Some examples of FAB and mild pro-p-group with trivial cup-product

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trivial cup-product

Christian Maire

To cite this version:

Christian Maire. Some examples of FAB and mild pro-p-group with trivial cup-product. Kyushu

Journal of Mathematics, Kyushu University, 2014, pp.à venir. �hal-00922614�

WITH TRIVIAL CUP-PRODUCT

CHRISTIAN MAIRE

Abstract.

Let

be the Galois group of the maximal pro-p-extension

of

unramified outside a finite set

of places of

not containing the prime

examples of mild (and thus FAB) pro-p-group

for which some rela- tions are of degree

(according to the Zassenhaus filtration). The key computation are done in some Heisenberg extensions of

of degree

. With the help of GP-Pari we produce some examples for

Introduction

Let p > 2 be an odd prime number. Let S = {ℓ

, · · · , ℓ

} be a finite set of prime numbers ℓ

, with ℓ

≡ 1(mod p). Consider Q

the maximal pro-p- extension of Q unramified outside S and put G

= Gal( Q

/ Q ).

In the 60’s, Koch (see [8]) gave a description of the pro-p-group G

by gen- erators and relations. Thanks to this description, in 2006 Labute in [9] gave the first examples of pro-p-groups G

with cohomological dimension 2. By class field theory these groups have the FAB property: every open subgroup U of G

has a finite abelianization. And then the strict cohomological di- mension of these pro-p-groups G

is 3 (see for example [12], chapter III).

To produce such examples, Labute used a criteria for a pro-p-group to be mild (this one is related to a criterion of Anick [1]): in some favorable situa- tions the initial terms of the relations satisfy some very special combinatorial properties such that the graded algebra built on the lower p-central series of G

has a very nice description in terms of the corresponding free graded algebra. In the examples of Labute, the relations are of degree 2 according to the Zassenhaus filtration.

Very recently, the arithmetic aspect of the work of Labute has been improved by a serie of papers of Schmidt [14], [15].

In [16], [17], when p = 2, Vogel has given a way to produce mild pro-2-groups G

where the relations are of degree 3. This method uses the Rédei symbol [13]. With this, Gärtner [7] has produced an arithmetic example of mild pro-2-group G where the relations are of degree 3 and such that, assuming the Leopoldt conjecture, this group is FAB. The pro-2-group produced by Gärtner corresponds to the maximal pro-2-extension of Q unramified outside S = {2, 17, 7489, 15809} in which the place 5 splits completely. As the

Date

: December 28, 2013.

1991

11R21, 11Y40, 11R34, 20F05, 20F14, 20F40.

Key words and phrases.

Mild pro-p-groups, restricted ramification, Class Field Theory.

1

prime 2 is in S, it is necessary to force a place to split completlely so as to rule out the Z

-cyclotomic extension.

In [5] Forré has developed the approach of mild pro-p-group by looking at the Zassenhaus filtration in the noncommutative ring of formal power series F

[[X

, · · · , X

]]

with coefficients in F

. It is this approach that we will use here.

By considering the arithmetic in some Heisenberg extension of degree 3

over Q we produce some mild pro-p-groups G

for which some relations are of degree 3. Moreover, these pro-3-groups are FAB (unconditionally). Here we do not have the Rédei symbols but, it will be interesting to explore the equality of Proposition 2.23 in this way.

In the first section, we recall the basic facts about mild pro-p-groups (ac- cording to the Zassenhaus filtration). In section 2, we develop the arithmetic strategy and present the principle of the computation based on Class Field Theory. In the last part we produce the two following examples:

Example 0.1. The pro-3-group G

= G

can be described by the generators x

, x

and x

and by the relations

ρ

≡ [[x

, x

], x

][[x

, x

], x

][[x

, x

], x

] (mod F

), ρ

≡ [[x

, x

], x

]

(mod F

),

ρ

≡ [[x

, x

], x

]

[[x

, x

], x

][[x

, x

], x

] (mod F

).

This pro-3-group G

is mild and FAB. In particular:

(i) the pro-3-group G

is of cohomological dimension 2;

(ii) the Zassenhaus filtration of G

has 1

1 − 3t + 3t

as Poincaré series.

Example 0.2. Let S = {7, 13, 381, 11971}. The pro-3-group G

is mild and FAB with two relations of degree 2 and two relations of degree 3 with

1

1 − 4t + 2t

+ 2t

as Poincaré series.

All the computations have been done with GP-Pari [2].

Notation: For x, y in a group G, we denote by [x, y] = x

y

xy the com- mutator of x and y.

Acknowledgments. The author would like to thank Jan Minac and John Labute for making possible a visit to McGill and to the University of Western Ontario during summer 2012. The work of this paper started during these stays. He is also grateful to John Labute and to Jan Minac for many useful conversations and for their warm hospitality and their enthusiastic support.

He would like to thank to Farshid Hajir and Jochen Gärtner for their interest in this work.

1. Relations and mild pro- p -groups

For this section, we refer to [4], [5] and [8].

1.1. The Zassenhaus filtration. Let F

(d) := F

[[X

, · · · , X

]]

be the noncommutative ring of formal power series in variables X

, · · · , X

over the finite field F

. Denote by I the two sided-ideal generated by the X

: it is the augmentation ideal of F

(d), i.e. the kernel of the natural morphism F

(d) ։ F

:

I = ker F

(d) −→ F

.

