HAL Id: hal-03155190
https://hal.archives-ouvertes.fr/hal-03155190v4
Preprint submitted on 19 May 2021 (v4), last revised 12 Aug 2021 (v5)
HAL
is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire
HAL, estdestinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
ON THE p-CLASS GROUP STABILITY ALONG CYCLIC p-TOWERS OF A NUMBER FIELD
Georges Gras
To cite this version:
Georges Gras. ON THE p-CLASS GROUP STABILITY ALONG CYCLIC p-TOWERS OF A NUM-
BER FIELD. 2021. �hal-03155190v4�
ON THE p-CLASS GROUP STABILITY
ALONG CYCLIC p-TOWERS OF A NUMBER FIELD
GEORGES GRAS
Abstract. Letkbe a number field,p≥2 a prime andSany set of finite places ofk. We callK/ka totallyS-ramified cyclicp-tower if Gal(K/k)≃Z/pNZ and ifS6=∅is totally ramified. Using analogues of Chevalley’s formula (Gras, Proc. Math. Sci. 127(1) (2017)), we give an elementary proof of a stability theorem (Theorem3.1) for generalizedp-class groupsXnof the layerskn⊆K:
letλ= max(0,#S−1−ρ) given in Definition1.1; then#Xn=#X0·pλ·nfor alln∈[0, N], if and only if#X1=#X0·pλ. This improves the caseλ= 0 of Fukuda (1994), Li–Ouyang–Xu–Zhang (2020), Mizusawa–Yamamoto (2020), whose techniques are based on Iwasawa’s theory or Galois theory of pro-p- groups. We deduce capitulation properties ofX0 in the tower. Finally we apply our principles to the torsion groupsTnof abelianp-ramification theory.
Numerical examples are given.
Contents
1. Introduction 1
2. Around Chevalley’s ambiguous class number formula 4 3. Introduction of Hasse’s norm symbols – Main theorem 5
4. Numerical examples 7
4.1. Cyclicp-towers overk=Q(√
±m) 7
4.2. Cyclicp-towers overQwith modulus 9
4.3. Cyclicp-towers over a cyclic cubic fieldk 9
5. Other examples of applications 10
6. Behavior inp-cyclic towers of other invariants 11 6.1. Case of the cyclotomicZp-extensionsek/kofkreal 13 6.2. Case of real cyclicp-towers with tame ramification 13
References 14
1. Introduction
The behavior of p-class groups in a Zp-extension gives rise to many theoreti- cal and computational results; a main observation is the unpredictability of these groups in the first layers of the Zp-extension, then their regularization from some uneffective level, Iwasawa’s theory giving the famous formula of the orders by means
Date: May 19, 2021.
2020Mathematics Subject Classification. Primary 11R23; 11R29; 11R37.
Key words and phrases. p-class groups, class field theory, Chevalley’s formula, cyclic p- extensions.
1
of the invariants λ, µ, ν; nevertheless, this algebraic context is not sufficient to es- timate these parameters (e.g., Greenberg’s conjecture [10]). Our purpose is to see, more generally, what is the behavior of the p-class groups (and some other arith- metic invariants) in cyclicp-extensions in which one allows tame ramification.
Let k be a number field, p ≥ 2 a prime number and K a cyclic extension of degreepN ofk,N ≥2, and letkn,n∈[0, N], be the degreepn extension ofkinK.
We speak of cyclicp-towersK/kand putG= Gal(K/k). LetSbe the set of places of k, ramified in K/k. We will assume S 6=∅ and S totally ramified in K/k; in general,Scontains tame places, except ifKis contained in aZp-extension ofk, in which caseS =Sp, a set ofp-places ofk. Due to the very nature of a tower, there will never be “complexification=ramification” of infinite real places (i.e.,S∞=∅).
Letm 6= 0 be an integer ideal ofk, of supportT disjoint from S; let Ck,m and Ck
n,m, denoted simply C0,m and Cn,m, be the p-Sylow subgroups of the ray class groups modulom, ofkandkn, respectively (forn >0,mis seen as extended ideal inkn). The class of an ideala ofkn is denotedcℓn,m(a).
Let NK/knbe the arithmetic norm inK/kn, defined on class groups from norms of ideals. Sincemis prime toS6=∅totally ramified, the correspondingp-Hilbert’s ray class fields Hn,m, of the kn’s are linearly disjoint from the relativep-towersK/kn; since, by class field theory, NK/kn corresponds to the restriction of automorphisms Gal(HN,m/K)→Gal(Hn,m/kn), we get NK/kn(CN,m) =Cn,m, for alln∈[0, N].
Let HN be a sub-Z[G]-module of CN,m; we represent a minimal set of Z[G]- generators of HN with prime ideals Q1, . . . ,Qt ∈/ S∪T. Let IN be the Z[G]- module generated by these ideals; thus NK/k(IN) is of minimal Z-rank t since NK/k(IN)⊗Q=hq1, . . . ,qti ⊗Q). We set, for all n∈[0, N]:
(1.1) In:= NK/kn(IN), Hn:= NK/kn(HN), Xn,m:=Cn,m/Hn.
This defines the family{Xn,m}n∈[0,N] (“generalizedp-class groups”) such that:
NK/kn(XN,m) =Xn,m, for alln∈[0, N].
Any placeQofK, such thatcℓN,m(Q)∈HN, totally splits in the subfield ofHN,m
fixed by the image of HN in Gal(HN,m/K); so this general definition allows to enforce decomposition conditions in the ray class fields.
To simplify, we shall remove the indicesm, except necessity.
One speaks of “λ-stability” of theseXnalong the cyclicp-towerK/kwhen there existsλ≥0 such that#Xn =#X0·pλ·n, for all n∈[0, N]. Of course, this is not a workable definition, both theoretically (classical context of Zp-extensions or less familiar case ofp-towers with tame ramification) and computationally (inability to use PARI/GP [25] beyond some leveln= 3 or 4, and almost nothing forp >5). So we intend to get an accessible criterion likely to give information in all the tower.
