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J. Inst. Math. Jussieu (2017)16(1), 121–154

doi:10.1017/S1474748015000134 ©Cambridge University Press 2015

121

THE ELEMENTARY THEORY OF LARGE FIELDS OF TOTALLY S-ADIC NUMBERS

ARNO FEHM

University of Konstanz, Department of Mathematics and Statistics, 78457 Konstanz, Germany(arno.fehm@uni-konstanz.de)

(Received 3September2014; accepted25March2015;

first published online 23 April 2015)

Abstract We analyze the elementary theory of certain fieldsKS(σ)of totallyS-adic algebraic numbers that were introduced and studied by Geyer and Jarden and by Haran, Jarden, and Pop. In particular, we provide an axiomatization of these theories and prove their decidability, thereby giving a common generalization of classical decidability results of Jarden and Kiehne, Fried, Haran, and V¨olklein, and Ershov.

Keywords: totally S-adic numbers; absolute Galois group; model theory of profinite groups; decidability of fields

1. Introduction

LetSbe a finite set of absolute values on a number field K. By KS we denote the field of totallyS-adic numbers – the maximal Galois extension of K in which the elements ofS are totally split. For an integere0and an e-tupleσ =1, . . . , σe)∈Gal(K)e of elements of the absolute Galois group ofK, we let KS(σ)be the fixed field of the group σ1, . . . , σeGal(K)inside KS.

These fields KS(σ) were studied by Jarden and Razon [25], Geyer and Jarden [18], and recently in a series of papers by Haran, Jarden, and Pop [20, 21]. In particular, these authors prove that, for almost all σ∈Gal(K)e, in the sense of Haar measure on the compact groupGal(K)e, the fieldKS(σ)satisfies a local–global principle for rational points on varieties, and its absolute Galois group has a nice description as a free product of local factors.

Combining these results, we are able to give an axiomatization of the theoryTalmost,S,e of first-order sentences (in the language of rings with constants from K) that hold in almost all KS(σ)(Theorem10.11), and we prove the decidability of this theory.

This work builds on Chapters 3 and 4 of the author’s Ph.D. thesis [13] supervised by M. Jarden at Tel Aviv University. As such, it was supported by the European Commission under contract MRTN-CT-2006-035495.

Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-0-331514

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Theorem 1.1. Let Sbe a finite set of absolute values on a number field K, and lete0.

Then the first-order theory Talmost,S,e of almost all KS(σ),σ ∈Gal(K)e, is decidable.

This theorem is a common generalization of classical decidability results of Jarden and Kiehne [23] (the case S= ∅), Fried, Haran, and V¨olklein [15] (the case K =Q, e=0, and S consisting only of the Archimedean absolute value), and Ershov [11] (the case K =Q,e=0, andS consisting only of p-adic absolute values).

In fact, we prove a more general and stronger statement; see Theorem11.12. However, although we can work for example with more general fields K, it seems difficult to allow also infinite sets of absolute valuesS; see the discussion in Remark11.14.

The main part of the proof consists of an analysis of the absolute Galois group of KS(σ)together with local data, and the model theory of such structures.

2. Preliminaries on profinite groups and spaces

We assume that the reader is familiar with the basic theory of profinite groups, as presented in [16, Ch. 1] and [35, Ch. 2].

We always consider profinite groups as topological groups, so in particular homomorphisms between profinite groups are continuous group homomorphisms. By H G(respectively,H G) we indicate thatH is a closed (respectively, normal closed) subgroup ofG. If XG, we denote byXthe closed subgroup generated byX inG. We use the symbol1to denote both the unit element ofG and the trivial subgroup{1}G.

In the category of profinite groups, direct products, inverse limits, and fiber products exist [16, 22.2.1]. For the notion of therank rk(G)of a profinite group Gand the notion of a free profinite group, see [16, Ch. 17]. We denote by Fˆe the free profinite group of ranke.

Lemma 2.1. Let π:GH be an epimorphism of profinite groups. Let e0, let N G be a closed normal subgroup with rk(G/N)e, and let h1, . . . ,heH such that H = h1, . . . ,he, π(N). Then there exist g1, . . . ,geG such that G= g1, . . . ,ge,N and π(gi)=hi,i =1, . . . ,e.

Proof. Let G¯ =G/N, H¯ =H/π(N), and let π: ¯¯ G→ ¯H be the induced epimorphism.

ThenH¯ =h¯1, . . . ,h¯e

, so Gasch¨utz’ lemma [16, 17.7.2] implies that there areg1, . . . ,geG such that G¯ = ¯g1, . . . ,g¯e and π(¯ g¯i)= ¯hi, i =1, . . . ,e. So, G= g1, . . . ,ge,N, and there aren1, . . . ,neNsuch thatπ(gi)=hiπ(ni),i=1, . . . ,e. Thus, settinggi=gin−1i , we have G=

g1, . . . ,ge,N

andπ(gi)=hi,i =1, . . . ,e.

A profinite space is a totally disconnected compact Hausdorff space. Profinite spaces can be characterized as inverse limits of finite discrete spaces, or as zero-dimensional compact Hausdorff spaces [35, 1.1.12]. Any product and any finite coproduct (i.e., direct sum) of profinite spaces is a profinite space, and a subspace of a profinite space is profinite if and only if it is closed. Since profinite spaces are compact Hausdorff, any continuous map between profinite spaces is closed, and any continuous bijection of profinite spaces is a homeomorphism.