The ring F

(d) is a topological local ring where the family ( I

)

is a neighborhood basis of 0.

Now consider the free prop-p-group F of rank d generated by the elements x

, · · · , x

. Denote by Λ(F) the complete algebra

Λ(F) := lim

←

U⊂F

F

[F/U],

where U runs through open normal subgroups of F. Let I(F) = ker (Λ(F) → F

) ,

be the augmentation ideal of Λ(F). Then it is well-know that the map (the Magnus expansion)

ϕ : Λ → F

(d) x

7→ 1 + X

is an isomorphism of topological rings. Remark that ϕ(I(F)) = I . Now consider the map ι from F to F

(d) defined by

ι(x) = ϕ(x − 1),

and put F

= {x ∈ F, ι(x) ∈ I

}. The sequence (F

)

is a neighborhood basis of 1: it is the Zassenhaus filtration of F.

We recall some basic facts (see [16], [4]).

Proposition 1.1. (i) The elements [x

, x

], i < j, form a F

-basis of F

/F

.

(ii) For p = 3, the elements

x

, i = 1, · · · , d [[x

, x

], x

], i < j, k ≤ j

form a F

-basis of F

/F

. For p > 3, one has to omit the p-powers x

.

Example 1.2. Suppose p > 2. When F is the free pro-p-group on two generators, then F/F

is a non-abelian group of order p

and of exponent p (because F

⊂ F

): this quotient is isomorphic to the Heisenberg group

H

=< x, y, x

= 1, y

= 1, [[x, y], x] = [[x, y], y] = 1 > . 1.2. Strongly free sequence.

Definition 1.3. Let S = {P

, · · · , P

} be some series in I ⊂ F

(d) and

let S be the two-side ideal generated by the elemens P

, · · · , P

. Then the

family S is called strongly free if the quotient S / S I is a F

(d)/ S -left-free

module on the images of P

, · · · , P

.

For P ∈ F

(d), P 6= 0, denote by P

its term of degree i. If i

is the smallest integer such that P

6= 0, then P

is called the initial form of P and is noted by ω(P). The integer i

is the degree of P and is noted by i

:= deg(P ). We put deg(0) = ∞.

Definition 1.4. If x ∈ F, the degree of x is the degree of ι(x) and is noted by deg(x). For a subgroup H of F, the degree of H, noted by deg(H), is the minimum of the degree of x, for all x ∈ H.

Definition 1.5 (Anick, [1]). A family M

, · · · , M

of monomials in I ⊂ F

(d), M

6= 1, is said to be combinatorially free if:

(1) no M

is a submonomial of any M

, j 6= i;

(2) for every i, j, the beginning of M

is not the same as the ending of M

.

Now let us fix a total order < on the set {X

, · · · , X

} and then consider the lexicographic ordering on F

(d) deduced from <. If P is a sum of homogeneous monomials, we denote by L (P) the leading term of P . Definition 1.6. A family P

, · · · , P

of series in I ⊂ F

(d) is called com- binatorially free (after ordering) if the family of monomials

L (ω(P

)), · · · , L (ω(P

)) is combinatorially free.

Theorem 1.7 (Forré, [5]). If the family S = {P

, · · · , P

} ⊂ F

(d) is combinatorially free then S is strongly free.

1.3. Mild pro-p-groups. Let

1 −→ R −→ F −→ G −→ 1,

be a minimal presentation of a finitely presented pro-p-group G. The p-rank of G is finite and equal to the p-rank of the free pro-p-group F and these two groups are topogically generated by d generators x

, · · · , x

.

Let ρ

, · · · , ρ

∈ R ⊂ F be a basis over F

of R/R

[F, R] ≃ H

(G, F

) (the elements ρ

are a basis of the relations of G).

The notion of strongly free sequence will give us a sufficient condition for a pro-p-group to be of cohomological dimension 2. The key criterion is the following:

Theorem 1.8 (Brumer, [3]). The pro-p-group G is of cohomological dimen- sion at most 2 if and only if the F

[[G]]-module R/R

[R, R] is free.

Now, with the previous theorem, it is possible to give criteria in the algebra F

(d) for a pro-p-group G to be of cohomological dimension at most 2.

Theorem 1.9 (Forré, [5]). The pro-p-group G is of cohomological dimension

at most 2 if and only if R / RI is a free left F

(d)/ R -module, where R =

ι(R).

We can then define the notion of mild pro-p-group.

Definition 1.10.

If a pro-p-group G has a presentation with relations ρ

, · · · , ρ

, then G is called mild (following the Zassenhaus filtration) if the family ι(ρ

), · · · , ι(ρ

) is combinatorially free.

Thanks to the previous results, one obtains:

Theorem 1.11. If G is mild then the cohomological dimension of G is at most 2.

Remark 1.12 (The Poincaré series). See [5], [9]. For n ≥ 1, denote by G

the quotient F

R/R and put a

= dim

G

/G

. Then the Poincaré series P (t) of G (associated to Zassenhauss filtration) is the formal series

P (t) = 1 + X

n≥1

a

t

.

When the relations ρ

, · · · , ρ

of G are combinatorially free then the Poincaré series of G satisfies:

P (t) = 1

1 − dt + P

i=1

t

. 1.4. The relations in F

(d).

