then the system e_{v} = φ_{v} − φ_{v_{0}} (all v ≠ v_{0}) is an

ℓ-adically independent ℤ_{(ℓ)}-basis of Div_{𝔗,(ℓ)}^{0}.

Thus finally, Div_{𝔗,(ℓ)} is a lattice in Hom(T, Z), and Div_{𝔗,(ℓ)}^{0} is a lattice in Hom(T, Z).

The dependence of Div_{𝔗,(ℓ)} on 𝔗 = (τ_{v})_{v} is as follows. Let 𝔗′ = (τ′_{v})_{v} = 𝔗^{ϵ} be another distinguished system of inertia generators. If 𝔅′ = (φ′_{v})_{v} is the dual basis to 𝔗′, then ϵ · 𝔅′ = 𝔅. Thus 𝔅 and 𝔅′ are ℓ-adically equivalent, and we have: Div_{𝔗,(ℓ)} = ϵ · Div_{𝔗′,(ℓ)} and Div_{𝔗,(ℓ)}^{0} = ϵ · Div_{𝔗′,(ℓ)}^{0}.

Therefore, all the subgroups of Hom(T^{ab}, Z) the form Div_{𝔗,(ℓ)}, respectively of the form Div_{𝔗,(ℓ)}^{0}, are ℓ-adically equivalent (all distinguished 𝔗). Hence the ℓ-adic equivalence classes of Div_{𝔗,(ℓ)} and Div_{𝔗,(ℓ)}^{0} do not depend on 𝔗, but only on G.

Fact 3.5. In the above context, denote by L_{𝔗,(ℓ)} the pre-image of Div_{𝔗,(ℓ)}^{0} in L̂_{G}. Further consider all the finite co-rank submodules Δ of L̂_{G} containing Û_{G}. Then the following assertions are equivalent:

(i) L_{𝔗,(ℓ)} is a Û_{G}-lattice in L̂_{G}.

(ii) Δ ∩ L_{𝔗,(ℓ)} is a Û_{G}-lattice in Δ (all Δ as above).

Moreover, if (i), (i), are satisfied, then ȷ^{v}(L_{𝔗,(ℓ)}) = ℤ_{(ℓ)}φ_{v} (all v).

Proof. Clear.

Definition 3.6. In the context of Fact above, suppose that the equivalent conditions (i), (ii) are satisfied. Then we define Div_{G} to be any of the lattices Div_{𝔗,(ℓ)} ⊂ D̂iv_{G}, and call it an abstract divisor group of G.

We further say that Div_{𝔗,(ℓ)}^{0} is the divisor group of degree 0 in Div_{𝔗,(ℓ)}. And remark that any two abstract divisor groups Div_{G} and Div′_{G} are ℓ-equiv-alent latices in D̂iv_{G}, and the same is true for Div_{G}^{0} and Div′_{G}^{0}.

The case: δ > 0.

We begin by mimicking the construction from the case δ = 0, and by induction on δ conclude the construction. Thus let 𝔗 = (τ_{v})_{v} be any system generators for the inertia groups T_{v} (all v). Further let F_{𝔗} be the Abelian pro-ℓ free group on the system 𝔗 (written multiplicatively). Then T is a quotient F_{𝔗} → T → 1. Thus taking ℓ-adic Homs we get an exact sequence

0 → Hom(T, Z) → Hom(F_{𝔗}, Z) ,

and remark that Hom(F_{𝔗}, Z) ≅ D̂iv_{G} in a canonical way. Next consider the system 𝔅 = (φ_{v})_{v} of all the functionals φ_{v} ∈ Hom(F_{𝔗}, Z) defined by φ_{v}(τ_{w}) = 1 if v = w, and 0, otherwise (all v, w). We denote by

Div_{𝔗,(ℓ)} = < 𝔅 >_{(ℓ)} ⊂ Hom(F, Z)

the ℤ_{(ℓ)}-submodule of Hom(F, Z) generated by 𝔅. Then 𝔅 is an ℓ-adic basis of Hom(F, Z), i.e., Div_{𝔗,(ℓ)} is ℓ-adically dense in Hom(F, Z), and there are no non-trivial ℓ-adic relations between the elements of 𝔅. We will call 𝔅 = (φ_{v})_{v} the “dual basis” to 𝔗. Thus finally, Div_{𝔗,(ℓ)} is a lattice in Hom(T, Z).

Now let 𝔗′ = (τ′_{v})_{v} be another system of inertia generators. And suppose that for some ϵ ∈ Z we have 𝔗′ = 𝔗^{ϵ}. If 𝔅 = (φ′_{v})_{v} is the dual basis to 𝔗′, then ϵ φ′_{v} = φ_{v} inside Hom(T, Z). Thus ϵ · 𝔅′ = 𝔅. In other words, 𝔅 and 𝔅′ are ℓ-adically equivalent. And we have: Div_{𝔗,(ℓ)} = ϵ · Div_{𝔗′,(ℓ)}.

Finally, we fix notations as follows:

For a system of inertia generators 𝔗 as above, and the corresponding lattice Div_{𝔗,(ℓ)} in D̂iv_{G}, let L_{𝔗,(ℓ)} be its pre-image in L̂_{G}. Next, for every v we denote by G_{v} the corresponding v-residual pre-divisorial Galois formation, etc.. In particular, G_{v} is both (δ − 1)-residually complete curve like, and ample up to level (δ − 1). Now let Δ be a finite co-rank Z-submodule of L̂_{G} such that Δ ∩ Û_{G} = 1. For every v such that Δ ⊂ Û_{v}, we set Δ_{v} = ȷ_{v}(Δ). Since G is ample up to level δ, there exists some v such that:

(∗) Δ ⊂ Û_{v}, and Δ_{v} has finite co-rank in L̂_{G_{v}}, and Δ_{v} ∩ Û_{G_{v}} = 1. In particular, Δ and Δ_{v} have the same Z-rank.

Fact 3.7. For every v and the corresponding G_{v}, let a Û_{G_{v}}-lattice L_{v,(ℓ)} in L̂_{G_{v}} be given. Then up to ℓ-adic equivalence, there exits at most one Û_{G}- lattice L_{(ℓ)} in L̂_{G} such that for every finite co-rank Z-module Δ ⊂ L̂_{G} with Δ ∩ Û_{G} = 1 and v as at (∗) above the following hold:

i) L_{Δ} := Δ ∩ L_{(ℓ)}, L_{Δ_{v}} := Δ_{v} ∩ L_{v,(ℓ)} are lattices in Δ, respectively Δ_{v}.

ii) The lattices ȷ_{v}(L_{Δ}) and L_{Δ_{v}} are ℓ-adically equivalent in Δ_{v}.

Moreover, if the Û_{G}-lattice L_{(ℓ)} exists, then its ℓ-adic equivalence class does depend only on the ℓ-adic equivalence classes of the Û_{G_{v}}-lattices L_{v,(ℓ)} (all v).

