1.1. Notation and definitions. Let K be a field and (G

ON THE PROFINITE ABELIAN BECKMANN-BLACK PROBLEM

Nour Ghazi

Abstract. The main topic of this paper is to generalize the problem of Beckmann-Black for profinite groups. We introduce the Beckmann- Black problem for complete systems of finite groups and for unramified extensions. We prove that every Galois extension of profinite abelian group over a ψ-free field is the specialization of some tower of regular Galois extensions of the same group.

1. Presentation

1.1. Notation and definitions. Let K be a field and (G

, s

)

be a complete system, i.e., a projective system of finite groups G

(n ∈ N) and epimorphisms s

: G

→ G

(n > 0).

An abelian complete system is a complete system (G

, s

)

such that G

is an abelian group for every n ∈ N.

Denote by K an algebraic closure of K and let G

be the absolute Galois group of K. Denote by K(T ) the field of rational functions in one variable T with coefficients in K.

A finite extension F/K(T ) is said to be a regular Galois extension of group G if F/K(T ) is a Galois extension of group G and the function field F/K is regular. Recall that regular means F ∩ K = K.

In this paper, we want to generalize one of open questions in Inverse Galois Theory known as the Beckmann-Black problem. More precisely, if K is a field and G is a finite group, then the Beckmann-Black problem asks whether each Galois extension E/K of group G is the specialization of some regular Galois extension F/K(T ) of group G at some unramified point t

∈ P

(K).

The Beckmann-Black problem for finite groups is known to have a positive answer in some situations. For example:

• G is a symmetric group (Beckmann [Be] if K is a number field, Black [Bl2] for an arbitrary field).

• G is the dihedral group D

of order 2n when n is odd (Black [Bl1]).

• G is an abelian group (Beckmann [Be] and Black [Bl1] if K is a number field, and D` ebes [D1] if K is an arbitrary field).

Mathematics Subject Classification. Primary 12F12, Secondary 12E25, 14H30.

Key words and phrases. Inverse Galois theory, algebraic covers.

233

• G is a finite group and K is P(seudo) A(lgebraically) C(losed)

(D` ebes [D1]).

Throughout this paper, we assume K is a perfect field.

1.2. Profinite Beckmann-Black Problem. Let K be a field and (G

, s

)

be a complete system. Fix a tower of Galois extensions (E

/K)

of group (G

, s

)

; this means that E

/K is a Galois ex- tension of group G

(n ⩾ 0) such that E

= E

for every n ⩾ 1.

The question of Profinite Beckmann-Black over K asks to find:

(i) a tower of regular Galois extensions (F

/K(T ))

realizing the complete system (G

, s

)

.

(ii) an unramified point t

∈ P

(K) such that the specialization (F

)

/K of F

/K(T ) at this point t

is a Galois extension isomorphic to E

/K for every n ∈ N.

Note that the specialization of F

/K(T ) is independent of the selected point over the point t

. So we can define, without problem, the specialization of a tower of regular Galois extensions at the point t

.

1.3. Main result. Our purpose in this paper is to study the Profinite Beckmann-Black Problem for abelian complete systems. We will prove, in

§3, the following result.

Theorem 1.1. Let K be an uncountable regular ψ-free field and let (G

, s

)

be an abelian complete system. For every tower of Galois ex- tensions (E

/K)

of group (G

, s

)

, there exists a tower of regu- lar Galois extensions (F

/K(T ))

(geometrically) unramified at a point T = t

∈ P

(K) with specialization ((F

)

0

/K(T ))

n∈N

isomorphic to the tower (E

/K)

.

A typical example of K satisfying the assumption of the above theorem is the field of formal Laurent series Q

((x)). See §2.1 below.

2. Preliminary reminders

2.1. Regular ψ-free field. A field K is said to be a regular ψ-free field if, for every complete system (G

, s

)

, there exists a tower of regular Galois extensions (F

/K(T ))

realizing regularly the complete system (G

, s

)

. This means that, for every n ∈ N, there exists a regular Galois extension F

/K(T ) of group G

such that there exists an epimorphism

ε

: Gal(F

/K(T )) → G

1Recall that a field K is PAC if each geometrically irreducible variety V defined over K has infinitely many K-rational points.

making the following diagram commute:

Gal(F

/K(T ))

εn+1



Gal(F

/K(T ))

εn



G

G

.

Moreover, if K is a henselian field of characteristic 0 such that [K(µ

) : K] < ∞, then K is a regular ψ-free field [DD2, theorem (3.4)].