Definition 1.13. Let I = (i

, · · · , i

) be a multi-index with i

∈ {1, · · · , d}.

One denotes by n = deg(I) the degree of I.

For Z ∈ F

(d), we denote by ε

(Z ) to be the X

· · · X

-coefficient of Z.

For y ∈ F, let us denote by abuse of notation, ε

(y) to be ε

(ι(y)).

Proposition 1.14. Let x, y ∈ F. Write ϕ(x) = 1 + X and ϕ(y) = 1 + Y , with X, Y ∈ F

(d).

(i) if deg(x) > deg(I ), then ε

(x) = 0;

(ii) ε

(xy) = X

JK=I

ε

(x)ε

(y), where the sum is taken over all subsets J, K of I such that the concatenation JK of J and K equals to I;

(iii) if min(deg(x), deg(y)) > deg(I), then ε

(xy) = 0;

(iv) if max(deg(x), deg(y)) ≥ deg(I), then ε

(xy) = ε

(x) + ε

(y);

(v) ϕ(x

) = 1 − X + X

− X

+ · · · ;

(vi) ϕ([x, y]) = 1 + XY − Y X + degree > 2;

(vii) if deg(y) ≥ 2, then ϕ([x, y]) = 1 + XY − Y X + degree > 3;

(viii) ϕ([[x, y], z]) = 1 + XY Z − Y XZ + −ZXY + ZY X + degree > 3.

Proof. Easy computation.

Now, we are interested in the image in F

(d) of the relations of G. If ρ

∈ F is a such relation, then let us write (by proposition 1.1)

ρ

≡ Y

i<j

[x

, x

]

(mod F

),

(1)

and if moreover ρ

∈ F

: ρ

≡ Y

j

x

Y

i<j,k≤j

[[x

, x

], x

]

(mod F

), (2)

with a

, e

(m) ∈ F

.

Proposition 1.15. For i < j < k, we have:

e

(m) = ε

(ρ

), e

(m) = −ε

(ρ

), e

(m) = ε

(ρ

), a

(m) = ε

(ρ

), e

(m) = −ε

(ρ

).

Remark 1.16. For p > 3, a

(m) = 0.

Proof. By proposition 1.14, we have:

ι([[x

, x

], x

]) = X

X

X

− X

X

X

− X

X

X

+ X

X

X

+ degree > 3, ι([[x

, x

], x

]) = X

X

X

− X

X

X

− X

X

X

+ X

X

X

+ degree > 3, and for i < k < j:

ι([[x

, x

], x

]) = X

X

X

− X

X

X

− X

X

X

+ X

X

X

+ degree > 3, ι([[x

, x

], x

]) = X

X

X

− X

X

X

− X

X

X

+ X

X

X

+ degree > 3.

Hence

e

(ρ

) = ε

(ρ

) = ε

(ρ

) = − 1

2 ε

(ρ

), e

(m) = 1

2 ε

(ρ

) = −ε

(ρ

) = −ε

(ρ

),

e

(m) = −ε

(ρ

) = −ε

(ρ

), e

(m) = −ε

(ρ

) = −ε

(ρ

) and

e

(m) + e

(m) = ε

(ρ

) = ε

(ρ

).

2. The principle of the computation

2.1. The arithmetic context. Let p ≥ 3 be a prime number and let S = {ℓ

, · · · , ℓ

} be a set of primes such that ℓ

≡ 1(mod p).

Let G

= Gal( Q

/ Q ), where Q

is the maximal pro-p-extension of Q un- ramified outside S.

For i = 1, · · · , d, denote by x

a generator of the inertia group in G

of a place l

|ℓ

along Q

/ Q such that its restriction to the maximal abelian subextension Q

/Q of Q

corresponds, via Class Field Theory, to the idèle where all components are 1 except the ℓ

-component which is a primitive root of 1 modulo ℓ

.

Then the pro-p-group G

is topologically generated by the elements x

, i = 1, · · · , d.

Let

1 −→ R −→ F −→ G

−→ 1,

be a minimal presentation of G

on the elements x

. For i = 1, · · · , d, we

identify x

with one of its preimages in F. The free pro-p-group F is generated

by the elements x

, · · · , x

.

We need also some particular lifts of Frobenius elements. For i = 1, · · · , d, let us fix a prime l

|ℓ

along Q

/Q. Consider y

a lift in G

of the Frobenius of the place l

such that the restriction of y

to Q

/Q corresponds, via Class Field Theory, to the idèle where all components are 1 except the ℓ

- component which is ℓ

.

As before, we identify y

with one of its preimage in F.

Remark 2.1. By the choice of y

, one has the following fact: if L/Q is a p- elementary subextension of Q

/Q in which the inertia degree of ℓ

is trivial, then y

= 1.

Definition 2.2. Denote by Q

i

/Q the maximal elementary p-extension over Q unramified outside ℓ

. This extension is of degree p in which ℓ

is totally ramified.

Remark 2.3. As the maximal pro-p-extension of Q unramified outside ℓ

is cyclic and totally ramified, then the p-class group of Q

i

is trivial.

Remark 2.4. Let q be a prime such that

(i) q

∈ F

is of order p (or equivalently, q is inert in Q

i

/Q);

(ii) for j 6= i, q

= 1 in F

(or equivalently, q splits in Q

j

/Q).