Definition 1.1. Let Λ := {x ∈ k×, x ≡ 1 (modm), (x) ∈ NK/k(IN)}. For instance, IN = 1 yields {Xn} = {Cn} and Λ = E, the group of units ε ≡ 1 (modm) ofk. We have the exact sequence 1→Λ/E→I0→H0→1. Since Λ/E is free (ofZ-rankt becauseH0 is finite), one can write, with representativesαj of Λ/E:
Λ = torZ(E)L
hε1, . . . , εr;α1, . . . , αtiZ=: torZ(E)L
X, εi∈E,
r:=r1+r2−1, where (r1, r2) is the signature ofkand whereX isZ-free ofZ-rank ρ:=r+t; we define the parameterλ:= max (0,#S−1−ρ).
We shall see that the norm properties, inK/k, of torZ(E)⊗Zp are specific and may give some obstructions that are elucidated by the following Lemma.
Lemma 1.2. (Albert’s Theorem for K/k; for a proof, see[6, Exercise I.6.2.3]). If k contains the group µpe of peth roots of unity, thenµpe ⊂NK/k(K×) if and only if there exists a degreepecyclic extension L/K such thatL/k is cyclic.
Remark 1.3. We shall restrict ourselves to the case torZ(E)⊗Zp ⊂NK/k(K×).
If necessary, it suffices to restrict the tower K/k to the sub-tower K′/k such that [K : K′] =#µp(k) (µp(k) being the group of pth roots of unity of k) or to notice that L/K/kdoes exist (recall that no arithmetic condition is required onL/K).
Thus, the norm properties ofΛ are assigned toX.
We put to simplify C0 =: C, H0 =: H, I0 =: I, X0 =: X, and so on in the sequel, m being implied. Under the assumptions on S, m, we will prove the following result (see Theorem3.1and Corollary3.2for more complete statements):
Main Result. Assume thattorZ(E)⊂NK/k(K×)(Remark1.3). Letρ≥0be the Z-rank of theZ-moduleX and letλ:= max (0,#S−1−ρ)(Definition1.1). Then
#Xn=#X ·pλ·n for alln∈[0, N]), if and only if#X1=#X ·pλ. If the criterion applies withλ= 0 (i.e.,#X1=#X) and ifpN ≥#X, thenX capitulates inK.
Remark 1.4. (a) Ifλ= max (0,#S−1−ρ)fulfills the criterion ofλ-stability from the base fieldk, it fulfills the sameλ-stability criterion in any relativep-towerK/k′, k′ :=kn0,n0∈[0, N[, since (withXn′ :=Xn
0+n,n≥0) we get#Xn′=#X′·pλ·n since #X′
n =#Xn
0+n=#X ·pλ(n0+n)= (#X ·pλ n0)·pλ n =#X′·pλ n. (b) If λ= max (0,#S−1−ρ) is not suitable for the criterion in k1/k (which shall be equivalent to#X1>#X ·pλ due to Chevalley’s formula), we can consider a relative tower K/k′; then #S′ =#S and λ′ := max (0,#S−1−ρ′)is a strictly decreasing function of[k′ :k](indeed, from Definition1.1, we compute thatρ′−ρ= r′−r= (r1+r2)·([k′ :k]−1)), so that two cases may arise:
(i) For somek′⊂K, we have#X′
1 =#X′·pλ′ giving theλ′-stability inK/k′ with regular linear increasing orders fromk′.
(ii) Whatever k′ ⊂ K, λ′ is not suitable (i.e., #X′
1 > #X′ ·pλ′), which means that #Xn is strictly increasing from some level n0, which is illustrated by many numerical examples and suggests non-linear increasing (in a Zp-extension this means “µ 6= 0”). It is important to note that a λ-stability may exist frome some level, witheλ > λ′; this is the case in aZp-extension with non-zero Iwasawa’s invariants eλ,eµ, the parameterλ′ being0 in somek′ high enough in the tower.
Numerical experiments are out of reach as soon as n >3 or 4, but a reasonable heuristic is a tendency to stabilization in totally real p-towers, by analogy with the study of Greenberg’s conjecture carried out in [9]. In a different framework, letkn
be thenth layer of the cyclotomicZp-extensionek ofk and letCn be the whole class group ofkn; then, forℓ6=p,#(Cn⊗Zℓ)stabilizes inek(a deep result in the abelian case[28], extended in[3] to Iwasawa context in theZp×Zℓ-extension ofk).
(c) For k fixed, λ is unbounded (which only depends on the tower via S) and the condition #C1 =#C of the literature (λ= 0, thus #S ≤r1+r2) is empty as soon as λ ≥1, whence the interest of the factor pλ to get examples whatever S.
For known results (all relative toλ= 0), one may cite Fukuda[4]using Iwasawa’s theory, Li–Ouyang–Xu–Zhang [20,§3]working in a non-abelian Galois context, in
Kummer towers, via the use of the fixed points formulas[5,7], then Mizusawa [22]
above Z2-extensions, and Mizusawa–Yamamoto [23] for generalizations, including ramification and splitting conditions, via the Galois theory of pro-p-groups.
To conclude, we will show that the behavior, inK/k, of the finite torsion groups Tn of p-ramification theory (as “dual” invariants of p-class groups) gives much strongly increasing groups with possibly non-zeroµ-invariant ine Zp-extensions.
2. Around Chevalley’s ambiguous class number formula
The well-known Chevalley’s formula [1, p. 402] is the pivotal element for a great lot of “fixed points formulas”. For the ordinary sense, it takes into account the complexification of real infinite places, which does not occur for us as explained in the Introduction. Let L/F be a cyclicp-extension of Galois groupG, S-ramified, non-complexified at infinity if p = 2, and let ev(L/F) be the ramification index of v ∈S; we have #CG
L = #CF·
Q
v∈Sev(L/F)
[L:F]·(EF:EF∩NL/F(L×)), where CF, CL denote the p-class groups ofF, L, respectively, andEF the group of global units ofF.
Various generalizations of this formula were given (Gras [5, Th´eor`eme 2.7, Corol- laire 2.8], [7, Theorem 3.6, Corollaries 3.7, 3.9], Jaulent [13, Chap. II,§3]); mention their recent idelic proof by Li–Yu [21, Theorem 2.1 & §2.3, Examples]. We shall use the following one for the layers of ap-cyclic towerK/k:
Proposition 2.1. AssumeS6=∅totally ramified inL/F. Letmof supportT be a prime-to-S modulus ofF and letCF,CL, be thep-Sylow subgroups of the ray class groups modulom, ofF andL, respectively. For anyZ[G]-moduleIL, of prime-to- T∪S ideals of L, defining HL:=cℓL(IL),HF := NL/F(HL) :=cℓF(NL/F(IL)), we have withΛL/F :={x∈F×, x≡1 (modm), (x)∈NL/F(IL)}:
(2.1) #(CL/HL)G =#(CF/HF)× (#G)#S−1
(ΛL/F : ΛL/F∩NL/F(L×)).