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3. Group piles

The notion of group piles was introduced in [20] to enrich profinite groups with extra local data (see Remark3.6 for some history concerning such structures). We recall this notion and extend it. Our main innovation is the introduction of a certain quotient G¯ that measures the failure of a deficient group pileGto be self-generated.

Fix a finite setS not containing the symbol0, and lete0.

Definition 3.1. Let G=lim←−NG/N be a profinite group, where N runs over all open normal subgroups ofG. Then the setSubgr(G)of all closed subgroups ofG is equipped with a profinite topology, induced by Subgr(G)=lim←−NSubgr(G/N). The group G acts continuously onSubgr(G)by conjugation. A homomorphismα:GHof profinite groups induces a mapSubgr(α):Subgr(G)→Subgr(H)given byα().

Lemma 3.2. The map Subgris a covariant functor from the category of profinite groups (with homomorphisms) to the category of profinite spaces (with continuous maps).

Proof. It is easy to check that, ifα:GH is a homomorphism of profinite groups, then the induced mapSubgr(G)→Subgr(H)is continuous.

Lemma 3.3. If H is a closed subgroup of a profinite group G, then Subgr(H)is a closed subspace ofSubgr(G).

Proof. By Lemma 3.2, the inclusion ι: Subgr(H)→Subgr(G)is continuous. Since both spaces are compact Hausdorff,ιis closed, and is thus a topological embedding.

Definition 3.4. Agroup pileis a structureG=(G,G0,Gp)p∈S consisting of (1) a profinite groupG,

(2) a nonemptyG-invariant closed subsetG0⊆Subgr(G)such that the elements ofG0

are pairwise conjugate in G, and

(3) aG-invariant closed subsetGp⊆Subgr(G)for eachp∈S.

Theorder ofGis the order of G. The rank rk(G)of Gis the rank rk(G)of G. Afinite group pile is a group pile of finite order. Let G=

p∈SGp. We call Gself-generated if there existsG0G0such thatG= G0,G, i.e.,Gis generated by G0 and the groups in Gp,p∈S. It is calledbareifG= {1}, anddeficientifG0= {1}. Thedeficient reductofG isG=(G,{1},Gp)p∈S. Instead of(G,{1},Gp)p∈S, we also write(G,Gp)p∈S. We call G separated if the sets Gp, p∈ {0} ∪S, are disjoint, andreduced if there are no nontrivial inclusions among the elements ofG.

Remark 3.5. Note that the notion of a group pile depends on the fixed set of primesS.

Also note that, if Gis self-generated, then G= G0,G for any G0G0. Condition (2) says thatG0consists of a single G-orbit inSubgr(G); i.e., there existsG0G0such that G0=(G0)G := {(G0)g:gG}. Hence, our notion of group piles coincides with the group piles of [20], except for a small difference in notation concerning G0.

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Remark 3.6. The idea of using structures similar to group piles has a long history, going back for example to the Artin–Schreier structures in [19], the involutory structures in [15], or the-groupsin [9], [10], and [12].

Definition 3.7. Ahomomorphismof group piles

f:(G,G0,Gp)p∈S(H,H0,Hp)p∈S

is a homomorphism of profinite groups f:GHsuch that f(Gp)Hpfor eachp∈ {0} ∪ S. It is anepimorphismif f:GHis surjective and f(Gp)=Hpfor eachp∈ {0} ∪S. It is anisomorphismif in addition f:GH is an isomorphism. The homomorphism f is calledrigidif f|is injective for eachG. IfNis a closed normal subgroup ofG, define the quotient G/N =(G/N,G0,N,Gp,N)p∈S by Gp,N = {N/N: Gp} ⊆Subgr(G/N).

This is again a group pile, the quotient map GG/N extends to an epimorphism of group pilesG→G/N, and every epimorphism of group piles is of this form.

Remark 3.8. We identify the category of bare deficient group piles (with homomorphisms) with the category of profinite groups (with homomorphisms) via the forgetful functor (G,G0,Gp)p∈SG.

Lemma 3.9. In the category of group piles with epimorphisms, inverse limits exist.

Proof. For a directed set I and an inverse familyGi =(Gi,Gi,0,Gi,p)p∈S,iI, of group piles,G=(G,G0,Gp)p∈S withG:=lim←−iIGi andGp:=lim←−iIGi,p⊆Subgr(G),p∈S∪ {0}, is an inverse limit.

Definition 3.10. For a group pileG=(G,G0,Gp)p∈S, letG:= Gbe the closed subgroup generated by the subgroups inGp,p∈S. LetG:=(G,Gp)p∈S andG¯ :=G/G. We say that G is e-generated ifrk(G)¯ e, and e-bounded if Gis self-generated and rk(G0)e for allG0G0.

Lemma 3.11. G is a self-generated and deficient group pile, andis a bare group pile.

Proof. By Lemma3.3,G is a group pile. The other claims are obvious.

Lemma 3.12. If ϕ:G→H is an epimorphism of group piles, then ϕ(G)=H, so ϕ induces epimorphismsϕ:G→H andϕ: ¯¯ G→ ¯H.