Then, we can choose x

such that its restriction to the maximal p-elementary subextension Q

/Q of Q

/Q is equal to the restriction of the Frobenius f

of q. Indeed, the principal idèle q has only two nontrivial component via the Artin map in Gal(Q

/Q): the ℓ

-component and the q-component.

2.2. A first principle. Let I = (i

, · · · , i

) be a multi-index, i

∈ {1, · · · , d}.

We want to estimate ε

(z) for some z ∈ F. The strategy is the following: to look at the restriction of z to some quotients of G

, i.e. in some p-extensions of Q unramified outside S.

Let Γ be a quotient of G

. We can assume that Γ is generated by the images of the x

, i = 1, · · · , d

, with d

≤ d.

Denote by F

the free pro-p-groups on d

-generators x

, · · · , x

and let α : F → F

be the natural morphism sending x

, · · · , x

to the generators of F

and such that α(x

) = 1 for i > d

.

By the universal property of F

, there exists a section γ from F

to F such that α(γ (α(x))) = α(x), ∀x ∈ F. One then has the following natural commutative diagramm

1 ^{/ /} R ^{/ /}

F ^{/ /}

G

/ / 1

1 ^/ / R

^/ / F

^/ /

γ

A A

Γ ^/ / 1

Here ker(α) is the the smallest normal subgroup of F generated by the ele- ments x

, · · · , x

and ker(β ◦ α) =< γ(ker(β)), ker(α) >.

Lemma 2.5. If I ⊂ {d

+ 1, · · · , d} and if deg(I ) < deg(ker(β)), then ε

(z)

does not depend on the lift of β(α(z)) in F.

Proof. The section γ induces the injection

F

[[X

, · · · , X

]]

֒ → F

(d)

and the degree of ι(γ(ker(β))) ⊂ F

[[X

, · · · , X

]]

is the same as the degree of ker(β ). Now, the kernel of α is the smallest normal subgroup containing x

, · · · , x

. Hence, ι(ker(α)) = (X

, · · · , X

), i.e. the two-sided ideal of F

(d) generated by the elements X

, · · · , X

.

In conclusion, for all J ⊂ I, ε

(ker(β ◦ α)) = 0. Hence for z, z

∈ F, such that β(α(z)) = β(α(z

)), one finally has ε

(z) = ε

(z

).

Let us give two key examples useful for what will follow.

Example 2.6. Consider Q

1

/ Q the maximal p-elementary extension of Q unramified outside ℓ

. Put Γ = Gal(Q

1

/Q) and let F

be the free pro-p- group on x

. Then, ker(β) =< x

>.

Now, let z ∈ F such that β(α(z)) = x

∈ Γ. Then ε

(z) = a and ε

(z) = a(a − 1)/2. In particular, ε

(z) = 0 if β(α(z)) = 1.

In this example, the computation of ε

(z) is reduced to look at the restriction of z to Q

1

/Q.

Example 2.7. Let T = {ℓ

, ℓ

} and let F

be the free-p-group generated by x

and x

. Suppose that the relations of G

are of degree 3. Then, G

/(G

)

≃ F

/F

≃ H

, where H

is he Heisenberg group. Then ker(β) is the smallest normal subgroup of F

generated by x

, x

, [[x

, x

], x

] and [[x

, x

], x

]. Hence, ker(β ) ⊂ F

. Hence for z ∈ F such that β(α(z)) = [x

, x

]

∈ Γ one obtains ε

(z) = a.

In this example, the computation of ε

(z) is reduced to look at the restriction of z to a Heisenberg extension of Q.

For what will follow, we introduce the following notation:

Definition 2.8. Let I = (i

, · · · , i

). Put µ(I) = ε

(y

), where we identify y

with one of its preimage in F.

The quantity µ(I ) was firstly introduced as arithmetic analogues of Milnor invariants of links by Morishita in [10] and [11]. See also [16].

2.3. The Koch computation. One has the following description of G

: Theorem 2.9 (Koch [8]). The group G

can be described by generators x

, · · · , x

and by the relations ρ

, · · · , ρ

where for m = 1, · · · , d:

ρ

= x

[x

, y

].

This description comes from the fact that the relations are all local: they

are coming from the maximal pro-p-extension of the local fields Q

. Let us

be a little more precise:

Proposition 2.10. In the previous arithmetic situation:

H

(G

, F

) ≃

d

M

i=1

H

(Γ

, F

) where Γ

= Gal( Q

i

/ Q ) and the natural map H

(G

, F

) →

d

M

i=1

H

(G

, F

)

is an isomorphism, where G

= Gal( Q

/ Q

) and where Q

is the maximal pro-p-extension of the complete field Q

.

For i = 1, · · · , d, let χ

be a character such that H

(Γ

, F

) =< χ

>.

Look at the cup product χ

∪ χ

∈ H

(G

, F

). Then χ

∪ χ

= 0 and for k different from i and j, χ

∪ χ

is zero in the ℓ

-component H

(G

, F

) because χ

and χ

are unramified at ℓ

.

Lemma 2.11. χ

∪ χ

= 0 in H

(G

, F

) if and only if ℓ

splits in Q

i

/Q.

Proof. It follows from a local computation.