Lemma 2.2. Let K/k be a cyclicp-tower of degreepN, of Galois groupG=:hσi. Let HN =:cℓN(IN)⊆CN and Λ :={x∈k×, x≡1 (modm), (x)∈NK/k(IN)} (see Definition1.1):
(i)Λ ={x∈k×, x≡1 (modm),(x)∈Nkn/k(In)} (i.e.,ΛK/k= Λkn/k), (ii) Λ⊆Λ1:={x1∈k×1, x1≡1 (modm),(x1)∈NK/k1(IN)}.
Proof. Point (i) comes fromIn = NK/kn(IN). Letx∈Λ seen ink1; then (x) = NK/k(A) = NK/k1(AΘ), where Θ = 1 +σ+· · ·+σp−1, andAΘ ∈IN sinceIN is a Z[G]-module, whence (ii). Notice that (x)∈NK/k(IN) expresses thatxis local norm in K/k at every v /∈ S; if moreover xis local norm at S it is a global one
(Hasse norm theorem).
In what follows, throughout the article, only the invariantρ= rkZ(X) is needed (Definition 1.1 giving λ := max (0,#S−1−ρ)) and never IN, Λ,. . . However, ρ=r+tmay be unknown because of the minimal number t of generators of the Z[G]-moduleIN if it is too general; but, in practice,ρis known (e.g.,ρ=r1+r2−1 for ray class groups modulom).
Lemma 2.3. Let X = hx1, . . . , xρiZ be a free Z-module of Z-rank ρ ≥ 0, let Ω := (Z/pnZ)δ,n≥1, δ≥0, and letf :X →Ωbe an homomorphism such that
the image of X →Ω/Ωp is ofFp-dimension min (ρ, δ). Then #f(X) =pmin (ρ,δ)·n
and for allm∈[0, n],#fpm(X) =pmin (ρ,δ)·(n−m) (wherefpm(x) :=f(x)pm).
Proof. From f :X →Ω, let f : X/Xp →Ω/Ωp; by assumption, dimFp(Im(f)) = min (ρ, δ). LetM =he1, . . . , eδiZp be a freeZp-module ofZp-rankδ, and replace Ω byM/Mpn. Letq :X →X/Xp, πn :M →M/Mpn, π: M/Mpn →M/Mp, and π1=π◦πn :M →M/Mp, be the canonical maps. Then, letF :X →M be any map such thatπn◦F =f. We have, sincen≥1, the commutative diagram:
❅❅❅f ց
X≃Zρ
X/Xp
M≃Zδp
Ω/Ωp≃M/Mp Ω≃M/Mpn F
f
πn
π
q π1
(i) Case ρ ≥ δ (surjectivity of f). Lety ∈ M; there exists x ∈ X such that π1(y) = f(q(x)) = π1(F(x)), whence y = F(x)· y′p, y′ ∈ M; so, by a finite induction,M =F(X)·Mpn andπn(M) = Ω =f(X) is of orderpδ·n.
(ii) Caseρ < δ (injectivity off). Lety∈M such that yp=F(x),x∈X; then 1 =π1(yp) = π1(F(x)) =f(q(x)), whence q(x) = 1 and x=x′p, x′ ∈ X, giving yp =F(x′)p in M thusy =F(x′), proving that F(X) is direct factor inM; thus f(X) =πn(F(X)) is a direct factor, in Ω, isomorphic to (Z/pnZ)ρ.
Sincefpm(X) =f(X)pm, this givesf(X)pm = Ωpm ≃((Z/pnZ)δ)pm in case (i) and ((Z/pnZ)ρ)pm in case (ii). Whence the orders.
3. Introduction of Hasse’s norm symbols – Main theorem LetK/k be a cyclicp-tower of degreepN, N ≥2. Letωkn/k be the map which associates with x ∈ Λ the family of Hasse’s norm symbols x , kn/k
v
∈ Iv(kn/k) (inertia groups ofv∈S), where Λ :={x∈k×, x≡1 (mod m),(x)∈NK/k(IN)}. Since x is local norm at the places v /∈ S, the image of ωkn/k is contained in Ωkn/k :=
(τv)v∈S ∈L
v∈SIv(kn/k), Q
v∈Sτv = 1 (from the product formula);
then Ker(ωkn/k) = Λ∩Nkn/k(kn×).
Assume S6=∅ totally ramified; so, fixing any v0∈S, Ωkn/k ≃L
v6=v0Gn, with Gn:= Gal(kn/k). We consider formulas (2.1) inkn/k, using Lemma2.2(i):
(3.1) #XGn
n =#X · pn(#S−1)
(Λ : Λ∩Nkn/k(kn×)) =#X ·#pωn(#S−1)
kn/k(Λ),
whereXn :=Cn/Hnfor the family{Hn}(cf. (1.1) with a prime-to-Smodulusm).
Theorem 3.1. Set Λ =: torZ(E)L
X, with X = hx1, . . . , xρiZ free of Z-rank ρ, and letλ:= max (0,#S−1−ρ). Assume thattorZ(E)⊂NK/k(K×)(see Lemma1.2 and Remark 1.3). We then have the following properties of λ-stability:
(i) If#X1=#X·pλ, then,#Xn=#X·pλ·nandXnGn=Xn for alln∈[0, N].
(ii) If#X1=#X·pλ, thenJkn/k(X) =Xpn
n andKer(Jkn/k) = Nkn/k(Xn[pn]), for all n∈[0, N], where the Jkn/k’s are the transfer maps in kn/k. If λ= 0, the norm maps Nkn/k :Xn →X are isomorphisms.