Proof. Let L_{(ℓ)}, L′_{(ℓ)} be Û_{G}-lattices in L̂_{G} satisfying i), ii), above. For Δ ≠ 0 as in Fact above, set L′_{Δ} = Δ ∩ L′_{(ℓ)}. Then by hypothesis ii) it follows that both lattices ȷ_{v}(L_{Δ}) and ȷ_{v}(L′_{Δ}) are ℓ-adically equivalent to the lattice L_{Δ_{v}} inside Δ_{v}. Thus they are equivalent. After replacing L′_{(ℓ)} by some multiple ϵ · L′_{(ℓ)} with ϵ ∈ ℤ_{ℓ}^{×}, we can suppose that L_{Δ} = L′_{Δ} ≠ 0. Now consider some arbitrary Δ′ containing Δ. In the notations from above -correspondingly- it follows that ȷ_{v′}(L_{Δ′}) and ȷ_{v′}(L′_{Δ′}) are ℓ-adically equivalent to the lattice L_{Δ′_{v′}} inside Δ′_{v′}. Since both contain ȷ_{v′}(L_{Δ}) = ȷ_{v′}(L′_{Δ}), we therefore must have ȷ_{v′}(L_{Δ′}) = ȷ_{v′}(L′_{Δ′}). Therefore, L_{Δ′} = L′_{Δ′}. Thus finally, as Δ′ was arbitrary, it follows L_{(ℓ)} = L′_{(ℓ)}.

Definition 3.8. Let G be a pre-divisorial Galois formation which is both δ-residual complete curve like and ample up to level δ > 0. We define an abstract divisor group Div_{G} ⊂ D̂iv_{G} —if it exists— inductively as follows: First, suppose that for all v, an abstract divisor group Div_{G_{v}} for G_{v} does exist. In particular, we have by definition, see hypothesis i) below: Div_{G_{v}} is a lattice in D̂iv_{G_{v}}, and its pre-image L_{v,(ℓ)} via ȷ^{G_{v}} is a Û_{G_{v}}-lattice in L̂_{G_{v}}.

Then an abstract divisor group of G —if it exists— is any lattice of the form Div_{G} = Div_{𝔗,(ℓ)} in D̂iv_{G} such that its pre-image L_{(ℓ)} in L̂_{G} satisfies the following conditions:

j) L_{(ℓ)} is a Û_{G}-lattice in L̂_{G}.

jj) L_{(ℓ)} satisfies the conditions i), ii) from Fact 3.7.

Remark/Fact 3.9. Let G be a pre-divisorial Galois formation which is both δ-residual complete curve like and ample up to level δ > 0. Suppose that abstract divisor groups Div_{G} = Div_{𝔗,(ℓ)} for G do exist. For such an abstract divisor group Div_{G}, let L_{(ℓ)} be its pre-image in L̂_{G}. Then for all v one has:

ȷ^{v}(L_{(ℓ)}) = ℤ_{(ℓ)} φ_{v} .

Indeed, by condition i) of the ampleness, see Definition/Remark 3.2, 7), it follows that ȷ^{v}(L̂_{G}) = ℤ_{ℓ} φ_{v}. Further, since L_{(ℓ)} is ℓ-adically dense in L̂_{G}, it follows that ȷ^{v}(L_{(ℓ)}) is dense in ℤ_{ℓ} φ_{v}. Thus the assertion.

2) In particular, the Û_{G}-lattice L_{(ℓ)} determines Div_{G}, as being the additive subgroup

Div_{G} = ⊕_{v} ȷ^{v}(L_{(ℓ)})

Therefore, giving an abstract divisor group Div_{G}, is equivalent to giving a Û_{G}-lattice L_{(ℓ)} in L̂_{G} such that:

j) L_{(ℓ)} satisfies the conditions i), ii) from Fact 3.7 with respect to the pre-images L_{v,(ℓ)} of some abstract divisor groups Div_{G_{v}} (all v).

jj) ȷ^{v}(L_{(ℓ)}) ≅ ℤ_{(ℓ)} (all v).

3) Finally, for an abstract divisor group Div_{G} for G, and its pre-image L_{(ℓ)} in L̂_{G}, we set ℭ𝔩_{L_{(ℓ)}} = Div_{G}/ȷ^{G}(L_{(ℓ)}), and call it the abstract ideal class group of L_{(ℓ)}. Thus one has a commutative diagram of the form

(∗)

0 → Û_{G} ↪ L_{(ℓ)} ⟶^{ȷ^{G}} Div_{L_{(ℓ)}}  ⟶^{can} ℭ𝔩_{L_{(ℓ)}} → 0
↓  ↓  ↓  ↓^{ȷ_{ℭ𝔩}}
0 → Û_{G} ↪ L̂_{G}  ⟶^{ȷ^{G}}  D̂iv_{G}  ⟶^{can}  ℭ̂𝔩_{G} → 0

where the first three vertical morphisms are the canonical inclusions, and the last one is the ℓ-adic completion homomorphism.

Proposition 3.10. Let G be a pre-divisorial Galois formation which is both δ-residual complete curve like and ample up to level δ > 0. Then any two abstract divisor groups Div_{G} and Div′_{G} for G are ℓ-adically equivalent as lattices in D̂iv_{G}. Or equivalently, their pre-images L_{(ℓ)} and L′_{(ℓ)} in L̂_{G} are ℓ-adically equivalent Û_{G}-lattices in L̂_{G}.

Proof. We prove this assertion by induction on δ. For δ = 0, the uniqueness is already shown, see Fact 3.5, and Definition in case δ = 0. Now suppose that δ > 0. Let Div_{G_{v}} and Div′_{G_{v}} be abstract divisor groups for G used for the definition of Div_{G}, respectively Div′_{G} (all v). By the induction hypothesis, Div_{G_{v}} and Div′_{G_{v}} are ℓ-adically equivalent. Thus their pre-images L_{v,(ℓ)} and L′_{v,(ℓ)} in L̂_{G,v} are ℓ-adically equivalent Û_{G,v}-lattices. Therefore, by Fact 3.7, the lattices L_{(ℓ)} and L′_{(ℓ)} (which are the pre-images of Div_{G} respectively Div′_{G} in L̂_{G}) are ℓ-adically equivalent. Finally, use 2) above to conclude.

Definition 3.11. A level δ > 0 divisorial Galois formation is by definition every pre-divisorial Galois formation G which is both (δ − 1)-residually complete curve like and ample up to level (δ − 1) and has abstract divisor groups Div_{G}. If this is the case, we will denote by L_{(ℓ)} the pre-image of Div_{G} in L̂_{G}, and call it a divisorial Û_{G}-lattice in L̂_{G}.

C) Example: Geometric Galois formation Let ℓ be a prime number. Let 𝔎|𝔨 be a field extension with 𝔨 an algebraically closed field, char(𝔨) ≠ ℓ. Consider a maximal Abelian pro-ℓ field extension 𝔎′ = 𝔎^{ℓab} of 𝔎, and set G = Gal(𝔎′|𝔎). Then we get:

Fact 3.12. G = (G, / O) is a level δ = 0 pre-divisorial Galois formation.