Hence, one obtains:

Corollary 2.12. The cup-product H

(G

, F

) ∪ H

(G

, F

) is zero if and only if for all i, j, the prime number ℓ

splits in Q

i

/ Q . Now, by using the principle of the section 2.2:

Lemma 2.13. One has y

≡ x

in Gal(Q

j

/Q).

Proof. It is an application of example 2.6.

With the notations of the section 1.4, one has:

Proposition 2.14. Let i < j. One has: e

(i) = µ(j, i) and e

(j) =

−µ(i, j). In the other case, e

(k) = 0.

Proof. Let I = (i, j). Then as x

is at least of degree 2:

ε

(ρ

) = ε

(x

[x

, y

])

= ε

(x

, y

])

= ε

(X

Y

) − ε

(Y

X

),

where Y

= ϕ(y

). The conclusion is then obvious.

Finally, one obtains the two following lemmas:

Corollary 2.15 (Fröhlich [6]). For m = 1, · · · , r, one has:

ρ

= Y

i6=m

[x

, x

]

(mod F

).

Corollary 2.16. The following are equivalent:

(i) the relation ρ

is in F

; (ii) for all i, ℓ

splits in Q

i

/Q ;

(iii) for all i, χ

∪ χ

= 0 in H

(G

, F

) ;

(iv) χ

∪ H

(G

, F

) ⊂ H

(G

, F

) is zero.

2.4. A key formula. For what will follow, we use the description of G

by Koch: ρ

= x

[x

, y

].

Proposition 2.17 ([16], Theorem 2.1.7). Let I = (i

, i

, i

). Suppose that ℓ

splits in Q

i1

/Q, Q

i2

/Q, and Q

i3

/Q. Then one has:

ε

(ρ

) = α(p, I) (ℓ

− 1)

p + δ

µ(i

, i

, m) − δ

µ(i

, i

, m), where α(p, I) = 0 if p > 3 or if I 6= (m, m, m), and is 1 otherwise.

Proof. Let Y

= ι(y

). The degree of x

is at least 3 and by example 2.6, the coefficients of Y

in which appear at least one of the X

, X

and X

are at least of degree 2. Then (by using proposition 1.14):

ε

(ρ

) = ε

(x

[x

, y

])

= ε

(x

) + ε

[x

, y

]

=

ε

(x

) + ε

(X

Y

) − ε

(Y

X

)

=

ε

(x

) + δ

µ(i

, i

, m) − δ

µ(i

, i

, m) Remark here that as an application of the example 2.6, we have:

Proposition 2.18. One has µ(i, i, i) = 0 and if ℓ

splits in Q

i

/Q, then µ(i, i, j) = 0.

2.5. Computation in some Heisenberg extensions. Let i 6= j be indices such that µ(i, j) = µ(j, i) = 0.

We want to compute the quantities µ(i, j, k) when k satisfies µ(i, k) = µ(j, k) = 0. To do this we use the principle of example 2.7.

Put T = {ℓ

, ℓ

} ⊂ S. By corollary 2.16, the conditions for the places of T imply that the relations of G

are in F

, where

1 −→ R

−→ F

−→ G

−→ 1,

is a minimal presentation of G

. Here F

is the free-pro-p-group generated by x

and x

: as usual, as G

/ / / / G

, we identify the elements x

and x

in G

with its preimages in G

, F

and F. By hypothesis, F

⊂ R

and then:

G

/(G

)

≃ F

/F

≃ H

, where (G

)

≃ R

∩ F

/R

and where

H

=< x, y, x

= 1, y

= 1, [[x, y], x] = [[x, y], y] = 1 >

is the Heisenberg group of order p

. Let K

= Q

i,ℓj)

be the p-extension associated by Galois theory to the group (G

)

and put M

= Q

i

Q

j

. Then Gal(K

/M

) =< [x

, x

] >.

Proposition 2.19. One has µ(i, j, k) = −µ(j, i, k). Moreover

µ(i, j, k) = 0 ⇐⇒ l

splits in K

/M

,

where l

is a prime of M

above ℓ

.

Proof. It is an application of example 2.7. Thanks to the conditions above ℓ

, ℓ

and ℓ

, and the remark 2.1, the restriction of the element y

to Gal(K

/Q) is in the subgroup < [x

, x

] >:

y

≡ [x

, x

]

(mod Gal(Q

/K

)).

Then ε

(y

) = ε

([x

, x

]

) = a and ε

(y

) = ε

([x

, x

]

) = −a.

2.6. The use of Class Field Theory. First, let us observe that:

Proposition 2.20. The extension K

/M

is unramified.

Proof. The non trivial elements of the Galois group of K

/Q are of order p.

Hence, if a prime above ℓ

is ramified in K

/M

, then Gal(K

/M

) is the inertia group in K

/Q of all primes above ℓ

which contradicts the fact that Q

i

/ Q is totally ramified at ℓ

.

Let C

:= Cl

/(Cl

)

be the elementary p-quotient of the class group of M

. By Class field theory, C

is isomorphic to the the Galois group G

of the maximal abelian unramified elementary p-extension H

of M

. Put

∆

= Gal(M

/ Q ).