Proof. Put Nn := Nkn/k, Jn := Jkn/k, Gn =: hσni, Ωn := Ωkn/k, ωn := ωkn/k, s:=#S. By assumption, ωn(Λ) =ωn(X) for alln∈[1, N]. From (3.1),#XG1
1 =
#X · #pωs−1
1(X) ≥#X ·pλsince #ω1(X)≤pmin (ρ,s−1); thus, with#X1=#X ·pλ, we obtainXG1
1 =X1, whence #XG1
1 =#X ·pλ and#ω1(X) =pmin (ρ,s−1). By restriction of Hasse’s symbols,ω1=π◦ωn, withπ: Ωn →Ωn/Ωpn ≃Gs−11 . Since we have proven that dimFp(ω1(X)) = min (ρ, s−1), Lemma 2.3 applies to X =hx1, . . . , xρiZ,δ=s−1,f =ωn; so,#ωn(X) =pmin (ρ,s−1)n and, from (3.1):
(3.2) #XGn
n =#X ·pλ·n, for alln∈[0, N].
Consider the extensionkn/k1, of Galois groupGpn =hσpni, and the corresponding map ω′n on k1× with values in Ωkn/k1 ≃ Ωpn; then #XG
p
n n = #X1 · p(n−1) (s−1)
#ω′n(Λ1) , Λ1 := {x ∈ k1×, x ≡ 1 (modm), (x) ∈ NK/k1(IN)}. Since Λ ⊆ Λ1 (Lemma 2.2(ii)), the functorial properties of Hasse’s symbols onX imply (where v1 |v in k1):
ωn′(xi) =x
i, kn/k1
v1
v=N
1(xi), kn/k v
v=xp
i, kn/k v
v=x
i, kn/k v
p
v, givingω′n=ωnp onX. Then,ω′n(X)⊆ωn′(Λ1) yields:
#XnGpn=#X1·p(n−1) (s−1)
#ωn′(Λ1) ≤#X1·p(n−1) (s−1)
#ω′n(X) =#X1·p(n−1) (s−1)
#ωnp(X) ;
Lemma2.3, forωpn, gives#ωpn(X) =p(n−1) min (ρ,s−1), then, with #X1=#X ·pλ:
#XG
p
n n≤#X1·p(np−(n1) min (ρ,s−1) (s−1)−1) =#X1·pλ·(n−1)=#X·pλ·n. SinceXG
p
n n⊇XnGn and#XnGn =#X·pλ·n from (3.2), we getXnGn =XG
p
n n, equivalent to Xn1−σn = X1−σ
p
n n =Xn(1−σn)·θ, whereθ = 1 +σn+· · ·+σnp−1 ∈ (p,1−σn), a maximal ideal ofZp[Gn] sinceZp[Gn]/(p,1−σn)≃Fp; soXn1−σn = 1, thusXn=XGn
n . Whence (i).
From Nn(Xn) = X, Xn = XGn
n and Jn◦ Nn = νn (the algebraic norm), one obtains Jn(X) = Jn(Nn(Xn)) = Xnνn = Xnpn. Let x ∈ Ker(Jn) and put x= Nn(y),y∈Xn; then 1 = Jn(x) = Jn(Nn(y)) =ypn, so Ker(Jn)⊆Nn(Xn[pn]) and ifx= Nn(y),ypn = 1, then Jn(x) = Jn(Nn(y)) =ypn= 1. Whence (ii).
Corollary 3.2. Let pe be the order of X in k and let K be a totallyS-ramified p-cyclic tower of degree pN, with N ≥ e, such that λ = 0 (i.e., #S ≤ 1 +ρ, namely #S ≤r1+r2 for class groups). If #X1 =pe, then X capitulates in ke. In particular, this applies in a Zp-extension ek/k with λ = 0 and S = Sp totally ramified.
Proof. Since λ = 0, Xe ≃ X from the isomorphism induced by Ne : Xe → X, whence Ker(Je) = Ne(Xe[pe])≃X[pe] =X. Section4will give many examples of such capitulations ofp-class groups of real fields k in K ⊆ k(µℓ); for instance, for k = Q(√
4409), the capitulation of the 3-class groupC ≃Z/9Z occurs ink2 for ℓ∈ {19, 37, 73, 109, 127, 271, 307, 379, 397, 523, 541, 577, 739, 883, 919, 937, . . .}. The 3-class groupC ≃Z/9Z×Z/3Z of the cyclic cubic field of conductor 5383, defined byP =x3+x2−1794x+ 17744, capitulates in in k3 for ℓ = 109,163,919, . . . The 2-class group C ≃ Z/16Z of
Q(√
2305) capitulates in ink4 forℓ = 97,193,353,449,929, . . . The 5-class group C ≃Z/25ZofQ(√
24859) capitulates in ink2 forℓ= 101,151,251,401, . . . Totally real fields k give λ = 0 whatever S = Sℓ, but some non-totally real fields may give λ = 0; for instance, let P = x6+x5−5x4−4x3+ 6x2+ 2x+ 7 defining k, of signature (0,3), of Galois group S6; so λ = max(0,#S −1− (r1+r2−1)) = max(0,#S −3). For p = 3, C ≃ Z/3Z capitulates in k1 for ℓ= 37,61,67,73,97,103,109,127,181,193, . . .For all these primes,#S ∈ {1,2,3}, givingλ= 0 as expected. This criterion allows to recover many numerical examples of abelian capitulations given in the literature with intricate methods (Gras (1997), Kurihara (1999), Bosca (2009), Jaulent (2017, 2019); see [15] for more history about these references).
The existence of primes ℓ ≡ 1 (mod 2pe) such that #C1 = #C, in the cor- responding p-cyclic tower of a totally real fieldk, enforce the capitulation of the p-class group of k in k(µℓ)/k. Since this only depends on norm symbols having obviously uniform distributions, we propose the following conjecture which would be obtained after some efforts with the techniques used in the classical approaches and especially those of [26], [18] and their subsequent articles):
Conjecture 3.3. Letkbe a totally real number field with generalizedp-class group Xk,m =Ck,m/Hk (cf. (1.1)), of order pe. There are infinitely many primesℓ≡1 (mod 2pe), prime tom and unramified ink, such that Xk,m capitulates ink(µℓ).
Remark 3.4. To simplify, let’s consider the family{Cn} assuming the context of Lemma1.2and Remark 1.3; soλ= max (0,#S−1−ρ)withρ=r1+r2−1.
(a) The condition of λ-stability, #C1 = #C ·pλ, implies CnGn = Cn for all n ≥ 1, but is not equivalent. Indeed, let k = Q(√m), m > 0, and let K = kL, where L is the subfield of Q(µ17) of degree 8, with 17 split in k; whence ρ = 1, λ= 0. Assume thatω1(ε) = 1 (ε norm in k1/k); then, from Chevalley’s formula,
#CG1
1 =#C ·p6=#C. Form= 26,ε= 3 +√
26≡52 (mod 17) in the completion Q17; one obtains the structures[2],[2,2],[4,2]giving#C = 2and#C1=#CG1
1 = 4.