Nevertheless, in the case 𝔎|𝔨 is a function field with d = td(𝔎|𝔨) > 0, we can refine the pre-Galois formation above by making it into a pre-divisorial Galois formation of level δ for every 0 ≤ δ ≤ d as follows: First, let us endow every function field 𝔎|𝔨 as above with a geometric set 𝔇 = 𝔇_{X} of Zariski prime divisors of 𝔎|𝔨. Here X → 𝔨 is a quasi-projective normal model of 𝔎|𝔨, and 𝔇_{X} is its set of Weil prime divisors. Second, since 𝔎′|𝔎 is Abelian, if v′ is a prolongation of v ∈ 𝔇 to K′, then T_{v′} ⊂ Z_{v′} do not depent on v′, but on v.

Then one can endow G with the family (Z_{v})_{v} of the decomposition groups of all the v ∈ 𝔇. Clearly, if T_{v} ⊂ Z_{v} is the inertia group at v, then T_{v} ≅ ℤ_{ℓ} is pro-ℓ cyclic. Further, if v ≠ w, then T_{v} ∩T_{w} = {1}. Finally, let U_{i} be a basis of (small enough) affine open subsets in X, and for every i, set 𝔘_{i} = U_{i} ∩𝔇. Then by general facts about geometric fundamental groups, it follows that (𝔘_{i})_{i} is a co-final system in the sense of Axiom II), ii), from Subsection A). Thus finally, we have a pre-divisorial Galois formation supported by G as follows:

G_{𝔇} := (G, (Z_{v})_{v}) =: G_{X}

Moreover, for every v ∈ 𝔇 we have: The residue field 𝔎v is again a function field over 𝔨 with td(𝔎v | 𝔨) = (d−1). Further, by general decomposition theory, the residue field extension 𝔎′v | 𝔎v is a maximal Abelian pro-ℓ extension of 𝔎v.

Thus if d > 1, each such function field 𝔎v|𝔨 comes equipped with a geometric set of Zariski prime divisors 𝔇_{v} = 𝔇_{X_{v}}, where X_{v} a quasi-projective normal model of 𝔎v | 𝔨.

Now consider any δ with 0 ≤ δ ≤ d. Then by induction on the transcendence degree over 𝔨, we can suppose that each G_{v} endowed with the set of decomposition groups defined by 𝔇_{v} is a pre-divisorial Galois formation G_{𝔇_{v}} of level (δ − 1). Thus we have the following:

Proposition/Definition 3.13. Let G_{𝔇} = (G, (Z_{v})_{v}) be as constructed above. Then G_{𝔇} is in a canonical way a level δ pre-divisorial Galois formation, for every δ satisfying 0 ≤ δ ≤ td(𝔎|𝔨).

A pre-divisorial Galois formation of the form G_{𝔇} will be called a geometric Galois formation of level δ.

Remarks 3.14. Let G_{X} = (G, (Z_{v})_{v}) be a geometric Galois formation as constructed/defined above. Then 𝔎̂ = Hom(G𝔎^{′} , Z) = L̂_{G_{X}} by Kummer Theory. In order to compute Û_{G_{X}}, D̂iv_{G_{X}}, and ℭ̂𝔩_{G_{X}}, we do the following: First, let H(X) denote the group of principal divisors on X, and consider the canonical exact sequence 0 → H(X) ⟶ Div(X) ⟶^{pr} ℭ𝔩 (X) → 0 .

1) Passing to ℓ-adic completions, we get an exact sequence of ℓ-adic complete groups of the form: 0 → T_{ℓ,X} ⟶ Ĥ(X) ⟶ D̂iv(X) ⟶ ℭ̂𝔩 (X) → 0, where T_{ℓ,X} = lim_{⟵} _{ℓ^{n}}ℭ𝔩 (X) (with multiplication as homomorphisms), and the last three objects the corresponding ℓ-adic completions.

2) On the other hand one has Div(X) = ⊕_{v∈𝔇_{X}} v𝔎. For every v, consider the commutative diagram from subsection A), 3). We get a commutative diagram of the following form:

𝔎̂  → D̂iv(X) = ˆ⊕_{v} v𝔎 → ℭ̂𝔩 (X) → 0
↓δ̂  ↓^{⊕θ^{v}}  ↓
Hom(Gℓ_{𝔎} , ℤ_{ℓ}) ⟶^{ȷ^{G_{X}}} ˆ⊕_{v} Hom(T_{v}, ℤ_{ℓ}) ⟶^{can} P̂_{X} → 0

where the vertical maps are isomorphisms, and P̂_{X} is simply the quotient of the middle group by the second. We remark/recall that ⊕θ^{v} is defined as follows: For every v, let γ_{v} = 1·v be the unique positive generator of v𝔎 = ℤv. Then there exists a unique generator of the arithmetical inertia τ_{v} ∈ 𝔗_{v} such that ȷ_{v}(γ_{v})(τ_{v}) = 1. Hence, the commutativity of the diagram above follows directly from the definition of the homomorphisms.

3) From this we deduce: D̂iv_{G_{X}} = D̂iv(X) and ℭ̂𝔩_{G_{X}} = ℭ̂𝔩(X).

4) Concerning Û_{G_{X}}, we recall that Û_{G_{X}} = Hom(π_{1}^{ℓ,ab}(X), ℤ_{ℓ}), see Definition/Remark 4.2, 2). Let T ⊂ G be the closed subgroup generated by the decomposition groups of all the divisorial valuations of 𝔎. We set π_{1,𝔎}^{ℓ,ab} = G/T, and call it the fundamental group to 𝔎, as it is a birational invariant of 𝔎. It equals the fundamental group of any complete regular model X_{0} → 𝔨 of 𝔎|𝔨, if any such do exist...

We will say that a normal model X → 𝔨 for 𝔎|𝔨 is regular complete like if π_{1}^{ℓ,ab}(X) = π_{1,𝔎}^{ℓ,ab}. From the structure theorem for (the Abelian pro-ℓ quotient of the) fundamental groups of a normal curve it follows that such a curve is regular complete like if and only if it is a complete one.

Proposition 3.15. Let 𝔎|𝔨 and a geometric Galois formation G_{X} be as introduced/defined in Proposition above. Consider some δ ≤ td(𝔎|𝔨), and let us view G_{X} as a pre-divisorial Galois formation of level δ. We denote by G_{X_{v}} the residual Galois formations of G_{X} with v multi-indices of length ≤ δ. In particular, X_{v} → 𝔨 are quasi-projective normal varieties. Then one has:

(1) G_{X} is δ-residually complete curve like if (and only if ) δ = td(𝔎|𝔨) − 1, and all the δ-residual varieties X_{v} are complete normal curves.

(2) Suppose that δ < td(𝔎|𝔨). Further suppose that X_{v} is regular complete like (all v as above). Then G_{X} is ample up to level δ.

(3) Suppose that G_{X} is δ-residually curve like, and ample up to level δ. Then Div(X)_{(ℓ)} := Div(X) ⊗ ℤ_{(ℓ)} inside D̂iv(X) is an abstract divisor group of G_{X}. We will denote by L_{X,(ℓ)} its pre-image in 𝔎̂.

Thus for every geometric Galois formation G_{X} as above, Div(X)_{(ℓ)} is an abstract divisor group of G_{X}, and L_{X,(ℓ)} is a divisorial Û_{G}-lattice of G_{X}.

Proof. To (1): Clear. It is nevertheless more/quite difficult to prove the “only if” part of (1), which we will not directly use, thus omit the proof here...