2.2. Specialization. Let G be a finite group, K be a field and F/K(T ) be a regular Galois extension of group G with a K-rational branch divisor t. This extension corresponds to some epimorphism ϕ : π

( P

\ t)

→ G and the extension F K/K(T ) corresponds to the restriction ϕ : π

( P

\ t)

→ G of ϕ to π

( P

\ t)

; it is still surjective as F/K is a regular extension. Let t

∈ P

(K) \ t be a K-rational point and consider the section, s

, corresponding to this point.

1

π

( P

\ t )

ϕ

π

( P

\ t)

ϕ

G

s_t0

1.

G G

The specialization, F

, of F/K(T ) at T = t

is the residue field of F at some prime above t

in the extension F/K(T ). The specialization F

corresponds to the homomorphism ϕ ◦ s

([D2, proposition(2.1)]). More precisely, F

is the fixed field in K of ker(ϕ ◦ s

). In particular, the spe- cialization F

/K is a Galois field extension of group Im(ϕ ◦ s

) ⊂ G. The morphism ϕ ◦ s

is called the specialization morphism of F/K(T ) at t

. For more details, we refer to [D2, chapter 3] and to [D1].

3. Proof of theorem 1.1

Let K be an uncountable regular ψ-free field and (G

, s

)

be an abelian complete system. Denote by K((T )) the field of formal Laurent series in T with coefficients in K.

To prove our theorem, we have three stages. Firstly, we show that our hypothesis implies that:

Proposition 3.1. There exist a point t

∈ P

(K) and a tower of regu-

lar Galois extensions ( b F

/K(T ))

of group (G

, s

)

such that b F

⊂

K((T − t

)) for every n ∈ N.

For second stage, let (H

, γ

)

be a sub-system of (G

, s

)

; this means that H

is a sub-group of G

and the restriction of s

on H

is γ

, for each n ∈ N. We will prove that Proposition 3.1 implies the following conclusion:

Proposition 3.2. Let (G

, s

)

be an abelian complete system and (H

, γ

)

be a sub-system of (G

, s

)

. For each tower of Galois exten- sions (E

/K)

of group (H

, γ

)

, there exists a tower of regular Galois extensions (F

/K(T ))

of group (G

, s

)

such that its specialization at the point T = t

is a tower of Galois extension of group (H

, γ

)

isomorphic to (E

/K)

.

At third stage, taking H

= G

in the proposition 3.2 gives the conclusion of our result.

• 1

stage “Proof of Proposition 3.1”

As K is a regular ψ-free field, there exists a tower of regular Galois extensions (L

/K(T ))

of group (G

, s

)

. Since there are at most countably many branch points in ∪

n

L

/K(T ), there is an unramified point T = t

∈ P

(K) in the function field tower (L

/K(T ))

.

Denote by ϕ

: π

( P

\ t

)

→ G

the regular representation correspond- ing to the regular Galois extension L

/K(T ), where t

is the branch point set of this extension. Let r

: π

(P

\ t

)

→ G

be the natural restric- tion and s

: G

→ π

(P

\ t

)

the section corresponding to the point T = t

.

On the other hand, for each n ∈ N, we have t

⊆ t

. So there exists a natural morphism i

: π

( P

\ t

)

→ π

( P

\ t

)

such that i

(x) = 1 for every x ∈ t

\ t

. The regular Galois extension L

/K(T ) is an unramified extension over t

\ t

, so the morphism s

◦ ϕ

factors through i

to give s

◦ ϕ

= ϕ

◦ i

for every n ∈ N. Furthermore, we have r

= r

◦ i

and i

◦ s

= s

.

Fix n ∈ N. Let b ϕ

: π

( P

\ t

)

→ G

be a map defined as follows: for x ∈ π

( P

\ t

)

, we pose

ϕ b

(x) = ϕ

(x).(ϕ

◦ s

◦ r

(x))

.

This map b ϕ

is a group homomorphism (because G

is an abelian group) with image group equal to G

. Thus b ϕ

defines a regular Galois extension F b

/K(T ) of group G

. Furthermore, b ϕ

and ϕ

coincide over K(T ). this implies that

L

K = b F

K.

For every n ∈ N, we have:

s

◦ b ϕ

= s

◦ (ϕ

.(ϕ

◦ s

◦ r

)

)

= (s

◦ ϕ

).(s

◦ ϕ

◦ s

◦ r

)

= (ϕ

◦ i

).(ϕ

◦ i

◦ s

◦ r

)

= (ϕ

◦ i

).(ϕ

◦ s

◦ r

◦ i

)

= ((ϕ

).(ϕ

◦ s

◦ r

)

) ◦ i

= b ϕ

◦ i

.

Thus s

◦ b ϕ

= b ϕ

◦ i

. This implies that the extension b F

/K(T ), n ∈ N, form a tower of regular Galois extensions of group (G

, s

)

. Furthermore, we have b ϕ

◦ s

= (ϕ

◦ s

).(ϕ

◦ s

)

= 1, hence b F

⊆ K((T − t

)) for every n ∈ N.