(b) When #C1 = #C ·pλ does not hold (whence #C1 > #C ·pλ), we observe that orders and p-ranks are unpredictable and possibly unlimited in the tower; in- deed, put Ci
n :={c ∈ Cn, c(1−σn)i = 1} =: cℓn(Ii
n), i ≥ 1; we have, from (2.1),
# Cni+1/Cni
= #C
#Nn(Cni)×p#nω(#S−1)
n(Λi), where Λi := {x ∈ k×,(x) ∈ Nn(Ini)}; in general these two factors are non-trivial giving #Cni+1 > #Cni and the filtration increases.
(c) If Lemma1.2does not apply, some counterexamples can arise. For instance, let k = Q(√
−55), p = 2, K = k(µ17) of degree 24 over k (−1 ∈/ NK/k(K×),
#S = 2, λ = 1); the successive 2-structures are [4],[8],[16],[32],[32], for which
#Cn=#C ·2n holds forn∈[0,3], since−1 is norm ink3/k, but not forn= 4.
4. Numerical examples 4.1. Cyclic p-towers over k=Q(√
±m). Consider, forℓ≡1 (modpN) (inell), the cyclicp-tower contained in k(µℓ)/k, for k =Q(√s m), m > 0 square free (in m), s = ±1. The following PARI/GP program computes Ck
n,m =: Cn for any prime-to-ℓmodulus m (in mod). The polynomialPn (inPn) defines the layerkn. The computations up to 3 or 4 (in N) are only for verification when a λ-stability
exists; otherwise some examples do notλ-stabilize in the interval considered and no conclusion is possible (the execution time becomes unaffordable for large degrees).
Ifs=−1 (resp. s= 1),λ=#S−1−ρ∈ {0,1}(resp. λ= 0) inkbut decreases in the tower (Remark1.4). Thenvndenotes the p-valuation of the order ofCn whose structureCknis given by the listL.
{p=2;ell=257;mod=5;s=-1;N=3;bm=2;Bm=10^3;for(m=bm,Bm,if(core(m)!=m,next);
P=x^2-s*m;lambda=0;if(s==-1 & kronecker(-m,ell)==1,lambda=1);\\lambda print();print("p=",p," mod=",mod," ell=",ell," m=",s*m," lambda=",lambda);
for(n=0,N,Pn=polcompositum(polsubcyclo(ell,p^n),P)[1];kn=bnfinit(Pn,1);\\layer kn knmod=bnrinit(kn,mod);v=valuation(knmod.no,p);Ckn=knmod.cyc;\\ray class field (mod) C=List;e=matsize(Ckn)[2];for(k=1,e,c=Ckn[e-k+1];\\ray class groups Cn
w=valuation(c,p);if(w>0,listinsert(C,p^w,1)));print("v",n,"=",v," ",C)))}
IMAGINARY QUARATIC FIELDS, p=2, ell=257, mod=1:
m=-2,lambda=1 m=-11,lambda=1 m=-14,lambda=0 m=-17,lambda=1 m=-15,lambda=1
v0=0 [] v0=0 [] v0=2 [4] v0=2 [4] v0=1 [2]
v1=3 [8] v1=2 [4] v1=4 [4,4] v1=5 [8,2,2] v1=2 [4]
v2=7 [16,4,2] v2=7 [8,8,2] v2=8 [4,4,4,4] v2=10[16,2,2,2,2,2,2] v2=3 [8]
v3=13 [32,8,2, v3=10 [16,16,4] v3=16 [8,8,8,4, v3=19[32,2,2,2,2,2, v3=4 [16]
2,2,2,2] 4,2,2,2] 2,2,2,2,2,2,2,2,2]
m=-782,lambda=1 m=-858,lambda=0
v0=3 [4,2] v0=4 [4,2,2]
v1=8 [4,4,4,2,2] v1=12 [128,2,2,2,2,2]
v2=22[32,32,4,4,4,2,2,2,2,2,2] v2=22 [256,16,8,2,2,2,2,2,2,2]
v3=35[64,64,8,8,8,2,2,2,2,2,2,2,2,2,2,2,2,2,2]v3=32 [512,32,16,4,4,4,2,2,2,2,2,2,2,2]
m=-15,mod=5,lambda=1 m=-35,mod=5,lambda=1 m=-253,mod=5,lambda=1
v0=2 [4] v0=2 [4] v0=4 [4,2,2]
v1=3 [4,2] v1=3 [4,2] v1=12 [64,4,2,2,2,2]
v2=4 [8,2] v2=4 [8,2] v2=22 [128,8,4,4,2,2,2,2,2,2,2,2]
v3=34 [256,16,8,4,4,4,4,4,4,2,2,2,2,2,2,2]
IMAGINARY QUARATIC FIELDS, p=3, ell=163, mod=1:
m=-2,lambda=1 m=-293,lambda=1 m=-983,lambda=1 m=-3671,lambda=0
v0=0 [] v0=2 [9] v0=3 [27] v0=4 [81]
v1=1 [3] v1=3 [27] v1=4 [81] v1=7 [243,3,3]
v2=2 [9] v2=4 [81] v2=5 [243] v2=11 [729,9,9,3]
REAL QUARATIC FIELDS, p=2, lambda=0:
mod=7,ell=12289:
m=2 m=19 m=89 m=15 m=39
v0=0 [] v0=1 [2] v0=0 [] v0=2 [2,2] v0=2 [2,2]
v1=1 [2] v1=2 [2,2] v1=0 [] v1=5 [2,2,2,2,2] v1=5 [2,2,2,2,2]