To (2): We make induction on d = td(𝔎|𝔨). In the notations from Definition/Remarks 3.2, 3), let Δ ⊂ 𝔎̂ be a co-finite rank submodule such that Δ ∩ Û_{G_{X}} = 0. Then there exists an open affine subset X′ ⊂ X such that for all v ∈ 𝔇_{X′} one has: Δ ⊂ Û_{v}. In particular, the canonical projection π_{1}(X′) → π_{1}(X) gives rise to an embedding Û(X) ↪ Û(X′), and Δ ⊂ Û(X′). Using de Jongs’s alterations, and the inclusion-norm maps, we can suppose that X is a smooth 𝔨-variety. Finally, let X_{1}, …, X_{n} be the finitely many Weil prime divisors in X\X′, and S := {x_{1}, …, x_{n}} their generic points. Thus the inertia groups at these finitely many points generate T_{X′,X} := ker(π_{1}(X′) → π_{1}(X)).

Now let ı : X → ℙN_{𝔨} be some 𝔨-embedding. For a general hyper-plane H ⊂ ℙN_{𝔨}, we set Y = H ∩ X, Y′ = H ∩ X′, Y_{i} = X_{i} ∩ H (all i). Then by Bertini, each Y_{i} is a prime divisor of Y. Let T := {y_{1}, …, y_{n}} is the set of their generic points of the Y_{i} (all i). Then by Bertini we have:

i) The canonical projections π_{1}(Y) → π_{1}(X) and π_{1}(Y′) → π_{1}(X′) are surjective.

ii) Set T_{Y′,Y} := ker(π_{1}(Y′) → π_{1}(Y)). Then T_{Y′,Y} is generated by the inertia groups at all the y_{i} (all i).

iii) Finally, T_{Y′,Y} is mapped surjectively onto T_{X′,X} under the projection π_{1}(Y′) → π_{1}(X′) from i) above.

This is now exactly the translation of the fact that for the Zariski prime divisor v of 𝔎 defined by the Weil prime divisor Y = H ∩X of X, the assertion from Definition/Remarks 3.2, 3), holds for Δ at v.

To (3): It follows immediately from (1) and (2) above.

D) The case k is the algebraic closure of a finite field

In this subsection we will discuss the case K|k is a function field with td(K|k) > 1 and k an algebraic closure of a finite field. Let G_{K}^{ℓ} = Gal(K^{ℓ}|K) be as usually defined. The aim of this subsection is to show that the system of all the geometric Galois formations G_{X} are actually group theoretically encoded in G_{K}^{ℓ}. Moreover, the extra information concerning such a pre-divisorial Galois formation, e.g., δ-residually complete curve like, and/or ample up to level δ, is also encoded in the G_{K}^{ℓ}. First some general preparation as follows:

Fact 3.16. Let 𝔎|𝔨 be some field extension with 𝔨 algebraically closed, char(𝔎) ≠ ℓ. Let 𝔎^{ℓ} and Gℓ_{𝔎} as usually define. Denote by

Gℓ_{𝔎} → Gℓ,ab_{𝔎} =: G

the canonical projection. For every subgroup (·)^{ℓ} of Gℓ_{𝔎} we will denote by (·)^{ℓ,ab} the image of (·) in Gℓ,ab_{𝔎}. By general decomposition theory we have:

1) For every valuation v^{ℓ} of 𝔎^{ℓ}, and its inertia, respectively decomposition groups T_{v}^{ℓ} ⊂ Z_{v}^{ℓ} in Gℓ_{𝔎} one has: T_{v}^{ℓ,ab} and Z_{v}^{ℓ,ab} are exactly the inertia, respectively decomposition groups of v^{ℓ} in Gℓ,ab_{𝔎}.

Second, the groups of the form T_{v}^{ℓ,ab} and Z_{v}^{ℓ,ab} are exactly all the inertia, respectively decomposition groups in Gℓ,ab_{𝔎}.

2) Further remark that T_{v}^{ℓ,ab} and Z_{v}^{ℓ,ab} do depend only on the restriction v of v^{ℓ} on 𝔎, and not on v^{ℓ} itself (which is one of the prolongations of v to 𝔎^{ℓ}).

3) T_{v}^{ℓ} → T_{v}^{ℓ,ab} is an isomorphism, and second, G_{Kv}^{ℓ,ab} is canonically isomorphic to Z_{v}^{ℓ,ab}/T_{v}^{ℓ,ab}.

4) The Theorem of F. K. Schmidt from Pop [P], Proposition 1.3, holds for the Galois extension 𝔎^{ℓ,ab}|𝔎. In particular, if two valuations v and w equal their cores respectively, then Z_{v}^{ℓ,ab} ∩ Z_{w}^{ℓ,ab} ≠ {1} if and only if v = w.

Fact 3.17. Let K|k be a function field with k the algebraic closure of a finite field. In order to simplify notations let us denote G := G_{K}^{ℓ,ab}, and for every valuation v^{ℓ} and T_{v}^{ℓ} ⊂ Z_{v}^{ℓ} as above, set T_{v} = T_{v}^{ℓ,ab} and Z_{v} = T_{v}^{ℓ,ab}. Then we have the following:

1) d = td(K|k) is encoded in G_{K}^{ℓ}, by Section 2, A).

2) The divisorial inertia groups T_{v} and the divisorial decomposition groups Z_{v} in G are known, by Section 1.

The same is true for all the defectless inertia elements, by Section 1), Theorem 1.11.

For given such subgroups Z_{v} and Z_{w} we have: Z_{v} ∩ Z_{w} is non-trivial if and only if v = w.

3) If π_{1,K}^{ℓ} = G_{K}^{ℓ}/T_{K}^{ℓ} is the fundamental group to K, then π_{1,K} := π^{ℓ,ab} is also encoded in G_{K}^{ℓ}, by 2) above.

4) Furthermore, the geometric sets of divisorial decomposition groups, say Z = {Z_{v}}_{v} of G are encoded in G_{K}^{ℓ} → G, see Section 2, C).

5) Let 𝔇 = 𝔇_{X} be a geometric set of Zariski prime divisors, say defined by a set Z = {Z_{v}}_{v} of decomposition groups. Then the fact that X is complete regular like is encoded in G_{K}^{ℓ} and the set Z = {Z_{v}}_{v}.

Therefore we finally have the following:

Proposition 3.18. Let K|k be a function field with td(K|k) > 1, and k the algebraic closure of a finite field. Then in the notations from above we have the following: The geometric Galois formations G_{𝔇}, say 𝔇 = 𝔇_{X}, of some level δ ≤ td(K|k) are encoded in G_{K}^{ℓ} along the lines indicated above.

Moreover, the fact that such a geometric Galois formation G_{𝔇} is δ-residual-ly complete curve like and/or that all residual geometric Galois formations G_{v} are regular complete like (all v) is also encoded in the group theoretic information carried by G.

Let G_{𝔇} be a geometric Galois formation which is δ-residually complete curve like and residually geometric complete like up to level δ. Then one has: δ = td(K|k)−1, and viewing G_{𝔇} as a pre-divisorial Galois formation, we have: Div(X)_{(ℓ)} is a representative for its divisorial lattices, and its pre-image L_{X,(ℓ)} in K̂ is a representative for the divisorial Û_{X}-lattices in K̂.