• 2

stage “Proof of Proposition 3.2”.

Let (H

, γ

)

be any sub-system of (G

, s

)

. Suppose given a tower of Galois extensions (E

/K)

of group (H

, γ

)

. By virtue of Propo- sition 3.1, we find a tower of regular Galois extension ( b F

/K(T ))

of group (G

, s

)

such that b F

⊆ K((T − t

)) for some unramified point t

∈ K. We want to replace the above tower ( b F

/K(T ))

by another tower (F

/K(T ))

of regular Galois extensions of group (G

, s

)

so that its specialization at the point T = t

is a tower of Galois extension of group (H

, γ

)

isomorphic to (E

/K)

.

We give two arguments. The first one uses a similar process as in first stage. The second one is essentially equivalent but uses a different formalism.

[Argument 1.] Consider the representation b ϕ

: π

( P

\ t

)

→ G

cor- responding to the regular Galois extension b F

/K(T ). Recall that L

K = F b

K, in particular the branch point set of b F

/K(T ) is t

and t

is unrami- fied.

Still denote by r

: π

( P

\ t

)

→ G

the natural restriction, by s

: G

→ π

( P

\ t

)

the section corresponding to the point t

and by i

: π

( P

\ t

)

→ π

( P

\ t

)

the natural morphism given by t

⊆ t

. Recall that s

◦ ϕ

= ϕ

◦ i

, r

= r

◦ i

and i

◦ s

= s

, for every n > 0.

Let φ

: G

→ H

⊂ G

be a representation of the Galois extension E

/K (n ∈ N); we have E

= (K)

and γ

◦ φ

= φ

.

Fix n ∈ N. Let ϕ

: π

( P

\ t

)

→ G

be the map defined as follows:

ϕ

(x) = b ϕ

(x).(φ

◦ r

(x))

.

The map ϕ

is a group homomorphism (because G

is abelian) with

image group equal to G

, and coincides with b ϕ

and so with ϕ

as well on

π

( P

\ t

)

. Thus ϕ

defines a regular Galois extension F

/K(T ) of group

G

and such that F

K = F

K.

For every n > 0, we have:

s

◦ ϕ

= s

◦ (b ϕ

.(φ

◦ r

)

)

= (s

◦ b ϕ

).(γ

◦ φ

◦ r

)

= ( b ϕ

◦ i

).(φ

◦ r

◦ i

)

= ( b ϕ

.(φ

◦ r

)

) ◦ i

= ϕ

◦ i

.

Thus s

◦ϕ

= ϕ

◦i

. This implies that the extensions F

/K(T ) (n ∈ N) form a tower of regular Galois extensions of group (G

, s

)

. Further- more, we have ϕ

◦ s

= 1.φ

= φ

and so the specialized extension of F

/K(T ) at t

is K

= E

(n ∈ N).

Remark 3.3. Stage 1 and stage 2 could have been merged by defining directly ϕ

in terms of ϕ

as follows: for x ∈ π

(P

\ t

)

,

ϕ

(x) = ϕ

(x) · (ϕ

◦ s

◦ r

(x))

· (φ

◦ r

(x))

(n ∈ N).

[Argument 2.] Fix n ∈ N. Denote by φ

: b F

→ K the K-place asso- ciated to b F

at the point t

. By an extension of scalars from K to E

, the extension b F

E

/E

(T ) is a regular Galois extension of group G

. As E

/K is an algebraic extension, the point t

is still unramified in the extension F b

E

/K(T ): E

F b

⊂ E

((T − t

)) ⊂ K((T − t

)).

On the other hand, b F

/K(T ) and E

/K are two Galois extensions, so F b

E

/K(T ) is a Galois extension of group isomorphic to G

×H

, for every n ∈ N. Consider the map ρ

: G

× H

→ G

given by ρ

(g, h) = gh. This map ρ

is a group homomorphism because G

is an abelian group (n ⩾ 0).

Fix n ∈ N. Denote by F

the subfield of E

F b

fixed by Kerρ

. Thus F

/K(T ) is a Galois extension of group G

. First we prove that F

is a reg- ular extension over K. This means that we must verify that [F

: K(T )] = [F

K : K(T )]. In fact, the two extensions F

/K(T ) and E

(T )/K(T ) are linearly disjoint because

F

∩ E

(T ) = (E

F b

)

∩ (E

F b

)

= (E

F b

)

= (E

F b

)

= K(T ).