v2=1 [2] v2=2 [2,2] v2=0 [] v2=9 [8,4,4,2,2] v2=10 [8,4,2,2,2,2,2]
mod=17,ell=7340033
m=17 m=19 m=21 m=38 m=26 m=226
v0=2 [4] v0=6 [16,4] v0=4 [16] v0=5 [16,2] v0=6 [16,2,2] v0=5 [8,8]
v1=2 [4] v1=6 [16,4] v1=4 [16] v1=5 [16,2] v1=8 [16,4,2,2] v1=10 [32,8,8]
REAL QUARATIC FIELDS, p=3, lambda=0:
mod=1,ell=109,m=326 ell=109,m=4409 mod=7,ell=163,m=5 ell=163,m=6
v0=1 [3] v0=2 [9] v0=1 [3] v0=1 [3]
v1=1 [3] v1=2 [9] v1=1 [3] v1=2 [3,3]
v2=1 [3] v2=2 [9] v2=1 [3] v2=2 [3,3]
mod=7,ell=163,m=22 ell=163,m=15 mod=19,ell=163,m=2 ell=163,m=11
v0=2 [3,3] v0=1 [3] v0=2 [9] v0=3 [9,3]
v1=2 [3,3] v1=3 [3,3,3] v1=3 [27] v1=4 [27,3]
v2=2 [3,3] v2=5 [9,9,3] v2=3 [27] v2=4 [27,3]
One finds examples ofλ-stabilities taking for instance#S = 6 with three primes split ink=Q(√
±m) and any of the 1024 totallyS-ramified cyclic 2-towersK/kof
degree 26 contained ink(µ257·449·577) but, unfortunately, only computing ink and k1=k(√
257·449·577) with λ= 4 (resp. 5) fork real (resp. imaginary):
m=193 m=1591 m=2669 m=-527 m=-1739
v0=0 [] v0=1 [2] v0=2 [4] v0=1 [2] v0=2 [4]
v1=4 [2,2,2,2] v1=5 [2,2,2,2,2] v1=6 [4,2,2,2,2] v1=6 [4,2,2,2,2] v1=7 [8,2,2,2,2]
4.2. Cyclic p-towers over Q with modulus. In a cyclic p-towerK/QforS = {ℓ},m= 1, (2.1) shows that λ= 0 and#Cn = 1 for alln. For m6= 1, a stability may occur (possibly from a stage n0 > 0) as shown by the following examples (p= 2,3) analogous to [23, Example 4.1]:
{p=2;ell=257;mod=17;N=5;print("p=",p," ell=",ell," mod=",mod);for(n=0,N, Pn=polsubcyclo(ell,p^n);kn=bnfinit(Pn,1);\\layer kn
knmod=bnrinit(kn,mod);v=valuation(knmod.no,p);Ckn=knmod.cyc;\\ray class field C=List;e=matsize(Ckn)[2];for(k=1,e,c=Ckn[e-k+1];w=valuation(c,p);
if(w>0,listinsert(C,p^w,1)));print("v",n,"=",v," Ck",n,"=",C))}
p=2,ell=257,mod=17 ell=257,mod=4*17 p=3,ell=163,mod=19 ell=163,mod=9*19
v0=3 [8] v0=4 [16] v0=2 [9] v0=3 [9,3]
v1=4 [16] v1=5 [16,2] v1=3 [27] v1=5 [27,3,3]
v2=4 [16] v2=7 [16,2,2,2] v2=3 [27] v2=6 [81,3,3]
v3=4 [16] v3=11 [16,2,2,2,2,2,2,2] v3=3 [27] v3=6 [81,3,3]
v4=4 [16] v4=12 [32,2,2,2,2,2,2,2]
v5=4 [16] v5=12 [32,2,2,2,2,2,2,2]
4.3. Cyclic p-towers over a cyclic cubic field k. LetK⊂k(µℓ) be ap-tower.
An interesting fact is that, in most cases, the stability occurs (with λ= 0 in the caseS=Sℓ); we give an excerpt with the defining polynomialPofkof conductor f∈[bf,Bf]:
{p=2;ell=257;N=3;bf=7;Bf=10^3;for(f=bf,Bf,h=valuation(f,3);if(h!=0 & h!=2,next);
F=f/3^h;if(core(F)!=F,next);F=factor(F);Div=component(F,1);d=matsize(F)[1];for(j=1,d, D=Div[j];if(Mod(D,3)!=1,break));for(b=1,sqrt(4*f/27),if(h==2 & Mod(b,3)==0,next);
A=4*f-27*b^2;if(issquare(A,&a)==1,\\computation of a and b such that f=(a^2+27b^2)/4 if(h==0,if(Mod(a,3)==1,a=-a);P=x^3+x^2+(1-f)/3*x+(f*(a-3)+1)/27);
if(h==2,if(Mod(a,9)==3,a=-a);P=x^3-f/3*x-f*a/27);print();print("f=",f," P=",P);
\\end of computation of P defining k
for(n=0,N,Pn=polcompositum(polsubcyclo(ell,p^n),P)[1];kn=bnfinit(Pn,1);
v=valuation(kn.no,p);Ckn=kn.cyc;C=List;e=matsize(Ckn)[2];for(k=1,e,c=Ckn[e-k+1];
w=valuation(c,p);if(w>0,listinsert(C,p^w,1)));print("v",n,"=",v," Ck",n,"=",C)))))}
p=2,ell=257,lambda=0
f=63 f=163 f=277 f=279
P=x^3-21*x-35 P=x^3+x^2-54*x-169 P=x^3+x^2-92*x+236 P=x^3-93*x+217
v0=0 [] v0=2 [2,2] v0=2 [2,2] v0=0 []
v1=2 [2,2] v1=4 [4,4] v1=4 [2,2,2,2] v1=2 [2,2]
v2=2 [2,2] v2=4 [4,4] v2=6 [2,2,2,2,2,2] v2=4 [4,4]
v3=2 [2,2] v3=4 [4,4] v3=6 [2,2,2,2,2,2] ?
