arXiv:2108.04502v1 [math.NT] 10 Aug 2021 hsodri h ere over degree, the is order This in in tions das ntenro es,ayrycasfield class ray any sense, narrow the in ideals, let eitn ogv,fo h rtmtcof arithmetic the from give, to intend We ouoamodulus a modulo esalsetefield the see shall We Let K prtsb ojgto nGal( on conjugation by operates sAeinover Abelian is fteecnnclfils enn Gal( defining fields, canonical these of h eeaie ls group class generalized the lzdcass miuu lse;Cealysformula. Chevalley’s classes; ambiguous classes; alized 2020 Date e od n phrases. and words Key eg,teHletcasfield class Hilbert the (e.g., L K/k eafiieAeinetninof extension Abelian finite a be coe 4 2016. 14, October : ahmtc ujc Classification. Subject Mathematics ra iwo h subject. the rediscovered ma on which been view bibliography broad often incomplete) have certainly techniques (but these stantial of some Since n hc aebcm tnadi a in standard become have which and ncnlso,w ugs h td,i h aesii,o so of spirit, same the the in to study, attached the suggest we conclusion, relatives In classes de groupes des eso rm ubr”i ylcetnino degree of extension cyclic a in notio numbers” prime the of and sets theorem”, “principal Mazur–Wiles–Kolyvagin ruso bla xesoso rm to prime of extensions Abelian of groups nScin5 eapyti otesuyo h tutr frel of structure the of study the to this apply we 5, Section In ylcetnino degree of extension cyclic a xesoscciusrltvsd degr´e premier de relatives cycliques extensions lost banteodrad(oetal)tesrcueo structure the com (potentially) normic and local order of the algorithm the obtain recall to we allows Then (1973). 3 23, ls il hoy rmter opractice to theory accordance from in Theory: definitions Field and Class notations with and provements g´en´eralis´ees invariantes etdpitn 05 ercl,i eto ,sm structur some 4, Section in Z recall, We 2005. printing rected Abstract. K/k eacci xeso fagbacnme ed,wt aosgroup Galois with fields, number algebraic of extension cyclic a be TUTR HOESFOR THEOREMS STRUCTURE p [ G NAIN EEAIE DA CLASSES IDEAL GENERALIZED INVARIANT -oue ( ]-modules nepii oml for formula explicit an , k . egv,i etos2ad3 negihtasainof: translation english an 3, and 2 Sections in give, We m p G of rmfiainter a ulfr fnnrmfiaintheor non-ramification of form dual (as theory -ramification 1. L ubrfils ls edtheory; field class fields; number ≃ k #Gal( ie,vaCasFedTer,b oeAtngopof group Artin some by Theory, Field Class via given, nrdcin–Generalities – Introduction ntenro es,o oegnrlyaysubfield any generally more or sense, narrow the in , Z C /p ℓ .Mt.Sc aa,4,3(94,wt oeim- some with (1994), 3 46, Japan, Soc. Math. J. , IN K, + K Z bandin: obtained ) m ftemxmlsbedof subfield maximal the of , L/K p L/K H . ERE GRAS GEORGES p ≃ K + nae elIsiu ore,4,1(1993). 1 43, Fourier, l’Institut de Annales , -EXTENSIONS SURVEY A Gal( ) of ). K G H rmr 12;11R37. 11R29; Primary espoethat suppose we ; #( = K K, + H 1 p k ai framework. -adic m K, + soitdwt h ru fprincipal of group the with associated /L n lmnaylclnri computa- normic local elementary and p u les Sur m C M,Srne-elg 2 Springer-Verlag, SMM, , ℓ ere sn h Thaine–Ribet– the using degree, /K ymaso sub- a of means by ) K, + ℓ H nae elIsiu Fourier, l’Institut de Annales , m )). K, + p / ℓ CASGROUPS -CLASS m H cassdi´axdn les d’id´eaux dans -classes p casgroups; -class p ) soitdwt a group ray a with associated from: , G . hoesfrfinite for theorems e eue ohv a have to used be y L/k a f ede invariants deep me L f“admissible of n egv sub- a give we , uain which putations u astructure la Sur p ihorbook: our with casgopin group -class (denoted sGli,s that so Galois, is ative p etnin;gener- -extensions; G Classes nd p -module -class cor- L y) ab which ) G and , H of G L 2 GEORGES GRAS