Hi,j Di,j

Gi,j≃Ci,j

Ki,j=Q⁽³⁾₍

ℓi,ℓj)

s s s s s s s s s s

Mi,j

rr rr rr rr rr

∆i,j

▲ ▲

▲ ▲

▲ ▲

▲ ▲

▲ ▲

Q^p,el_ℓi

▲ ▲

▲ ▲

▲ ▲

▲ ▲

▲ ▲

▲ ▲

Q^p,el_ℓj

rr rr rr rr rr rr

Q

Then the extension H

/Q is Galois and ∆

acts on G

(and on C

) as follows

τ · (a, H

/M

) := τ (a, H

/M

)τ

= (a

, H

/M

), where (., H

/M

) : C

→ G

= Gal(H

/M

) is the Artin symbol.

As consequence of Proposition 2.19, one has:

Proposition 2.21.

µ(i, j, k) = 0 ⇐⇒ l

splits in K

/M

⇐⇒ (l

, H

/M

) ∈ D

, where l

is a prime of M

above ℓ

.

We finish this part with the question to find the subgroup D

.

Lemma 2.22. There exists an unique subgroup C of C

such that C

is

normal in Gal(H

/Q) and such that C

/C ≃ Z/pZ. Hence, D

is the

unique subgroup of C

of index p fixed by ∆

.

Proof. If C

is an another subgroup, then the quotient Gal(H

/Q)/C

is a group of order p

. Let K

be the fixed field by C

. The extension K

/M

is unramified. First, it is obvious that the group Gal(K

/Q) can not be the group ( Z /p Z )

. Now the groups Z /p

Z × Z /p Z and the nonabelian group of order p

different from H

have the same particularity: all the subgroups of order p

are cyclic, excepts one. Hence, if Gal(K

/Q) is different from H

, we can assume that Gal(K

/Q

i

) is cyclic. Then, as K

/M

is unramified, one deduces that K

/Q

i

is unramified. Contradiction. Hence, Gal(K

/Q) ≃ H

. The Galois group Gal(K

/Q) is a quotient of F

, the relations of this quotient are in F

, and by comparing the indexes, one

obtains that C

= C.

2.7. How to compute the relations modulo F

. Recall that S = {ℓ

, · · · , ℓ

}. Following the remark 2.4, for j = 1, · · · , d, let us choose some auxiliary primes q

such that:

(i) the prime q

is inert in Q

j

/ Q ;

(ii) for all i 6= j, the prime q

splits in Q

i

/Q.

For j = 1, · · · , d, there exist p

primes Q

in Q

above the auxiliary prime q

. Then, for j = 1, · · · , d, let us fix Q

|q

one of these primes and then let us choose x

∈ G

such that its restriction to Gal(Q

/Q) is equal to the inverse f

of the Frobenius f

of Q

.

Consider two primes ℓ

and ℓ

such that µ(i, j) = µ(j, i) = 0. Let ℓ

be a third prime (eventually ℓ

= ℓ

), such that µ(i, k) = µ(j, k) = 0.

We want to compute µ(i, j, k) when it is nonzero.

We use the notations of sections 2.5 and 2.6 for the primes ℓ

and ℓ

. First, the extension K

/Q is a Heisenberg extension and we know that

y

≡ [x

, x

]

(mod Gal(Q

/K

)) and then µ(i, j, k) = a.

The field Q

i

contains p primes l

, · · · l

above ℓ

and p primes q

, · · · , q

above q

. Now, in Gal(K

/Q), to fix the subgroup generated by the Frobe- nius f

q⁽_j^∗)

of a prime above q

is equivalent to fix the inertia group of a place l

. For what it follow, we assume that f

j

corresponds to l

, n = 1, · · · , p, and that moreover Q

∩ K

= q

:= q

. Then the restriction of x

to Gal(K

/Q) is equal to the inverse of the Frobenius f

of q

.

Consider the subfied N

of Q

i,ℓj)

/Q

i

fixed by the Frobenius f

of q

. Then:

[x

, x

] ≡ f

x

≡ f

qjx−1

i

x

(mod Gal(Q

/K

)).

Now the elements x

and f

qjx−1

i

are in Gal(Q

/Q

i

), and then [x

, x

] ≡ f

jfqi

∈ Gal(N

/ Q

i

),

where f

is the Frobenius of the auxiliary prime q

in Gal(Q

i

/Q).

Hence, as f

jfqi

is not trivial in N

/Q

i

,

y

≡ [x

, x

]

(mod Gal(Q

/K

)) if and only if

y

≡ f

qjfqi

∈ Gal(N

/ Q

i

), which makes still sense because y

∈ Gal( Q

/ Q

i

). Hence, to have a ∈ F

, it suffices to compare y

with f

jfqi

in Gal(N

/Q

i

).

The question for the next is how to find N

?

The Frobenius f

is associated to the inertia group of the prime l

j

above ℓ

. Hence the extension N

/Q

i

is of conductor dividing l

· · · l

.

Denote by C

(l

· · · l

) the p-elementary quotient of the ray class group of Q

i

of conductor l

· · · l

. Let B

be the p-elementary abelian extension of Q

i

of conductor l

· · · l

: by Class Field theory, C

(l

· · · l

) ≃ Gal(B

/Q

i

). As the p-class group of Q

i

is trivial, Gal(B

/Q

i

) is a quotient of (Z/pZ)

.