f=333 f=349 f=397 f=547
P=x^3-111*x+37 P=x^3+x^2-116*x-517 P=x^3+x^2-132*x-544 P=x^3+x^2-182*x-81
v0=0 [] v0=2 [2,2] v0=2 [2,2] v0=2 [2,2]
v1=4 [4,4] v1=4 [2,2,2,2] v1=2 [2,2] v1=4 [2,2,2,2]
v2=4 [4,4] v2=4 [2,2,2,2] v2=2 [2,2] v2=6 [4,4,2,2]
p=3,ell=109,lambda=0
f=1687 f=1897 f=2359 f=5383
x^3+x^2-562*x-4936 x^3+x^2-632*x-4075 x^3+x^2-786*x+7776 x^3+x^2-1794*x+17744
v0=1 [3] v0=1 [3] v0=2 [3,3] v0=3 [9,3]
v1=1 [3] v1=1 [3] v1=2 [3,3] v1=3 [9,3]
As for the quadratic case, we may use k1 =k(√
257·449·577) for p= 2 with
#S = 9, ρ = 2, λ = 6. The contribution of the 2-tower over Q is given by the 2-stability from the structures [ ],[2,2]. We find a great lot of λ-stabilities with the structures [ ],[2,2,2,2,2,2] or [2,2],[4,4,2,2,2,2] and very few exceptions for which one does not know if aλ-stabilization does exist forn >1 (case off=1531 below):
f=171,P=x^3-57*x-152 f=349,P=x^3+x^2-116*x-517 f=1531,P=x^3+x^2-510*x-567
Ck0=[] Ck0=[2,2] Ck0=[]
Ck1=[2,2,2,2,2,2] Ck1=[4,4,2,2,2,2] Ck1=[4,4,2,2,2,2]
5. Other examples of applications
Assume the total ramification ofS6=∅ in the cyclicp-towerK/kand the main condition#X1=#X ·pλ withλ= max (0,#S−1−ρ) (Definition1.1); remember that it is easy to ensure that torZ(E)⊂NK/k(K×) (Remark1.3):
(a) Letek/kbe aZp-extension with Iwasawa’s invariantsλ,e µ,e eν. We then have S=Sp with#Sp≤[k:Q] =r1+ 2r2 andλ= max (0,#Sp−r1−r2)∈[0, r2]; one gets the formula#Cn =#C ·pλ·n for all n, as soon as#C1 =#C ·pλ holds, and in that case,eλis equal toλandµe= 0; the Iwasawa formula being#Xn =pλ·n+ν withpν:=#X. For CM-fieldsk withptotally split ink,λ=12[k:Q].
Whenkis totally real, Greenberg’s conjecture [10] (eλ=µe= 0) holds if and only if #Cn
0+1 = #Cn
0 is fulfilled for somen0 ≥ 0 (λ-stabilization with λ = eλ = 0), which may be checked ifn0is not too large by chance (as in [19] for real quadratic fields,p= 3 non-split, or in various works of Taya [27], Fukuda, Komatsu).
(b) TakingHN =Cpe
N may give theλ-stability of the Cn/Cpe
n ’s, for someλ(e), whence rkp(Cn) = rkp(C) +λ(1)·nfor alln, as soon as this equality holds forn= 1 (where rkp(A) := dimFp(A/Ap)). For instance, letk=Q(√
−m) andK⊂k(µ257) and consider the stability of the 2-ranks when rk2(C)≥1 (λ(1) = 0 since #S≤2 andρ=t≥1 sinceC 6= 1); we find many examples:
Form∈ {-15, -35, -95, -111, -123, -159, -215, -235, -259, -287, -291, -303, -327, -355, -371, -415, -447,. . .} the 2-rank is 1 and stable fromn= 0.
Form∈ {-39, -87, -91, -155, -183, -195, -203, -247, -267, -295, -339, -395, -399, -403, -427, -435, -455, -471,. . .}the 2-rank (2 or 3) is stable fromn0= 1.
For m =−55 (successive structures [4],[4,4],[8,4,4,2],[16,8,8,4]), m = −115 (structures [2],[2,2],[2,2,2,2],[8,8,8,8]), we have two examples of stabilization from n0 = 2 (rank 4); the computations for the 4th stages are rapidly out of reach.
We see that the orders are not necessarilyλ′-stable fromn= 0 (λ′ =#S−1):
m=-15 m=-403 m=-259 m=-95 m=-195 m=-895
v0=1 [2] v0=1 [2] v0=2 [4] v0=3 [8] v0=2 [2,2] v0=4 [16]
v1=2 [4] v1=5 [8,2,2] v1=3 [8] v1=4 [16] v1=4 [4,2,2] v1=5 [32]
v2=3 [8] v2=8 [16,4,4] v2=4 [16] v2=5 [32] v2=9 [8,8,8] v2=6 [64]
v3=4 [16] v3=11 [32,8,8] v3=5 [32] v3=6 [64] v3=12 [16,16,16] v3=7 [128]
(c) If K/Q is abelian with p ∤ [k : Q], λ-stabilities may hold for the isotopic Zp[Gal(k/Q)]-components of theXn’s, using fixed points formulas [13,14].
6. Behavior in p-cyclic towers of other invariants
In relation with class field theory and the cohomology of the Galois groupGk,S
p
of the maximalp-ramified pro-p-extension of k and of its local analoguesGk
p, the fundamental Tate–Chafarevich groups [17, Theorem 3.74], [24,§1]:
IIIik(Sp) := Ker Hi(Gk,S
p,Fp)→L
p∈Sp Hi(Gk
p,Fp)
,i= 1,2,
can be examined in the same way, in towers, as for generalized class groups. This is possible for III1k(Sp)≃(Ck/cℓk(Sp))⊗Fp, but III2k(Sp) does not appear immediately as a generalized class group and is not of easy computational access (see [12] for a thorough study of III2k(S) whenSp 6⊂ S). Nevertheless, we have an arithmetic interpretation, much more easily computable and having well-known properties, with the key invariant, closely related to III2k(Sp),
Tk:= torZp(Gab
k,Sp)≃H2(Gk,S
p,Zp)∗ [24, Th´eor`eme 1.1]
(see other rank computations in [6, Theorem III.4.2 and Corollaries]). Nevertheless, Tk, linked to the residue at s = 1 of p-adic ζ-functions [2], gives rise to some analogies with generalized class groups (due to reflection theorems [6, Theorem II.5.4.5]). See in [8] the interpretation of Tk with the p-class group Ck and the normalizedp-adic regulatorRk. Since the deep aspects of these cohomology groups are due to the p-adic properties of global units we shall restrict ourselves to the totally real case under Leopoldt’s conjecture. For any fieldκ, leteκbe its cyclotomic Zp-extension.
Let K/k be a totally real, S-ramified, cyclic p-tower; denote by Sta ⊆ S the subset of tame places (Sta 6=∅ is assumed totally ramified, but the subset Sp of p-places may be arbitrary). For all extensionsL/F contained inK/k, the transfer maps JL/F :TF →TLare injective [6, Theorem IV.2.1] and the norm maps NL/F : TL→TF are surjective sinceL/e Feis totallySta-ramified. WhenL/F ⊂ek/k, NL/F
is also surjective (indeed,Fe=Le=ek).