Proof. Let G = (G, (Z_{v})_{v}) be a pre-divisorial Galois formation of level δ on G = G_{K}^{ℓ,ab}. Then G is a geometric Galois formation of level δ ≤ td(K|k) =: d if an only if by induction the following hold:

i) Z_{v} is a divisorial decomposition group as given by the Fact above for all “abstract valuations” v, and T_{v} is the inertia subgroup of Z_{v}.

ii) The set 𝔇 of all the valuations v of G is a geometric set of Zariski prime divisors.

(iii) By induction on d, each residual pre-divisorial Galois formation G_{v} on G_{v} = G_{Kv}^{ℓ} is a geometric Galois formation of level δ − 1. Remark that in the limit case when d = 2, all the inertia groups T_{a} ⊂ G_{Kv}^{ℓ} are encoded in G_{K}^{ℓ}, see Fact 2.4, thus “known”.

Now suppose that G = (G_{K}^{ℓ}, (Z_{v})_{v}) a geometric Galois formation of level δ ≤ td(K|k) is given. Then G is δ-residually complete curve like if and only if δ = d−1; and further for all the δ-residual Galois formations G_{v} = (G_{v}, (Z_{v})_{v}) one has: (Z_{v})_{v} is exactly the family of all the inertia groups of G_{v}. Remark that by induction on the length of the multi-index v′, and Fact 2.4, we know all the geometric inertia elements in G_{v′}. And that for a multi-index v of length δ, a subgroup of Z_{v} is inertia group if and only if it is maximal among the pro-cyclic subgroups generated by inertia elements of G_{v}.

Finally, let G = (G_{K}^{ℓ}, (Z_{v})_{v}) be a geometric Galois formation of level δ < d be given. Let G_{v} = (G_{v}, (Z_{v})_{v}) be the v-residual geometric Galois formation to some multi-index v of length ≤ δ. Denote by π_{1,Kv} the fundamental group attached to the residual function field Kv|k. Then by the Fact above, the fact that the normal model X_{v} → k of Kv|k is complete regular like if and only if π_{1,Kv} = π_{1,G_{v}}. Thus G is ample up to level δ if and only if π_{1,Kv} = π_{1,G_{v}} (all multi-indeces v of length ≤ δ).

4. Concluding the Proof of the Theorem (Introduction)

A) Detecting rational projections

First some definitions: Let 𝔎|𝔨 be an arbitrary function field, say with 𝔨 algebraically closed. For every non-constant function x ∈ 𝔎, let 𝔎_{x} be the relative algebraic closure of 𝔨(x) in 𝔎. Then 𝔎_{x}|𝔨 is a function field in one variable. For a given Galois extension 𝔎′|𝔎, the relative algebraic closure of 𝔎_{x} in 𝔎′ will be denoted by 𝔎′_{x}|𝔎_{x}. And the inclusion 𝔎′_{x} ↪ 𝔎′ gives rise to a surjection projection

p′_{x} : G𝔎^{′} → G𝔎_{x}^{′}

which we then call the 1-dimensional projection defined by or attached to x ∈ 𝔎 (for the Galois group G𝔎^{′}). We will say that a 1-dimensional projection p′_{x} is a rational projection, if K_{x} is a rational function field.

Our aim in this section is to show that in the case K|k with td(K|k) > 1 and k an algebraic closure of a finite field, the rational projections of G_{K}^{ℓ,ab} are group theoretically encoded in G_{K}^{ℓ}.

The strategy is as follows, compare with Bogomolov [Bo], Lemma 4.2. [But note also that in loc.cit. one gives/has a recipe for detecting the projections p_{x} under the hypothesis that ȷ(K^{×})_{(ℓ)} inside K̂ is known... whereas we cannot work under this hypothesis: We are actually looking for detecting ȷ(K^{×})_{(ℓ)} inside K̂.] First, for n = ℓ^{e}, ℓ ≠ char, consider the cup product:

ψ_{n} : 𝔎^{×}/n × 𝔎^{×}/n = H^{1}(Gℓ_{𝔎} , ℤ/n) × H^{1}(Gℓ_{𝔎} , ℤ/n) → H^{2}(Gℓ_{𝔎} , ℤ/n) = _{n}Br (𝔎).

Now Bogomolov’s idea from loc.cit. is to use the following fact: Let x, y be non-constant in 𝔎. Then 𝔎_{x} = 𝔎_{y} if and only if ψ_{n}(x, y) = 0 for all n. This means that if we know ȷ_{𝔎} (𝔎^{×}) inside 𝔎̂, then we know ȷ_{𝔎} (𝔎_{x}^{×}) too. Now Kummer theory gives an ℓ-adic perfect pairing < , >: 𝔎̂ × Gℓ,ab_{𝔎} → ℤ_{ℓ}. Thus the kernel of the projection p_{x} : Gℓ,ab_{𝔎} → Gℓ,ab_{𝔎_{x}}, is nothing but the orthogonal complement of ȷ_{𝔎} (𝔎_{x}^{×}):

N_{x} := ker(p_{x}) = {σ ∈ Gℓ,ab_{𝔎} | < x′, σ > = 0 for all x′ ∈ ȷ_{𝔎} (𝔎_{x}^{×}) }.

And this gives a description of all the kernels N_{x} of projections p_{x}. We will nevertheless need a modified version of the fact above as follows:

Fact 4.1. Let T ℓ_{𝔎} be the subgroup of Gℓ_{𝔎} generated by all the inertia subgroups of all Zariski prime divisors. For n = ℓ^{e} consider

Ψ_{n} : 𝔎^{×}/n × 𝔎^{×}/n ⟶^{ψ_{n}} _{n}Br (𝔎) ⟶^{res} H^{2}(T ℓ_{𝔎} , ℤ/n) .

Then 𝔎_{x} = 𝔎_{y} if and only if Ψ_{n}(x, y) = 0 for all n.

Proof. The implication “ ⇒ ” follows by Tsen’s Theorem, as 𝔨 is algebraically closed. To prove “ ⇐ ”, suppose that x, y are algebraically independent over 𝔨. Let t_{1} := x, t_{2} := y, …, t_{d} be a transcendence base of 𝔎|𝔨. Set Λ_{0} = 𝔨((t_{1})) … ((t_{d})), thus G_{Λ_{0}}^{ℓ} = T_{Λ_{0}} ≅ ℤ_{ℓ}^{d}, and t_{1} ∪ t_{2} has order n in H^{2}(Λ_{0}, ℤ/n). Now let Λ = Λ_{0}𝔎 be some compositum of Λ_{0} and 𝔎 over 𝔎_{0} := 𝔨(t_{1}, …, t_{d}). Then image of the canonical projection G_{Λ}^{ℓ} → G_{Λ_{0}}^{ℓ} is open, say of index δ. Therefore, the order of t_{1} ∪ t_{2} in H^{2}(Λ, ℤ/n) is at least n/δ. Finally, since the image of G_{Λ}^{ℓ} = T_{Λ}^{ℓ} → Gℓ_{𝔎} is contained in T ℓ_{𝔎}, it follows that for n = ℓ^{e} large enough we have: x ∪ y = t_{1} ∪ t_{2} ≠ 0 in H^{2}(T ℓ_{𝔎} , ℤ/n). Thus Ψ_{n}(x, y) ≠ 0.