Proof. By the definition of an epimorphism of group piles, ϕ(G)=H. Hence, since ϕ is continuous and closed,ϕ(G)= H, as claimed.

Lemma 3.13. The map G→G (respectively, G→ ¯G) is a covariant functor from the category of group piles with epimorphisms to the category of self-generated deficient group piles with epimorphisms (respectively, the category of bare group piles with epimorphisms).

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Proof. This follows from Lemma3.12.

Lemma 3.14. Let G=(G,Gp)p∈S be a deficient group pile and A=A a bare deficient group pile. Then the map ϕ→ ¯ϕ gives a bijection between the epimorphisms fromG to Aand the epimorphisms fromG¯ to A.

Proof. If ϕ:G→A is an epimorphism, then ϕ: ¯¯ G→ ¯A=A is an epimorphism.

Conversely, given an epimorphism ϕ: ¯¯ GA, the composition ϕ¯◦π: G→A, where π:G→ ¯G is the quotient map, is an epimorphism. These two operations are inverse to each other.

Remark 3.15. Note that a deficient group pile is self-generated if and only if it is 0-generated. Everye-bounded group pile ise-generated.

Lemma 3.16. Let ϕ:G→Hbe an epimorphism of group piles. If Gise-generated, then Hise-generated. IfGise-bounded, thenHise-bounded.

Proof. The induced mapϕ¯: ¯G→ ¯His an epimorphism by Lemma3.13, sork(H¯)rk(G¯).

IfGis self-generated, thenHis also self-generated. Sinceϕ(G0)=H0, for H0H0there exists G0G0 withϕ(G0)=H0, and thusrk(H0)rk(G0).

Proposition 3.17. A deficient group pile G is e-generated if and only if every finite quotient ofGise-generated.

Proof. IfGise-generated, then every finite quotient ofGise-generated by Lemma3.16.

Conversely, suppose thatGis note-generated. Then there is an epimorphismG¯ → Aonto a finite group Awithrk(A) >e(see, for example, [35, 2.5.3]), so A is a finite quotient of G(Lemma3.14) which is note-generated.

Lemma 3.18. LetAbe ane-bounded group pile, and letB˜ =(B,Bp)p∈Sbe ane-generated deficient group pile. For every epimorphism π: ˜B→A there exists an e-bounded self-generated group pileB withB= ˜B such thatπ:B→Ais an epimorphism.

Proof. Let A0A0, and choose a1, . . . ,aeA with A0= a1, . . . ,ae. By Lemma 3.12, A= a1, . . . ,ae,A and A=π(B˜), so Lemma 2.1 gives b1, . . . ,beB with B= b1, . . . ,be,B˜ and π(bi)=ai. Let B0= b1, . . . ,be and B=(B, (B0)B,Bp)p∈S. Then Bise-bounded, andπ:B→Ais an epimorphism.

4. Embedding problems

We recall the notion of embedding problems for group piles from [20,§4], and rephrase some results in terms ofe-bounded group piles.

Definition 4.1. LetGbe a group pile. Anembedding problemforGis a pair EP =(ϕ:G→A, α:B→A)

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of epimorphisms of group piles. It is calledfinite, self-generated, e-generated, e-bounded, deficient, or bare, if B has this property. It is called rigid if α is rigid. A solution of the embedding problem (ϕ, α) is an epimorphism γ:G→B such that αγ =ϕ.

The embedding problem EP is locally solvable if, writing G=(G,G0,Gp)p∈S and B= (B,B0,Bp)p∈S, the following holds for every p∈ {0} ∪S.

For every Gp there is aBp, and for everyBpthere is aGp,

such that there exists an epimorphism γ:withαγ =ϕ|. (L S) Lemma 4.2. If there exist G0G0 and B0B0 and an epimorphism γ0:G0B0 with αγ0=ϕ|G0, then(L S)holds forp=0.

Proof. IfgG and=(G0)gG0, choosebB withα(b)=ϕ(g), and let=(B0)b. IfbB and=(B0)bB0, choosegG withϕ(g)=α(b), and let=(G0)g. Define γ:byγ(x)=γ0(xg1)b. Thenα(γ(x))=ϕ(xg1)ϕ(g)=ϕ(x)for allx. Lemma 4.3. Every rigid embedding problem satisfies(L S)for every p∈S.

Proof. Suppose thatEP is rigid. IfGp, chooseBpwithα()=ϕ(). IfBp, chooseGpwithϕ()=α(). Sinceαis rigid,γ=(α|)−1ϕ|mapsontoand satisfiesαγ =ϕ|.

Proposition 4.4. Every rigid deficient embedding problem is locally solvable.

Proof. Suppose that EP is rigid and deficient. By Lemma 4.3, (L S) holds for p∈S.

SinceBis deficient, so isA; hence ifG0G0, thenϕ(G0)=1. Thus, (L S) is satisfied for p=0.

Definition 4.5. Letϕ:G→Aandα:B→Abe homomorphisms of deficient group piles.

Define the (asymmetric) rigid product of B and G over A as B×rigA G=(H,Hp)p∈S, where H =B×AG is the fiber product and

Hp= {∈Subgr(H):β()Gp, π()Bp, β| is injective}, withπ :HB andβ :HG the natural projection maps.