Lemma 6.1. LetK/k be a totally real cyclicp-tower of degreeq pN,p-ramified or else totally ramified atSta6=∅, whereq=por4; we assume thatk∩Qe =Q. Then, for anyL/F,k⊆F ⊆L⊆K andG:= Gal(L/F),#TG
F =#TL·#G#Sta. Proof. The fixed points formula is#TG
L =#TF·#G#Sta·pρ−r[6, Theorem IV.3.3, Exercise 3.3.1], where pr = [L : F], ρ = minl∈Sta(r;. . . , νl+ϕl+γl, . . .) (pνl ∼ q−1log(ℓ) wherel∩Z=ℓZ,pϕl= residue degree oflink/Q,γl= 0 by hypothesis).
WhenSta6=∅,the existence (see [6, V, Theorem 2.9] for a characterization using explicit governing fields) of the cyclic p-towerK/k impliesN (l)≡1 (modq pN), for alll∈Sta, whereN is the absolute norm; indeed, the inertia group of the local extension KL/kl, L | l, is isomorphic to a subgroup of the multiplicative group of the residue field Fl ≃FN(l) of l. So, we have νl+ϕl ≥N. This explains the limitation of the levelnto get the given fixed points formula in any case.
If aλ-stability does exist, we haveT1=TG1
1 of order#T ·pλ, whenceλ=#Sta which is compatible withTn =TGn
n of order#T ·pλ n, for alln∈[0, N].
However, the properties of the groups Tn will give non-trivial Iwasawa’s µ-e invariants in ek/k and very large ranks in cyclic p-towers with tame ramification;
moreovereλmay be much larger than#Stawhich explain that theλ-stability prob- lem is not pertinent in this framework. A reason (based on the properties, in
kn+m/kn, m≥1, of the norms Nmn (surjectivity) and transfer maps Jmn (injectiv- ity)) is the following, wheregnm:= Gal(kn+m/kn):
From 1→JmnTn→Tn+m→Tn+m/JmnTn→1, we get:
1→Tg
m n
n+m/JmnTn→(Tn+m/JmnTn)gmn →H1(gnm,JmnTn)→H1(gmn,Tn+m), where H1(gnm,JmnTn) = (JmnTn)[pm]≃Tn[pm] and:
#H1(gnm,Tn+m) =#H2(gnm,Tn+m) =#(Tg
m n
n+m/JmnTn) =pm·#Sta. We then obtain#Tn+m≥#Tgnm
n+m=#Tn·pm·#Sta, which seems less precise than the following one, because of the fact that#Sta is independent ofn:
(6.1) #Tn+m≥#Tn·#Tn[pm].
For m = 1 one gets #Tn+1 ≥ #Tn ·prkp(Tn). This tendency to give large orders and p-ranks is enforced by the presence of tame ramification (see §6.2 for numerical illustrations) and leads to the vast theory of pro-p-extensions and Galois cohomology of pro-p-groups in a broader context thanp-ramification theory, which is not our purpose (see [11] for more information). This explains that Corollary3.2 never applies for the invariantT (exceptT = 1). It would be interesting to give an analogous study for the Jaulent logarithmic class group [16] since its capitulation in the cyclotomicZp-extension ofk (real) characterizes Greenberg’s conjecture.
Remark 6.2. For p-class groups Cn in totally real p-towers, one obtains an in- equality similar to (6.1)provided one replacesCn byJmnCn, whereJmn is in general non-injective, a crucial point in this comparison. But for CM-fieldsk,Jnis injective on the “minus parts”, giving #Cn+m− ≥#Cn−·#Cn−[pm], which explains the results for imaginary quadratic fields in§4.1, especially form=−782andm=−858, for which we can predict#C4 larger than254 and246, respectively.
Remark 6.3. One can wonder about the contribution of the p-towers contained in Q(µℓ),ℓ ≡1 (modqpN), used in the compositum withk; inQ, it is clear thate Tn = 1 for all n≥0, which is not the case in a finite cyclic p-tower when tame ramification is allowed; we have the following examples of Tn for p = 2,3 and
#Sta = 1; let vn,rkn denote the p-valuation, the p-rank, of Tn, respectively; the parameterE must be chosen such thatE > e+ 1, wherepe is the exponent ofTn. Then the structure follows:
{p=3;ell=17497;E=8;print("p=",p," ell=",ell);for(n=0,4, Qn=polsubcyclo(ell,p^n);kn=bnfinit(Qn,1);\\ layer kn KpEn=bnrinit(kn,p^E);HpEn=KpEn.cyc;\\computation of Tn
T=List;e=matsize(HpEn)[2];Rn=0;vn=0;for(k=1,e-1,c=HpEn[e-k+1];w=valuation(c,p);
if(w>0,vn=vn+w;Rn=Rn+1;listinsert(T,p^w,1)));\\computation of vn and rkn print("v",n,"=",vn," rk",n,"=",Rn," ",T))}
p=2,ell=17 p=2,ell=7681 p=2,ell=257 v0=0 rk0=0 [] v0=0 rk0=0 [] v0=0 rk0=0 []
v1=1 rk1=1 [2] v1=1 rk1=1 [2] v1=3 rk1=1 [8]
v2=2 rk2=1 [4] v2=2 rk2=1 [4] v2=7 rk2=3 [16,4,2]
v3=3 rk3=1 [8] v3=3 rk3=1 [8] v3=13 rk3=7 [32,8,2,2,2,2,2]
v4=4 rk4=1 [16] v4=4 rk4=1 [16] v4=23 rk4=15 [64,16,2,2,2,2,2,2,2,2,2,2,2,2,2]
v5=38 rk5=15 [128,32,4,4,4,4,4,4,4,4,4,4,4,4,4]
p=3,ell=109 p=3,ell=163 p=3,ell=487 p=3,ell=17497 v0=0 rk0=0 [] v0=0 rk0=0 [] v0=0 rk0=0 [] v0=0 rk0=0 []
v1=1 rk1=1 [3] v1=1 rk1=1 [3] v1=2 rk1=2 [3,3] v1=3 rk1=2 [9,3]
v2=2 rk2=1 [9] v2=2 rk2=1 [9] v2=4 rk2=2 [9,9] v2=6 rk2=3 [27,9,3]