Now come back to the situation K|k is a function field with td(K|k) > 1, and k the algebraic closure of a finite field.

Let G_{X} be a geometric Galois formation, which is complete curve like and ample up to level δ = td(K|k) − 1. Further let Div_{X,(ℓ)} be an abstract divisor group for G_{X} and L_{X,(ℓ)} its pre-image in K̂. Then after multiplying Div_{X,(ℓ)} by an ℓ-adic unit ϵ, we can suppose that Div(X)_{(ℓ)} = Div_{X,(ℓ)}. In particular, the divisorial Û_{X}-lattice L_{(ℓ)} contains K_{(ℓ)}^{×} := ȷ_{K}(K^{×}) ⊗ ℤ_{(ℓ)} inside K̂. We now have the following

Lemma 1. In the above context, L_{X,(ℓ)} / (Û_{X} · K_{(ℓ)}^{×}) is a torsion group.

Proof. First remark that it is sufficient to prove the assertion of Lemma 1 after replacing X by a complete model, say X′ → k, and further after replacing X′ by any projective model X′′ → k which dominates X′. (Indeed, such changes just replace Û_{X} by smaller subgroups, and L_{X,(ℓ)} by possibly bigger ones.) Furthermore, since L_{X,(ℓ)} is the pre-image of Div_{X,(ℓ)} in K̂, the contention of Lemma 1 is equivalent to the following: ȷ^{G_{X}} (L_{X,(ℓ)})/div(K^{×})_{(ℓ)} is a torsion group. We have a commutative diagram of the form

(∗)

0 → div(K^{×})_{(ℓ)} ↪ Div(X)_{(ℓ)} ⟶^{can} ℭ𝔩(X)_{(ℓ)} → 0
↓_{incl}  ↓_{id}  ↓^{can}
0 → ȷ^{G_{X}} (L_{X,(ℓ)}) ↪ Div(X)_{(ℓ)} ⟶^{can} ℭ𝔩_{X,(ℓ)} → 0

where the first vertical map is an inclusion. So by the Snake Lemma it follows that ȷ^{G_{X}} (L_{X,(ℓ)})/div(K^{×})_{(ℓ)} ≅ ker(ℭ𝔩(X)_{(ℓ)} → ℭ𝔩_{X,(ℓ)}). Thus we have to show that this last group is a torsion one.

Now since k is the algebraic closure of a finite field, one has an exact sequence of the form:

(∗)  0 → (torsion group)_{(ℓ)} → ℭ𝔩(X)_{(ℓ)} → NS(X)_{(ℓ)} → 0

where NS(X) is a finitely generated Abelian group. (Actually, if X is a projective smooth, the assertions above are all well known. If not, then consider a smooth alteration X̃ of X, and conclude by using the inclusion-norm map.)

And remark that the ℚ-rank of NS(X)_{(ℓ)} is the same as its ℚ_{ℓ}-rank. And further, that this last rank is the same as the ℚ_{ℓ} -rank of ℭ̂𝔩_{X,(ℓ)}. Thus finally equal to the ℚ-rank of ℭ𝔩_{X,(ℓ)}. Therefore, ker(ℭ𝔩(X)_{(ℓ)} → ℭ𝔩_{X,(ℓ)}) is a torsion group, as contented.

Corollary. For every x̃ ∈ L_{X,(ℓ)} there exists some u ∈ Û_{X} and x ∈ K_{(ℓ)}^{×} such that x^{m} = u · x for some integer m > 0.

In other words we have the following: L_{X,(ℓ)} is exactly the relative divisible hull of Û_{X} · K_{(ℓ)}^{×} in K̂, i.e., one has:

L_{X,(ℓ)} = {x̃ ∈ K̂ | ∃ m such that x̃^{m} ∈ Û_{X} · K_{(ℓ)}^{×} }

Notations/Remark.

We set K′_{X,(ℓ)} = Û_{X} · K_{(ℓ)}^{×}. Further, for x ∈ K_{(ℓ)}^{×} we denote:

1) K′_{x,(ℓ)} = Û_{X} · ȷ_{K}(K_{x})_{(ℓ)}.

2) L′_{x,(ℓ)} the divisible hull of K′_{x,(ℓ)} in L_{X,(ℓ)}.

For x̃ ∈ L_{X,(ℓ)} let us consider some u ∈ Û_{X} and x ∈ ȷ_{K}(K^{×})_{(ℓ)} such that x̃^{m} = u · x for some m > 0. Remark that since ȷ_{K}(K_{x})_{(ℓ)} ∩ Û_{X} = {1}, the presentation above is unique in the following sense: If x̃^{m} = u · x and x̃^{m′} = u′ · x′, then u^{m′} = u′^{m} and x^{m′} = x′^{m}.

For x̃ and x̃^{m} = u · x as above but with x ∈ ȷ_{K}(K^{×}), we denote:

3) K′_{x̃,(ℓ)} := K′_{x,(ℓ)}, and L′_{x̃,(ℓ)} := L′_{x,(ℓ)}.

Remark that both K′_{x̃,(ℓ)} and L′_{x̃,(ℓ)} depend only on x̃, and not the representation x̃^{m} = u · x.

N.B., for all x̃ there exists presentations as above with x ∈ ȷ_{K}(K^{×}).

Lemma 2. In the above notations for all x̃ ∈ L_{X,(ℓ)} one has:

x̃^{⊥} := { ỹ ∈ L_{X,(ℓ)} | Ψ_{n}(x̃, ỹ) = 0 for all n = ℓ^{e} } = L′_{x,(ℓ)}

Proof. For ỹ ∈ x̃^{⊥}, consider presentations x̃^{m} = u · x and ỹ^{m′} = u′ · y as usual. Then we have: 0 = Ψ_{n}(x̃^{m}, ỹ^{m′}) = Ψ_{n}(x, y), as Û_{X} is contained in both the right and the left kernel of Ψ_{n} for all n. Thus Ψ_{n}(x, y) = 0 for all n. By Fact above, it follows that K_{x} = K_{y}. Equivalently we have K′_{x,(ℓ)} = K′_{y,(ℓ)}, thus also L′_{x,(ℓ)} = L′_{y,(ℓ)}. Hence by the definition of L′_{x,(ℓ)} it follows that ỹ ∈ L′_{x,(ℓ)}. Conversely, we show that L′_{x,(ℓ)} is contained in x̃^{⊥}. Indeed, each ỹ ∈ L′_{x,(ℓ)}, is of the form ỹ^{m′} = u′ · y for some integer m′ > 0, and some u′ ∈ Û_{X}, y ∈ K′_{x,(ℓ)}. Thus 0 = Ψ_{n}(x, y) = Ψ_{n}(u x, u′ y) = m m′Ψ_{n}(x̃, ỹ) for all n. Thus Ψ_{n}(x̃, ỹ) = 0 for all n, and finally ỹ ∈ x̃^{⊥}.