Lemma 4.6. Let EP=(ϕ:G→A, α:B→A)be a locally solvable embedding problem of finite deficient group piles. ThenrigA G is a deficient group pile, π :B×rigA G→B is an epimorphism, and β:B×rigA G→Gis a rigid epimorphism.

Proof. Since Gp is G-invariant and Bp is B-invariant, Hp is H-invariant. Since H is finite,Hpis closed. The projections β andπ are surjective and, by the definition ofHp, homomorphisms of group piles. The definition of Hp also gives that β is rigid. Given G1Gp, there isB1Bpand there is an epimorphismγ:G1B1withαγ =ϕ|G1. It defines a homomorphismγˆ:G1H with β◦ ˆγ =idG1 and π◦ ˆγ =γ. Let H1= ˆγ (G1).

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Thenβ(H1)=G1Gpand π(H1)=γ (G1)=B1Bp. Furthermore, sinceβ◦ ˆγ =idG1, β|H1is injective, soH1Hp. Similarly, givenB1Bp, there isH1Hpwithπ(H1)=B1. Therefore,β andπ are epimorphisms of group piles.

Remark 4.7. The rigid product can be seen as a canonical version of [20, Lemma-Construction 4.2]. One could define it as a subgroup pile of the fiber product in the category of deficient group piles, which always exists.

Lemma 4.8. Let (ϕ:G→A, α:B→A) be a locally solvable embedding problem. Then, for every normal subgroupNB, the induced embedding problem(G→A/α(N),B/N → A/α(N)) is also locally solvable.

Proof. Let B˜ =(B˜,B˜0,B˜p)p∈S=B/N and A˜ =(A˜,A˜0,A˜p)p∈S=A/α(N), and let π:B→ ˜B,π:˜ A→ ˜Abe the quotient maps andα: ˜˜ B→ ˜Athe induced epimorphism. Then

˜

πα= ˜απ. We have to prove that the embedding problem(π˜◦ϕ,α)˜ is locally solvable.

Let p∈ {0} ∪S, and let Gp be given. Then there is a Bp and there is an epimorphism γ: with αγ=ϕ|. Let =π()∈ ˜Bp. Thenπγ:is an epimorphism withα˜◦γ)= ˜παγ =˜◦ϕ)|.

Conversely, let∈ ˜Bpbe given. ChooseBpwithπ()=. Then there is aGp and there is an epimorphism γ: with αγ =ϕ|. Hence, πγ: is an epimorphism withα˜◦γ)= ˜παγ=˜ ◦ϕ)|.

Lemma 4.9. Let (ϕ:G→A, α:B→A) be a locally solvable finite embedding problem.

Then there exists an open normal subgroup NG withN Ker(ϕ)such that the induced embedding problem (G/N →A,B→A)is locally solvable.

Proof. This is a special case of [20, Lemma 4.1].

The following proposition is closely related to [20, Lemma 4.3], which we need to reprove because we have to takee-boundedness into consideration.

Proposition 4.10. Let G be an e-bounded group pile and let (ϕ:G→A, α:B→A) be a locally solvable e-bounded finite embedding problem for G. Then it can be dominated by a rigid e-bounded finite embedding problem; i.e., there exist epimorphisms α: ˆˆ B→ ˆA,

ˆ

ϕ:G→ ˆA,β: ˆˆ A→A, andβ: ˆB→B such thatϕ = ˆβ◦ ˆϕ andβˆ◦ ˆα=αβ, and(ϕ,ˆ α)ˆ is a rigide-bounded finite embedding problem.

Proof. By Lemma4.9, there is a finite group pileAˆ and there are epimorphismsϕˆ:G→ ˆA, β: ˆˆ A→Awithϕ= ˆβ◦ ˆϕ such that(β, α)ˆ is a locally solvable embedding problem. Since Gise-bounded,Aˆ is alsoe-bounded (Lemma 3.16).

Let B˜ =B×rigA be the rigid product, and let α: ˜˜ B→ ˆA and β: ˜˜ B→B be the projections. By Lemma 4.6,α˜ is a rigid epimorphism and β˜ is an epimorphism. Choose Aˆ0∈ ˆA0 and B0B0 and an epimorphism γ0: ˆA0B0 with αγ0= ˆβ|Aˆ0. Then γ0

defines a homomorphismγˆ0: ˆA0→ ˜Bwithα˜◦ ˆγ0=idAˆ

0 andβ˜◦ ˆγ0=γ0. LetB˜0= ˆγ0(Aˆ0), and note thatα(˜ B˜0)= ˆA0 andβ(˜ B˜0)=B0.

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We have thatrk(B˜0)rk(Aˆ0)e, and, sinceAˆ andBare self-generated, Aˆ= ˆA0,Aˆ and B= B0,B. Let Bˆ = ˜B0,B˜B˜ and Bˆ0=(B˜0)Bˆ. Then Bˆ =(Bˆ,Bˆ0,B˜p)p∈S is a self-generated group pile, and α(˜ Bˆ)= ˆA0,Aˆ = ˆA and β(˜ Bˆ)= B0,B = B by Lemma 3.12. Since Aˆ0=(Aˆ0)Aˆ and B0=(B0)B, α(˜ Bˆ0)= ˆA0 and β(˜ Bˆ0)=B0, so α|˜ Bˆ andβ|˜ Bˆ are epimorphisms of group piles. Therefore, withαˆ = ˜α|Bˆ andβ= ˜β|Bˆ,(ϕ,ˆ α)ˆ is a rigide-bounded finite embedding problem which dominates(ϕ, α).