We deduce that [F

: K(T )] = [F

E

: E

(T )]. But the field F

E

= b F

E

is regular over E

, so [F

E

: E

(T )] = [F

E

: E

(T )]. As E

/K is a finite extension, then E

= K. We conclude that

[F

: K(T )] = [F

E

: E

(T )] = [F

K : K(T )].

Hence the finite extension F

/K(T ) is a regular Galois extension of group G

.

Furthermore, the following diagram:

G

× H

ρn //

(sn,γn)



G



G

× H

G

is a commutative diagram because the restriction of s

on H

is equal to γ

(n > 0). We deduce that the family of regular Galois extensions (F

/K(T ))

form a tower of regular Galois extensions of group (G

, s

)

.

Now we must show that the specialization of (F

/K(T ))

at the point T = t

is isomorphic to (E

/K)

.

Let us fix n ∈ N and we will study the specialization of F

/K(T )) at the point T = t

. Firstly, as E

⊆ E

F b

, so φ

(E

) ⊆ φ

(E

F b

). Now E

/K is an extension geometrically unramified at t

, so φ

(E

) = E

. Thus E

⊆ φ

(E

F b

). Denote by D

the decomposition group of t

in E

F b

/K.

As E

⊆ φ

(E

F b

), so |D

| = [φ

(E

F b

) : K] ⩾ [E

: K] = |H

|.

Furthermore, we know that (E

F b

)

= b F

, so φ

((E

F b

)

) = φ

( b F

).

Now φ

being a K-place means that φ

( b F

) = K. Thus φ

((E

F b

)

) = K.

As the point t

is unramified in b F

E

/K(T ), so denote by (E

F b

)

the subfield of E

F b

fixed by D

. This field (E

F b

)

is the biggest subfield of E

F b

such that φ

((E

F b

)

) = K.

Indeed φ

((E

F b

)

) = K, then (E

F b

)

⊆ (E

F b

)

. Thus [ b F

E

: (E

F b

)

] ⩽ [ b F

E

: (E

F b

)

] = |H

|.

Thus

[φ

( b F

E

) : φ

((E

F b

)

)] ⩽ |H

| so [φ

( b F

E

) : K] ⩽ |H

|.

We deduce that |D

| = [φ

( b F

E

) : K] = |H

| = [E

: K] and E

⊆ φ

( b F

E

). Thus E

= φ

( b F

E

), so the specialization of b F

E

/K(T ) at the point t

is isomorphic to E

/K and D

= Gal(E

/K) = H

.

Finally, denote by φ c

the restriction of φ

on F

. The specialization

of F

/K(T ) at the point t

is an intermediate extension of E

/K (the

specialization extension of b F

E

/K(T )) of group equal to ρ

(D

). But

ρ

(D

) = ρ

(H

) = ρ

( {1} × H

) = H

. This implies that the special- ization extension of F

/K(T ) at the point t

is a Galois extension of group H

isomorphic to E

/K.

To sum up, we find a tower of regular Galois extensions (F

/K(T ))

of group (G

, s

)

such that the tower of specialization at the point T = t

is a Galois tower of group (H

, γ

)

isomorphic to (E

/K)

.

• 3

stage “Conclusion”.

Putting G

= H

for every n ∈ N in Proposition 3.2, we conclude that, for each tower of Galois extensions (E

/K)

of group (G

, s

)

, there exists a tower of regular Galois extensions (F

/K(T ))

such that its specialization at the point T = t

is a tower of Galois extension isomorphic to (E

/K)

.

Acknowledgment: I am indebted to an anonymous reviewer for insightful comments and a kind help and his interest in my paper.

References

[Be] S. Beckmann, Is every extension ofQ the specialization of a branched covering?, J. Algebra 164 (1994), 430–451.

[Bl1] E. Black, Arithmetic lifting of dihedral extensions, J. Algebra 203 (1998), 12–29.

[Bl2] E. Black, Deformations of Dihedral 2-group extensions of fields, Trans. Amer.

Math. Soc. 351 (1999), 3229–3241.

[D1] P. D`ebes, Galois Covers with Prescribed Fibers: The Beckmann-Black problem, Ann. Scuola Norm. Sup. Paris, Cl. Sci. 28 (1999), 273-286.

[D2] P. D`ebes, Arithm´etique des revˆetements de la droite., Cours DEA (chapters 1-8), Univ. de Lille 1, 2009; math.univ-lille1.fr/∼pde/.

[DD2] P. D`ebes and B. Deschamps, Corps ψ-libres et th´eorie inverse de Galois infinie , J. Reine Angew. Math., 574 (2004), 197–218.

Nour Ghazi University of Damascus,

Faculty of Sciences, Department of Mathematics,

Damascus, Syria

e-mail address: [email protected] (Received March 11, 2017 ) (Accepted August 23, 2017 )