Thus Lemma 2 above shows that the following first two sets are equal, and in a bijective correspondences with the third one:

{ x̃^{⊥} | x̃ ∈ L_{X,(ℓ)}\ Û_{X} } = { L′_{x,(ℓ)} | x ∈ K′_{X,(ℓ)}\ Û_{X} } ≅ { K′_{x,(ℓ)} | x ∈ K\k }

Now let some L′_{x̃,(ℓ)} be given. By the above correspondence, there exists a unique subfield K_{x} of K such that L′_{x̃,(ℓ)} = L′_{x,(ℓ)}, or equivalently, that L′_{x̃,(ℓ)} is the relative divisible hull of Û_{X} · K_{x,(ℓ)}. To simplify notations, let us set

G := G_{K}^{ℓ,ab}, G_{x} := G_{K_{x}}^{ℓ,ab}, and p_{x} : G → G_{x}

the resulting canonical projection. And let φ̂_{x} : K̂_{x} → K̂ be the embedding defined by p_{x} via Kummer theory. By ℓ-adic Kummer theory, the knowledge of p_{x} is equivalent to the knowledge of φ̂_{x}. We consider the unique complete smooth model X_{x} → k of K_{x}|k, and the corresponding geometric Galois formation G_{x}, etc.. We set K_{x,(ℓ)} = Û_{x} · K_{x,(ℓ)} inside K̂_{x}. Then we have:

K_{x,(ℓ)} := φ̂_{x}(Û_{x} · K_{x,(ℓ)}) = φ̂_{x}(K̂_{x}) ∩ K′_{x,(ℓ)} = φ̂_{x}(K̂_{x}) ∩ L′_{x,(ℓ)}

We next indicate how to identify K_{x,(ℓ)} inside L′_{x,(ℓ)}. Equivalently, by the discussion at the beginning of this Subsection, this gives then a recipe for detecting the 1-dimensional projection p_{x}.

First, let v be a Zariski prime divisor of K, and ȷ^{v} : K̂ → ℤ_{ℓ} as introduced/defined in Section 3, Definition/Remark 3.2, 3). Since ȷ^{v} is trivial on Û_{X}, it follows that ȷ^{v} is trivial on L_{x̃,(ℓ)} if and only if v is trivial on K_{x}.

Next let ȷ_{v} : Û_{v} → K̂v be the reduction map attached to a Zariski prime divisor v, see Section 3, Definition/Remark 3.2, 6). Then the restriction of ȷ_{v} to the subgroup K_{(ℓ)}^{×} coincides with the reduction map defined/introduced in Section 1, A), Definition/Remark 2.4, 3). In particular, since td(K_{x}|k) = 1, one has: ȷ^{v} is not trivial on K_{x} if an only if ȷ_{v} is trivial on K_{x}.

Lemma 3. In the above context we have:

K_{x,(ℓ)} = { ỹ ∈ L′_{x,(ℓ)} | ȷ_{v}(ỹ) = 1 for all v with ȷ^{v}(L′_{x,(ℓ)}) ≠ 0 }

Proof. The direct inclusion is clear by the discussion above. In order to prove the converse, we view the regular field extension K|K_{x} as the generic fiber of a flat family X → S, where X → k is a projective normal model of K|k, and S → k is a projective normal model of K_{x}|k. By the Specialisation Theorem for the fundamental group, it follows that for a “general point” s ∈ S, we have a canonical exact sequence

π_{1}(X_{s}) → π_{1}(X) → π_{1}(S) → 1 .

Now let v be the Zariski prime divisor of K defined by the generic point of the special fiber X_{s}. Then the translation of the above assertion via Kummer Theory means the following: The reduction homomorphism ȷ_{v} defines an exact sequence of the form:

1 → Û_{x} → Û_{X} ⟶^{ȷ_{v}} Û_{X_{v}} ⊂ K̂v .

In other words, ỹ ∉ Û_{x} implies ȷ_{v}(ỹ) ≠ 1.

Proposition 4.2. In the above notations, consider the perfect ℓ-adic pairing given by Kummer theory < , >: K̂ × G_{K}^{ℓ,ab} → ℤ_{ℓ}.

(1) The kernels of 1-dimensional projections p_{x}^{ℓ,ab} : G → G_{x} are exactly the closed subgroups N_{x} of G of the form:

N_{x} = {σ ∈ G | < ỹ, σ > = 0 for all ỹ ∈ K_{x,(ℓ)} }

Let p_{x} : G → G/N_{x} =: G_{x} be a 1-dimensional projection. Then one has:

(2) A procyclic subgroup T_{v_{x}} of G_{x} is an inertia group if and only if there exits an inertia group T_{v} of G such that p_{x}(T_{v}) has finite index in T_{v_{x}}.

3) Finally, p_{x} : G → G_{x} is a rational projection if and only if the inertia groups T_{v_{x}} (all v_{x}) generate G_{x}.

Clearly, all the facts above are group theoretically encoded in G_{K}^{ℓ}.

Proof. Clear by the observations above and the characterization of the 1-dimensional projections via K_{(ℓ)} mentioned above.

B) The proof of Theorem (Introduction)

By Proposition 3.18 it follows that the geometric Galois formations over K|k are encoded in G_{K}^{ℓ}. And it is clear that the group theoretic description of the geometric Galois formations is preserved under isomorphisms. Therefore, every isomorphism

Φ : G_{K}^{ℓ} → G_{L}^{ℓ}

defines a bijection from the set of all geometric Galois formations over K|k onto the set of all geometric Galois formations over L|l. Further, if G_{Y} corresponds to G_{X} under this bijection, then Φ gives rise to an isomorphism of divisorial Galois formations

Φ : G_{X} → G_{Y} .

Now suppose that X is complete regular like. Since this fact is encoded group theoretically in G_{K}^{ℓ} and the projection G_{K}^{ℓ} → G := G_{K}^{ℓ,ab}, it follows that the corresponding Y is complete regular like too. Consider the canonical commutative diagram

(∗)_{(ℓ)}  \begin{aligned}0 → Û_{Y} & → & L_{Y,(ℓ)} & → & Div(Y)_{(ℓ)} & → & ℭ𝔩_{Y,(ℓ)} → 0 \\ ↓ψ̂ & & ↓ψ̂ & & ↓ & & ↓ \\ 0 → Û_{X} & → & L_{X,(ℓ)} & → & Div(X)_{(ℓ)} & → & ℭ𝔩_{X,(ℓ)} → 0 \end{aligned}

where the vertical map ψ̂ is such a multiple ψ̂ = ϵ φ̂ of the Kummer homomorphism φ̂ : L̂ → K̂ that one has: ψ(L_{Y,(ℓ)}) = L_{X,(ℓ)}.

We next remark that by Proposition 4.2 above, Φ also defines a bijection from the set of all the rational projections of G_{X} onto the set of all the rational projections of G_{Y}.