Indeed, since G is cyclic, it is not difficult to see that the commutator subgroup 1 σ + 1 σ [Γ, Γ] of Γ := Gal(L/k) is equal to Gal(L/K) − ( ℓ / ) − , where σ is a ≃ C K,m H generator of G (or an extension in Γ). So we have the exact sequences 1 σ ab ab 1 Gal(L/K) − Γ Γ =Γ/[Γ, Γ] = Gal(L /k) 1, (1) −→ −−−→ −−−→ −→ G 1 σ 1 σ 1 Gal(L/K) Gal(L/K) − Gal(L/K) − 1. −→ −−−→ −−−→ −→ #Γab [Lab : k] Hence #Gal(L/K)G = [L : K] = [L : K] = [Lab : K]. The · #Γ · [L : K][K : k] study of the structure of Gal(L/K) as G-module (or at least the computation of its order) is based under the study of the following filtration:

Definition 1.1. Let M := Gal(L/K) and let (Mi)i 0 be the increasing sequence ≥ of sub-G-modules defined (with M0 := 1) by M /M := (M/M )G, for 0 i n, i+1 i i ≤ ≤ where n is the least integer i such that Mi = M. G 1 σ For i = 0, we get M1 = M . We have equivalently Mi+1 = h M, h − Mi . (1 σ)i n { ∈ ∈ } Thus M = h M, h − =1 and (1 σ) is the anihilator of M. i { ∈ } − If Li is the subfield of L fixed by Mi, this yields the following tower of fields, Galois i (1 σ) (1 σ)i over k, from the exact sequences 1 M M − M − 1 such that → i −−−→ −−−→ → [Li : Li+1] = (Mi+1 : Mi) which can be computed from local arithmetical tools in K/k as described in the Sections 3 and 4:

M ℓ+ / ≃C K,m H

M1 K =Ln Li+1 Li L1 L=L0 (1 σ)i+1 M − M G= σ ≃ i h i (1 σ)i Mi+1 M − k ≃ Γ

In a dual manner, we have the following tower of fields where Li′ is the subfield of (1 σ)i L fixed by M − , whence [Li′ : K] = #Mi:

M ℓ+ / ≃C K,m H

#M1 ab K =L0′ L1′ =L Li′ Li′+1 L=Ln′ #M (1 σ)i+1 G= σ i M − h i (1 σ)i #Mi+1 M − k Γ

Our method to compute #(Mi+1/Mi) differs from classical ones by “translating” the well-known Chevalley’s formula giving the number of ambiguous classes, see (28), Remark 3.10), by means of the exact sequence of Theorem 3.3 applied to a suitable = . H H0 The main application is the case where G is cyclic of order a prime p and when L/K is an Abelian finite p-extension defined via class field theory (e.g., various p-Hilbert class fields in most classical practices). So, when the Mi are computed, it is possible − 1+σ+ +σp 1 G to give, under some assumptions (like M ··· = 1 and/or #M = p), the structure of Gal(L/K) as Zp[G]-module or at least as Abelian p-group. INVARIANTGENERALIZEDIDEALCLASSES 3

In the above example, this will give for instance the structure of the p-class group in the restricted sense from the knowledge of the p-class group of k and some local normic computations in K/k. Remarks 1.2. (i) In some french papers, we find the terminology sens restreint vs sens ordinaire which was introduced by J. Herbrand in [H, VII, §4], and we have used in [Gr1] the upperscripts res and ord to specify the sense; to be consistent with many of today’s publications, we shall use here the words narrow sense instead of restricted sense and use the upperscript +. However, we utilize S-objects, where S is a suitable set of places (S-units, S-class groups, S-class fields, etc.), so that S = corresponds to the restricted sense and totally positive elements ; the ordinary (or∅ wide) sense corresponds to the choice of the set S of real infinite places of the field, thus, for the ordinary sense, we must keep the upperscript ord (see §§2.1, 2.2). We shall consider generalized S-class groups modulo m since any situation is avail- able by choosing suitable m and S (including the case p = 2 with ordinary and narrow senses). (ii) It is clear that the study of p-class groups in p-extensions K/k is rather easy compared to the “semi-simple” case (i.e., when p ∤ Gal(K/k)); see, e.g., an overview in [St1], and an extensive algebraic study in [L1] via representation theory, then in [Ku], [Sch1], [Sch2], [Sch3], [SW], and in [Wa] for cyclotomic fields. Indeed, the semi-simple case is of a more Diophantine framework and is part of an analytic setting leading to difficult well-known questions in Iwasawa theory [Iw], then in p-adic L-functions that we had conjectured in [Gr14, (1977)], and which were initiated with the Thaine–Ribet–Mazur–Wiles–Kolyvagin “principal theorem” [MW, (1984)] with significant developments by C. Greither and R. Kuˇcera (e.g., [GK1], [GK2], [GK3], [GK4]), which have in general no connection with the present text, part of the so called “genera theory” (except for the method of Section 5 in which we obtain informations on the semi-simple case).