5. Model theory of group piles

This section extends the comodel theory of profinite groups in [6] (see also [5]) to group piles. A similar construction can be found in [8]. We will give full definitions, but focus our proofs on the necessary extensions to the classical theory. For more on the comodel theory of profinite groups, see [2–4] or [17].

Definition 5.1. The colanguage Lco,S= {,,P, (Gn)n∈N, (Gp,n)p∈S,n∈N} consists of unary relation symbols Gn(n∈N), binary relation symbolsand , a ternary relation symbol P, andn-ary relation symbolsGp,n (p∈S,n∈N).

Definition 5.2. To a deficient group pile G=(G,Gp)p∈S assign an Lco,S-structure S(G)=(SG,G,G,PG, (GGn)n∈N, (Gp,Gn)p∈S,n∈N)as follows.

(1) SG=

· NG/N, where N runs over all open normal subgroups ofG.

(2) x1N1Gx2N2if and only if N1N2. (3) x1N1Gx2N2if and only if x1N1x2N2.

(4) (x1N1,x2N2,x3N3)PG if and only if N1=N2=N3and x1x2N1=x3N1. (5) x NGGn if and only if(G:N)n.

(6) (x1N1, . . . ,xnNn)Gp,nG if and only ifN1= · · · =NnGGn and there isGpsuch that N1/N1= {x1N1, . . . ,xnNn}.

Definition 5.3. AnLco,S-structureS=(S,,,P, (Gn)n∈N, (Gp,n)p∈S,n∈N)is aninverse system (of group piles)if the following statements hold.

(1) is a pre-order with a unique largest element.

(2) If∼denotes the equivalence relation defined by, and[x]denotes the equivalence class of xS with respect to ∼, then the induced partial order on{[x] :xS} = S/∼is directed downwards.

(3) Gn = {x∈ S: |[x]|n}.

(4) (x,y,z)P implies that[x] = [y] = [z], and, for eachxS, P∩ [x]3is the graph of a binary operation making[x]into a group.

(5) If (x1, . . . ,xn)Gp,n, then x1, . . . ,xnGn and [x1] = · · · = [xn]. If moreover y1, . . . ,ynGn and{y1, . . . ,yn} = {x1, . . . ,xn}, then (y1, . . . ,yn)Gp,n. If xGn, then, with

Gp,x = {{x1, . . . ,xn} ⊆ [x] :(x1, . . . ,xn)Gp,n}, [[x]] =([x],Gp,x)p∈S is a (finite) deficient group pile.

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(6) xy implies that x y, and, for each x,yS with xy, defines an epimorphism of group piles πx,y from[[x]] to[[y]], depending only on[x]and[y].

(7) For allxyz,πx,x =id[[x]] andπx,z =πy,zπx,y.

(8) If N is a normal subgroup of [x], then there is a unique [y] such that xy and N =kerx,y).

(9) S=

n∈NGn.

Remark 5.4. Note that, for S= ∅, (5) is vacant, and the remaining axioms are exactly the ones used in [6]. Also note that (1)–(8) areLco,S-elementary statements, but (9) is not.

Lemma 5.5. If Gis a deficient group pile, thenS(G)is an inverse system.

Proof. This can be checked directly from the definitions.

Definition 5.6. Ifϕ :G→His an epimorphism of deficient group piles,G=(G,Gp)p∈S, H=(H,Hp)p∈S, define a mapϕof S(H)into S(G)by

ϕ(h N)=1(N)G/ϕ1(N), where N H is open, hH, and gG satisfiesϕ(g)=h. Lemma 5.7. The mapϕ is an embedding ofLco,S-structures.

Proof. First of all note that, sinceϕ is a surjective homomorphism,ϕ is injective and it preserves the relations,, P, and(Gn)n∈N.

Let h1N1, . . . ,hnNnSH and p∈S. Then (h1N1, . . . ,hnNn)Gp,Hn if and only if N1= · · · =NnGHn and there isHp such thatN1/N1= {h1N1, . . . ,hnNn}. Since ϕ(Gp)=Hp, this is the case if and only if ϕ(N1)= · · · =ϕ(Nn)GGn and there is Gp such that ϕ−1(N1)/ϕ−1(N1)= {ϕ(h1N1), . . . , ϕ(hnNn)}. This is equivalent to (h1N1), . . . , ϕ(hnNn))Gp,nG , soϕ also preserves the relations(Gp,n)p∈S,n∈N. Definition 5.8. We assign to each inverse systemS=(S,,,P, (Gn)n∈N, (Gp,n)p∈S,n∈N) a deficient group pile G(S)as follows. By axioms (1), (2), and (4), the[x]are a family of groups, which by (3) and (9) are all finite. By (5), the[[x]] are finite groups piles. By axioms (6) and (7), the mapsπx,y turn these group piles into an inverse system in the category of group piles with epimorphisms. Let G(S)=(GS,GpS)p∈S:=lim←−xS/∼[[x]] be the inverse limit; cf. Lemma3.9.