On the other hand, K_{(ℓ)} is encoded in G_{K}^{ℓ} as follows: The multiplicative group K^{×} is generated by the set of all the K_{x}^{×} with K_{x} a rational function field. In other words, if (p_{x})_{x} is the family of all the rational projections as above, then K_{(ℓ)} ⊂ K̂ is generated by all the images K_{x,(ℓ)}. Therefore, in order to detect the ℓ-adic equivalence class of the lattice K_{(ℓ)} inside K̂, one can proceed as follows: Choose a divisorial lattice L_{K,(ℓ)} in K̂. For every rational projection p_{x}, consider the unique divisorial lattice L_{K_{x},(ℓ)} in K̂_{x} whose image K_{x,(ℓ)} under the completion morphism φ̂_{x} : K̂_{x} → K̂ is contained in L_{K,(ℓ)}. Then the subgroup L_{K,(ℓ)}^{0} of L_{K,(ℓ)} which is generated by all the K_{x,(ℓ)} is the unique lattice in L_{K,(ℓ)} which is equivalent to K_{(ℓ)}.

Since the description of L_{K,(ℓ)}^{0} above is given in pure group theoretic terms, it is clear that the Kummer homomorphism φ̂_{K} : L̂ → K̂ maps L_{(ℓ)} isomorphically onto a lattice L_{K,(ℓ)}^{0} in K̂ which is ℓ-adically equivalent to K_{(ℓ)}. Thus after multiplying φ̂ by a properly chosen ℓ-adic unit (unique modulo rational ℓ-adic units), we can suppose that φ̂ maps L_{(ℓ)} isomorphically on K_{(ℓ)}.

We finally apply Pop [P4], Theorem 5.11, and deduce that there exist finite pure inseparable extensions K_{0}|K and L_{0}|L such that φ̂_{K_{0}} : L̂_{0} → K̂_{0} is the ℓ-adic completion of a unique field isomorphism ı_{0} : K_{0} → L_{0}.

Moreover, let K′|K be a finite (Galois) extension of K inside K^{ℓ}, and via the isomorphism Φ the corresponding finite (Galois) extension L′|L. Then using the functoriality of Kummer theory, we have a commutative diagram of the form

L̂ ⟶^{φ̂_{K}}  K̂
↓^{can}  ↓^{can}
L̂′ ⟶^{φ̂_{K′}} K̂′

Suppose K′|K is Galois, and Φ′ : Gal(K′|K) → Gal(L′|L) is the isomorphism induced by Φ. Then for all σ ∈ Gal(K′|K), and τ = Φ(σ) we have:

σ ∘ φ̂_{K′} = φ̂_{K′} ∘ τ i.e., Φ(σ) = φ̂_{K′}^{−1} ∘ σ ∘ φ̂_{K′} .

So if φ̂_{K} maps L_{(ℓ)} into K_{(ℓ)}, then φ̂_{K′} automatically maps L′_{(ℓ)} into K′_{(ℓ)}, i.e., it is not necessary to “re-norm” φ̂_{K′} in order to have φ̂_{K′}(L′_{(ℓ)}) = K′_{(ℓ)}. Thus the above commutative diagram induces a corresponding one with K_{(ℓ)} in stead of K̂, etc..

Conclusion:

Now taking limits over all finite extensions K′|K_{0} inside K_{0}^{ℓ} = K^{ℓ}K_{0}, and the corresponding finite extensions L′|L_{0} inside L_{0}^{ℓ}, we finally get an isomorphism ϕ : L^{i}^{ℓ} → K^{i}^{ℓ} defining Φ, i.e., Φ(g) = ϕ^{−1} g ϕ for all g ∈ G_{K}^{ℓ}.

We still have to prove that any two automorphisms ϕ′ and ϕ′′ both defining Φ differ by a power of Frobenius. Indeed, setting ϕ := ϕ′′ ∘ ϕ′^{−1} we get: ϕ is an automorphism of L^{i}^{ℓ} which maps L^{i} onto itself, and induces the identity on G_{L^{i}}^{ℓ}. We claim that such an automorphism ϕ is a power of Frobenius. Indeed, let v be an arbitrary Zariski prime divisor of L^{i}^{ℓ}. Then w := v ∘ ϕ is also a Zariski prime divisor of L^{i}^{ℓ}. Moreover, by the usual formalism, we have Z_{w} = ϕ^{−1}Z_{v} ϕ inside G_{L^{i}}^{ℓ}. Since the conjugation by ϕ is the identity on G_{L^{i}}^{ℓ}, it follows that Z_{w} = Z_{v}. Thus by Proposition 1.4, (1), it follows that v and w are equivalent valuations on L^{i}^{ℓ}. In particular, v(x) > 0 if and only if w(x) > 0 (all x ∈ L^{i}^{ℓ}). Let char(k) = p. We claim that y := ϕ(x) is some p-power of x. First, if y and x are algebraically independent, then there exists a Zariski prime divisor v of L^{i}^{ℓ} such that v(x) = 1 and v(y) = 0. But v(y) = v ∘ ϕ(x) = w(x), contradiction! Therefore, y = ϕ(x) is always algebraic integer over k[x], and vice-versa: x is algebraic integer over k[y]. Now let f(X, Y) ∈ k[X, Y] be the minimal polynomial polynomial relation between x and y over k. We claim that f(X, Y) is pure inseparable in Y (and thus by symmetry, also in X). Let namely y_{a} =: y_{1}, …, y_{r} be the distinct roots of f(x, Y) over k(x). The for a “general a ∈ k we have: If x_{a} = x + a, then y_{a} =: ϕ(x_{a}) = y + ϕ(a). Thus x_{a} and y_{a} have the same zeros in 𝔇_{K^{ℓ}}^{1}. And clearly, the minimal polynomial polynomial relation satisfied by x_{a} and y_{a} over k is f_{a}(X, Y) = f(X − a, Y − ϕ(a)). Therefore, for a general a ∈ k, the polynomial f_{a}(0, Y) has s distinct roots b_{1}, …, b_{s}. And the specializations (x, y) ↦ (a, b_{i}) give rise to s different places v_{i} of k(x)^{ℓ} all of which are zeros of x_{a}. But then these places must also be zeros of y_{a}. Thus 0 = y_{a}(b_{i}) = ϕ(a)−b_{i} implies b_{i} = ϕ(a), thus s = 1. Thus finally f(X, Y) is purely inseparable in Y, and by symmetry also in X.

Thus finally we have: ϕ(x) = a_{x} x^{n_{x}}, where a_{x} ∈ k and n_{x} = p^{e_{x}} depend on x. We next show that a_{x} = 1 and that n_{x} = p^{e} do not depend on x. Indeed, since ϕ is a field automorphism, we have ϕ(x + y) = ϕ(x) + ϕ(y) for all x, y from L^{i}^{ℓ}. Or equivalently: a_{x+y}(x + y)^{n_{x+y}} = a_{x} x^{n_{x}} + a_{y}y^{n_{y}}. Considering several arbitrary y ∈ L^{i}^{ℓ} such that x and y are algebraically independent over k, we therefore must have n_{x+y} = n_{y} = n_{x} = p^{e} and a_{x+y} = a_{y} = a_{x} = a, thus independent of x and y. Since a_{1} = 1, we have a = 1. Finally, choosing a transcendence basis T of L over k (say, which contains x), we see that the restriction of ϕ to k(T ) is Frob^{e}. Therefore, ϕ = Frob^{e} on L^{i}^{ℓ}.

The Theorem (Introduction) is proved.

Mathematisches Institut, Universität Bonn

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