Bi,j

❈ ❈

❈ ❈

❈ ❈

❈ ❈

❈

Ci(l(2) j ···l(p)

j )

Ei,j Ki,j=Q^(p)₍

ℓi,ℓj)

Ni,j

s s s s s s s s s s

<fqj>

Mi,j

rr rr rr rr rr

▲ ▲

▲ ▲

▲ ▲

▲ ▲

▲ ▲

Q^p,el

ℓi

▲ ▲

▲ ▲

▲ ▲

▲ ▲

▲ ▲

▲ ▲

Q^p,el

ℓj

rr rr rr rr rr rr

Q

If Gal(B

/ Q

i

) is cyclic, there is nothing to do: B

= N

.

Let A be a prime for which the Frobenius f

generates Gal(K

/M

). Then the extension N

/Q

i

is such that:

(i) the restriction of f

is trivial, (ii) the prime q

= q

splits,

(iii) the primes q

are inert, n = 2, · · · , d.

These properties caracterize N

(and then the subgroup D

) but also the primes l

associated to q

:= q

. In conclusion:

Proposition 2.23. The quantity µ(i, j, k) ∈ F

is such that l

≡

q

∈ C

(l

· · · l

)/E

, where l

|ℓ

is a prime ideal of Q

i

above ℓ

not dividing l

· · · l

. In

particular when k = j, one has to take l

= l

.

3. Examples

3.1. Example. Take p = 3 and S = {ℓ

= 11863, ℓ

= 19, ℓ

= 9811}.

First, we note that ℓ

≡ 1(p

) and that for all i 6= j, the prime ℓ

splits in Q

j

/Q: µ(i, j) = 0. Now, thanks to Propositions 1.15 and 2.17, the relations of G

become:

e

e

e

e

ρ

−µ(1, 2, 1) 0 −µ(1, 3, 1) 0

ρ

0 −µ(1, 2, 2) 0 −µ(1, 3, 2)

ρ

0 0 0 −µ(1, 2, 3)

e

e

e

e

ρ

0 −µ(2, 3, 1) 0 0

ρ

0 0 −µ(2, 3, 2) 0

ρ

−µ(1, 3, 3) µ(1, 2, 3) 0 −µ(2, 3, 3) Notations. If ℓ

and ℓ

are two fixed primes, put M

= Q

i

Q

j

and let H

be the elementary unramified p-extension of M

.

If A is an ideal of M

, denote by σ

:= (A, H

/M

) the Artin symbol of A in H

/M

. If ℓ is a prime of Q, then L

will be a prime of M

above ℓ.

3.1.1. The extension Q

1,ℓ2

/ Q . The number field Q

1

= Q (θ

) is the unique subfield of Q(ζ

) of degree 3 over Q. It is defined by a root θ

of the equation: x

+ x

− 3954x + 39104 = 0. The field Q

2

= Q(θ

) is defined by a root θ

of the equation: x

+ x

− 6x − 7 = 0. The compositum M

= Q

1

Q

2

is generated by a root θ of the equation

x

− x

− 51408x

+ 137525x

+ 778957094x

+ 583863320x

−3310991579976x

− 29421274145536x

+ 1777568574652416x +20509622778724352 = 0.

The 3-class group C

of M

is isomorphic to Z/3Z and Gal(K

/M

) =<

σ

>=< σ

>. We remark that σ

= σ

. Hence, by Proposition 2.21, µ(1, 2, 1) 6= 0 and µ(1, 2, 2) 6= 0.

3.1.2. The extension Q

1,ℓ3

/ Q . The number field Q

3

is defined by the equa- tion x

+ x

− 3270x − 6904 = 0. The compositum M

= Q

1

Q

3

is generated by a root β of the equation:

x

− x

− 25866384x

+ 495245276x

+ 166553813929280x

−2186400407814976x

− 56279799218070071808x

+83890962452662796288x

+ 942384971138013179412480x +19677317846068743788036096 = 0.

The class group of M

is isomorphic to Z /3 Z and Gal(K

/M

) =<

σ

>=< σ

>. Moreover, σ

= 1. Hence, by Proposition 2.21,

µ(1, 3, 2) = 0, µ(1, 3, 1) 6= 0 and µ(1, 3, 3) 6= 0.

3.1.3. The extension Q

2,ℓ3

/Q. The compositum M

= Q

2

Q

3

is gener- ated by a root γ of the equation:

x

− x

− 42516x

+ 35249x

+ 535158074x

− 630338704x

−1724988572520x

+ 3634048124000x

+ 45824385358080x

−112874663383552 = 0.

The class group of M

is isomorphic to (Z/3Z)

. The p-group ∆

acts triv- ially on σ

and on σ

and then on < σ

, σ

>≃ (Z/3Z)

. Hence

< σ

, σ

>= D

and one verifies that Gal(K

/M

) =< σ

2,3

>.

The primes L

and L

split in K

/M

, and then: µ(2, 3, 3) = µ(2, 3, 2) = 0. To finish, one has σ

∈ / D

: µ(2, 3, 1) 6= 0.

3.1.4. The ordering. Consider now the ordering X

> X

> X

. Then by the above computation

ℓ(ω(ρ

)) = X

X

X

, ℓ(ω(ρ

)) = X

X

X

, ℓ(ω(ρ

)) = X

X

X

. To conclude, the family {ρ

, ρ

, ρ

} is combinatorially free, the pro-p-group G

is mild, and then by Theorem 1.11, the cohomological dimension of G

is 2.