Lemma 5.9. The maps S and G are quasi-inverse to each other; i.e., for every deficient group pileGthere is a natural isomorphismG(S(G))∼=G, and for every inverse system S there is a natural isomorphism S(G(S))∼=S.

Proof. LetG=(G,Gp)p∈Sbe a deficient group pile. SinceGpis closed,Gp=lim←−NGp,N, where N runs over all open normal subgroups ofG. ThereforeG(S(G))∼=G.

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Conversely, letSbe an inverse system. GivenxS, letNx be the kernel of the natural projection GS → [[x]]. Defineψ:S→ S(G(S)) to sendx to its image under the natural isomorphism [[x]] ∼=GS/NxSG(S). Then ψ is injective and it preserves the relations , , P, Gn, and Gp,n. By axiom (5.3), ψ is surjective. Thusψ gives an isomorphism S∼=S(G(S)).

Remark 5.10. Given the preceding lemma, we will sometimes identify an inverse system S with S(G(S)). In particular, we will treat elements of S as cosets x N of open normal subgroups of the group pileG(S).

Definition 5.11. Ifψ:T→Sis an embedding of inverse systems, define an epimorphism of profinite groupsψ:GSGTas follows.

Sinceψpreserves the relation, it also preserves the relation∼, i.e.,ψ([x])[ψ(x)].

Sinceψpreserves the relationsGn(n∈N),ψgives, for every xT, a bijection between [x] and [ψ(x)]. Since ψ preserves the relation P, this bijection is an isomorphism of groups. Sinceψpreserves the relation, the inverse system of finite groups([ψ(x)])x∈T/∼

with homomorphisms πψ(x),ψ(y) is isomorphic to the inverse system of finite groups ([x])x∈T/∼ with homomorphismsπx,y. This gives a natural epimorphism

ψ:GS=ylim←−S/∼[x] →xlim←−T/∼[ψ(x)] ∼=xlim←−T/∼[x] =GT.

Lemma 5.12. The map ψ induces an epimorphismψ:G(S)G(T)of group piles.

Proof. Let G(S)=(G,Gp)p∈S, G(T)=(H,Hp)p∈S, p∈S and Gp. For an open normal subgroup M of H one may check that ψ(ψ()M/M)=N/NGp,N, where ψ(H/M)=G/N. Since ψ preserves the relations Gp,n, this implies that ψ()M/MHp,M. Soψ()∈lim←−NHp,N =Hp.

Conversely, let Hp be given. The family ψ(M/M)Gp,N, M H open and ψ(H/M)=G/N, is compatible with respect to theπx,y; hence there existsGpwith ψ(M/M)=N/N for all M H open, whereψ(H/M)=G/N, and thusψ()=. Thusψ(Gp)=Hpfor allp∈S, so ψis an epimorphism of group piles.

Proposition 5.13. The map S defines an equivalence of categories between the category of deficient group piles (with epimorphisms) and the category of inverse systems (with embeddings), with quasi-inverseG.

Proof. Note that S andG are in fact functorial. Now apply the previous lemmas.

Definition 5.14. Acoformulais anLco,S-formula that is bounded; i.e., all quantifiers are of the form(∃vGn). Acosentenceis a coformula without free variables.

A group pileGcosatisfiesa setΣof coformulas with free variablesV (or is acomodelof Σ) if there are elements xvS(G), vV, such that S(G)|ϕ(xv, vV)for allϕΣ.

AcotheoryT is a set of cosentences.

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The cocardinalityof a group pile Gis the cardinality of S(G). A setΣ of coformulas with parameters in some inverse systemSisrankedif, for every variablevthat occurs in some formulaϕΣ, alsoGn(v)Σfor somen ∈N. A group pileGisκ-cosaturated, for a cardinal numberκ, if every ranked set Σ of coformulas with parameters in S(G)with

|Σ|< κ is cosatisfied inG, provided that every finite subset ofΣ is cosatisfied inG.

Remark 5.15. An inverse system S in the language Lco,S can also be viewed as an ω-sorted structure Sω in a language Lωco,S, where the nth sort consists of the sGnGn−1, and for eachk-ary relation relation RinLco,Swe have anωk-family ofk-ary relations Rn1,...,nk in Lωco,S; cf. [3,§1]. Clearly, everyLco,S-formula can be translated to a correspondingLωco,S-formula, and vice versa.

6. e-free C-piles

In this section, we generalize theCantor group piles of [20]. Recall that for a group pile G we defined the deficient reduct G, the subgroup G= G of G, and the quotient G¯ =G/G; see §3. Moreover, recall that Fˆe denotes the free profinite group of rank e;

cf.§2.

Definition 6.1. Ane-free C-pileis ane-generated deficient group pileGfor which every rigide-generated deficient finite embedding problem is solvable (cf. Definition3.10).

Lemma 6.2. If Gis ane-free C-pile, thenG¯ ∼= ˆFe.

Proof. If B is a finite group withrk(B)e, then(G→1,B→1)is a rigide-generated deficient finite embedding problem forG, so it has a solution by assumption. Therefore, by Lemma3.14, every finite group B withrk(B)eis a quotient ofG. Since¯ rk(G¯)e, this implies that G¯ ∼= ˆFe; cf. [16, 16.10.7].

Lemma 6.3. LetGbe ane-bounded group pile, and let(ϕ:G→ ˜A, α: ˜B→ ˜A)be a locally solvable e-generated deficient finite embedding problem. If G0∼= ˆFe for G0G0, then there existAandBwithA= ˜AandB= ˜Bsuch that(ϕ:G→A, α:B→A)is a locally solvablee-bounded finite embedding problem.

Proof. Let G0G0 and A0=ϕ(G0). Then G= G0,G implies that A˜= A0,A˜ (Lemma 3.12), so A=(A, (A˜ 0)A˜,A˜p)p∈S is e-bounded. By Lemma 3.18, there exists ane-bounded group pileB=(B, (B˜ 0)B˜,B˜p)p∈Ssuch that α:B→Ais an epimorphism.

Without loss of generality, assume that α(B0)= A0. We claim that EP =(ϕ:G→ A, α:B→A)is locally solvable. Clearly it satisfies (L S) forp∈S. SinceG0∼= ˆFe andB ise-bounded, there exists an epimorphismγ0:G0B0withαγ0=ϕ|G0, cf. [16, 17.7.3].

Thus, by Lemma4.2,EP satisfies (L S) forp=0.

Proposition 6.4. Every locally solvablee-generated deficient finite embedding problem for ane-free C-pileGis solvable.

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Proof. LetEP =(ϕ:G→ ˜A, α: ˜B→ ˜A)be a locally solvablee-generated deficient finite embedding problem forG. By Lemma6.2,G¯ ∼= ˆFe. LetG0Gbe a subgroup of rank at mostethat under the quotient mapG→ ¯G maps ontoG¯ ∼= ˆFe. Since every finite group generated byeelements is a quotient ofFˆe, it is also a quotient ofG0, and thusG0∼= ˆFe; cf. [16, 16.10.7]. Moreover,G=(G, (G0)G,Gp)p∈S ise-bounded. By Lemma 6.3, there existAandBwithA= ˜AandB= ˜Bsuch thatEP1=(ϕ:G→A, α:B→A)is locally solvable ande-bounded. By Proposition4.10,EP1can be dominated by a rigide-bounded finite embedding problemEP2. The deficient reduct ofEP2is a rigide-generated deficient finite embedding problem, and hence has a solution. It induces a solution ofEP. Example 6.5. For eachp∈S, letp be a profinite group andTp a profinite space, and let0= ˆFe be the free profinite group of ranke. Then [20,§5] constructs from this data a certain group pile GT, which we call the e-free semi-constant group pile of (p)p∈S over(Tp)p∈S. We do not repeat this definition but rely on the properties ofGT proven in [20].

Lemma 6.6. The e-free semi-constant group pile G of (p)p∈S over (Tp)p∈S is an e-bounded self-generated group pile.

Proof. By [20, Proposition 5.3(c)],Gis self-generated. By the construction, everyG0G0

is isomorphic to0= ˆFe; henceGise-bounded.

Proposition 6.7. LetGbe ane-free semi-constant group pile of nontrivial profinite groups (p)p∈S over perfect profinite spaces (Tp)p∈S. Then the deficient reduct G of G is an e-free C-pile.

Proof. By Lemma 6.6, G is e-bounded, so G is e-generated. Let EP =(ϕ:G→ A, α: ˜˜ B→ ˜A) be a rigid e-generated deficient finite embedding problem for G. By Lemma 4.4, EP is locally solvable. By Lemma 6.3, there exist A and B with A= ˜A andB= ˜Bsuch that(ϕ:G→A, α:B→A)is a locally solvablee-bounded (and hence self-generated) embedding problem. By [20, Proposition 5.3(h)], this embedding problem has a solution, which in turn induces a solution ofEP.

Definition 6.8. Let the cotheoryTCco,S,e consist of the following.

(1) For n∈N, a cosentence about a group pile G=(G,Gp)p∈S stating that, for each N G with(G:N)n, the finite quotientG/N ise-generated.

(2) For n,k∈N, a cosentence about a group pile G=(G,Gp)p∈S stating that, for every N G with(G:N)n and every rigid epimorphismα:B→G/N with B ane-generated deficient group pile of orderk, there is an MGwith(G: M)k and M N and an isomorphismβ :G/M→Bsuch thatαβ is the natural map G/M →G/N.

(Such sentences can easily be written down, since for every n and k there exist, up to isomorphism, only finitely many group pilesAandBof order at mostn (respectively,k) and only finitely many rigid epimorphismsα:B→A.)

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Proposition 6.9. A deficient group pile G is ane-free C-pile if and only if it cosatisfies TC,S,eco .

Proof. By Proposition3.17,Gcosatisfies (1) if and only ifGise-generated. And (2) just says that all rigide-generated deficient finite embedding problems forGare solvable.

Proposition 6.10. Let Gbe an1-cosaturatede-free C-pile. Then every rigide-generated deficient embedding problem :G→A, α:B→A)withrk(B)ℵ0 is solvable.