3.1.5. The computation of the relations modulo F

. Remind of that p = 3 and S = {ℓ

= 11863, ℓ

= 19, ℓ

= 9811}.

First, we compute some auxiliary primes following section 2.7: q

= 31, q

= 2, q

= 191.

• The quantity µ(2, 3, 1).

The computation will be done in the Heisenberg extension Q

2,ℓ3

/Q. Follow- ing the notations of section 2.7, we take i = 2 and j = 3.

Let O

be the ring of integers of Q

2

= Q(θ

). One has the decompositions:

191O

= l

l

l

, with l

= (191, 35 + θ

), l

= (191, 75 + θ

), l

= (191, 82 + θ

) and 9811O

= l

l

l

, with l

= (9811, −3147 + θ

), l

= (9811, −1158 + θ

), l

= (9811, 4306 + θ

). The p-part of the ray class group of Q

2

of conductor l

l

is isomorphic to (Z/3Z)

: C

(l

l

) =< (3), (θ

) >. The computation in this ray class group and the conditions (i-iii) of the section 2.7 allow us to verify that l

is associated to f

: in Gal(K

/Q) the Frobenius f

generates the inertia group of l

. One verifies that f

: θ

7→ −θ

+ 4 and that l

= l

. Then, following the computation of the section 2.7:

[x

, x

] ≡ f

∈ Gal(N

/ Q

2

).

(3)

To conclude, in the quotient C(l

l

)/E

, the ideals l

and l

are in the same class and then (thanks to (3)):

y

≡ f

≡ [x

, x

](mod Gal(Q

/K)), and µ(2, 3, 1) = 1.

• The quantities µ(1, 2, 2) and µ(1, 2, 1).

The computation will be done in the Heisenberg extension Q

2,ℓ1

/Q and fol- lowing the notations of section 2.7, we take i = 2 and j = 1. One has in Q

2

:

31O

= l

l

l

where l

= (31, −15 + θ

), l

= (31, 4 + θ

), l

= (31, 12 +

θ

), and 11863O

= l

l

l

, where l

= (11863, −3181 + θ

), l

= (11863, −382+θ

), l

= (11863, 3564+θ

). The ray class group of Q

2

of conductor l

l

is cyclic of degree 3: C

(l

l

) =< (θ

) >.

The computation allow us to see that f

generates the inertia group of l

and that l

= l

. Then:

[x

, x

]

≡ x

x

x

x

≡ f

∈ Gal(B

/Q

2

).

Now the restrictions of f

and of f

31

in B

/Q

2

are the same. In conclu- sion:

y

≡ f

≡ [x

, x

]

(mod Gal(Q

/Q

)) i.e. µ(1, 2, 2) = −1. By similar computation,

(i) f

= f

and then µ(1, 2, 1) = 1 ; (ii) f

= f

and then µ(1, 2, 3) = −1.

• By similar computation in the number field Q

3

, one also obtains µ(1, 3, 3) = µ(1, 3, 1) = 1.

To conclude, the computations above show the following:

Proposition 3.1. The pro-3-group G

can be defined by the gen- erators x

, x

and x

, and by the relations

ρ

≡ [[x

, x

], x

][[x

, x

], x

][[x

, x

], x

] (mod F

), ρ

≡ [[x

, x

], x

]

(mod F

),

ρ

≡ [[x

, x

], x

]

[[x

, x

], x

][[x

, x

], x

] (mod F

).

3.2. A second example. Take p = 3, S = {ℓ

= 13, ℓ

= 7, ℓ

= 11971, ℓ

= 181} and consider the ordering: X

> X

> X

> X

.

The relations ρ

and ρ

are of degree 2. Indeed, as µ(4, 1) 6= 0, thanks to Proposition 1.15 and Proposition 2.14, one has ℓ(ω(ρ

)) = X

X

.

Moreover µ(4, 2) = µ(3, 2) = 0 and µ(1, 2) 6= 0, and then ℓ(ω(ρ

)) = X

X

. Now for all i, µ(i, 3) = µ(i, 4) = 0: by Proposition 2.14, the relations ρ

and ρ

are in F

. Thanks to proposition 2.17 and example 2.6 the study of the relations ρ

and ρ

we will be done in some H

-extension of Q.

First, let us remark that as ℓ

≡ 1(mod p

). Hence ε

(ρ

) = 0.

By a computation in the extension Q

3,ℓ4

/Q, one obtains that µ(4, 3, 3) = 0 and that µ(3, 4, 4) 6= 0. By a computation in the extension Q

2,ℓ4

/Q, one obtains µ(2, 4, 3) 6= 0. Recall that µ(4, 4, 3) = 0 (see Proposition 2.18).

Hence: ε

(ρ

) = µ(4, 4, 3) = 0, ε

(ρ

) = µ(4, 3, 3) = 0, and ε

(ρ

) = µ(4, 2, 3) 6= 0. Then ℓ(ω(ρ

)) = X

X

X

.

Moreover, ε

(ρ

) = µ(4, 3, 4) 6= 0, and then ℓ(ω(ρ

)) = X

X

X

. One conclude that G

is mild by noting that the family

{X

X

, X

X

, X

X

X

, X

X

X

}

is combinatorially free.