Proof. Sincerk(B)0, there is a descending sequence of open normal subgroups Ni B,i ∈N, with

i∈NNi =1; cf. [16, 17.1.7(a)]. For eachi ∈N, letαi:B/Ni →A/α(Ni)be the epimorphism induced byα, and, for i j ∈N, let πi:A→A/α(Ni), ρi:B→B/Ni, andρji:B/Nj →B/Ni be the quotient maps. Thenαiρi =πiα.

G B α - A

?ϕ

B/Nj ρj

? α-j A/α(Nj) πj

?

B/Ni

ρji

? α-i

γi

...

A/α(N?i)

By Lemma 4.4, the rigid deficient embedding problem (ϕ, α) is locally solvable; hence the induced embedding problemiϕ, αi)is locally solvable by Lemma 4.8. SinceBis e-generated,B/Ni ise-generated by Lemma 3.16. Hence,iϕ, αi)is a locally solvable e-generated deficient finite embedding problem for G. Since G is an e-free C-pile, this embedding problem has a solutionγi:G→B/Ni by Proposition6.4.

For each i, fix an enumeration B/Ni = {bi,1, . . . ,bi,ni}, and let ai,ν =αi(bi,ν)∈ A/α(Ni)S(A). View S(A) as a subset of S(G) via ϕ, and let Σ be the following set of bounded Lco,S-formulas in the variables xi, i ∈N, 1νni, with constants from S(A).

(1) For eachi and each1νni, theLco,S-formulaGni(xi).

(2) For eachi, anLco,S-formula stating that[xi,1] = {xi,1, . . . ,xi,ni}and that the map xi,νbi,ν is an isomorphism of group pilesβi : [[xi,1]] →B/Ni.

(3) For eachi, anLco,S-formula with constants from S(A)stating thatxi,1ai,1and πxi,1,ai,1(xi,ν)=ai,ν for all1ν ni.

(4) For eachi j, anLco,S-formula stating thatxj,1xi,1andβiπxj,1xi,1 =ρjiβj. Every finite subset Σ0 of Σ is cosatisfied in G. Let j be the maximal index of a variable xj,ν appearing in Σ, and, for i j, 1νni, let gi,ν =γjji(bi,ν)). Then (gi)1ij,1νni satisfiesΣ0. By (1),Σ is ranked.

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Thus, since G is ℵ1-cosaturated, there are (˜gi)1i,1νni in G that satisfy Σ. For eachi, the mapbi,ν → ˜gi,ν gives an isomorphism ψi: B/Ni → [[ ˜gi,1]] by (2), and hence an epimorphism γ˜i =ψi: G→B/Ni, which satisfies αi◦ ˜γi =πiϕ by (3). By (4), these epimorphisms are compatible, giving rise to an epimorphism γ˜ =lim←−iγ˜i: G→ lim←−iB/Ni =B, which then satisfiesα◦ ˜γ =ϕ.

7. Model theory of PSCC fields

In the next section, we will let the finite set S be a set of primes, and associate to each fieldF a group pileGalS(F), which extends the absolute Galois groupGal(F)with S-local data. The notion ofprimeand the corresponding local–global principlePSCCwe use is the one developed in [14]. We now briefly recall the main definitions and results, but refer to [14,§§2–3] for further details,1 some history, and references to special cases of these results that were proven before. Basics on real closed and p-adically closed fields are summarized in AppendicesA andB.

Definition 7.1. For a field F of characteristic zero, we denote by F˜ a fixed algebraic closure of F, and byGal(F)=Gal(F˜/F)the absolute Galois group of F. For a subfield E of F, we let E˜ be algebraic closure of E contained in F˜.

Definition 7.2. Aprime2 of a fieldK is either an ordering ofK or an equivalence class of p-valuations on K, for some prime number p. It is local if the ordering is Archimedean (respectively, if the value group is isomorphic to Z). If P is a prime of K, we denote by CC(K,P) (forclassical closures) the set of all real (respectively, p-adic) closures of (K,P)inside K˜. If pis a prime of K and F/K is a field extension, we denote bySp(F) the set of all primes P of F that lie abovep (we write this as P|K =p) and are of the same type, and byCCp(F)the union of allCC(F,P),P∈Sp(F); cf. [14, Definitions 3.4, 3.7, 4.2].

Setting 7.3. For the rest of this work, let S be a finite set of local primes of a field K of characteristic zero, and let F/K be a field extension. For p∈S, fix a closure Kp∈ CCp(K).

Lemma 7.4. Let KEF, p∈S, Q∈Sp(F), and P=Q|ESp(E). If F∈ CC(F,Q), thenE:=F∩ ˜E ∈CC(E,P), andres:Gal(F)→Gal(E)is an isomorphism.

In particular, F∩ ˜E∈CCp(E)for any F∈CCp(F).

Proof. The field E is algebraically closed in the real closed (respectively, p-adically closed) fieldF, so it is real closed (respectively, p-adically closed) itself; see LemmasA.2 andB.1. LetQ be the unique prime of F overQ. ThenP=Q|E is the unique prime

1The reader who wants to check these details should be aware of the fact that, in the notation of [14], here we consider only the case of relative typeτ=(1,1), so for example thePSCCproperty is there calledPSτCCwithS=Sandτ=(1,1).

2This is called aclassical primein [14].

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