2. Class field theory – Generalized ideal class groups We use, for some technical aspects, the principles defined in [Gr2]; one can also use the works of Jaulent as [Ja1], [Ja2], of the same kind. For instance, for a real infinite place which becomes complex in an extension, we speak of complexification instead of ramification, and the corresponding inertia subgroup of order 2 is called the decomposition group of the place; in other words this place has a residue degree 2 instead of a ramification index 2. If the real place remains real by extension, we say as usual that this place splits (of course into two real places above) and that its residue degree is 1. The great advantage is that the moduli m of class field theory are ordinary integer ideals, any situation being obtained from the choice of S. A consequence of this viewpoint is that the pivotal notion is the narrow sense.

2.1. Numbers – Ideals – Ideal classes. Let F be any number field (this will apply to K and k). We denote by:

(i) PlF = PlF,0 PlF, , the set of finite and infinite places of F . The places (finite or infinite) are given∪ as∞ symbols p; the finite places are the prime ideals; the infinite places may be real or complex and are associated with the r1 + r2 embeddings of F into R and C as usual (with r1 +2 r2 = [F : Q]); (ii) T & S, two disjoint sets of places of F . We suppose that T has only finite places and that S =: S0 S , S0 PlF,0, S PlF, , where S does not contain any complex place; ∪ ∞ ⊂ ∞ ⊂ ∞ ∞ (iii) m, a modulus of F with support T (i.e., a nonzero integral ideal of F divisible by each of the prime ideals p T and not by any p / T ); ∈ ∈ 4 GEORGES GRAS

(iv) vp : F × Z is the normalized p-adic valuation when p is a prime ideal; if → p is a real infinite place, then vp : F × Z/2 Z is defined by vp(x) = 0 (resp. → vp(x) = 1) if σp(x) > 0 (resp. σp(x) < 0) where σp is the corresponding embedding F R associated with p; if p is complex (thus corresponding to a pair of conjugated → embeddings F C), then vp = 0. → + (v) F × = x F ×, vp(x)=0, p PlF, , group of totally positive elements; { ∈ ∀ ∈ ∞} + + U = x F ×, vp(x)=0, p T ; U = U F × ; F,T { ∈ ∀ ∈ } F,T F,T ∩ + + U m = x U , x 1 (mod m) ; U = U m F × ; F, { ∈ F,T ≡ } F,m F, ∩ S (vi) E = x F ×, vp(x)=0, p / S , group of S-units of F ; F { ∈ ∀ ∈ } ES = x ES , x 1 (mod m) ; F,m { ∈ F ≡ } EPl∞ =: Eord , group of units (in the ordinary sense) ε 1 (mod m); F,m F,m ≡ + E∅ =: E , group of totally positive units ε 1 (mod m); F,m F,m ≡ (vii) IF , group of fractional ideals of F ;

PF , group of principal ideals (x), x F × (ordinary sense); + ∈ + PF , group of principal ideals (x), x F × (narrow sense); ∈ + + I = a I , vp(a)=0, p T ; P = P I ; P = P I ; F,T { ∈ F ∀ ∈ } F,T F ∩ F,T F,T F ∩ F,T P m = (x), x U m , ray group modulo m in the ordinary sense; F, { ∈ F, } P + = (x), x U + , ray group modulo m in the narrow sense; F,m { ∈ F,m} ord (viii) ℓ = I /P m, generalized ray class group modulo m (ordinary sense); C F,m F,T F, ℓ+ = I /P + , generalized ray class group modulo m (narrow sense); C F,m F,T F,m S + ℓF,m := ℓF,m/ cℓ(S) Z, S-class group modulo m where cℓ(S) Z is the sub- C + C h i h i group of ℓF,m generated by the classes of p S0 and, for real p S , by the classes C m m ∈ ∈ ∞ of the principal ideals (xp ) where the xp F × satisfy to the following congruences and signatures: ∈ m m m xp 1 (mod m), σp(xp ) < 0 & σq(xp ) > 0 q PlF, p ; ≡ ∀ ∈ ∞ \{ } + m + we have PF = (xp) p Pl ∞ PF & PF,m = (xp ) p Pl ∞ PF,m. h i ∈ F, · h i ∈ F, · Taking S = , then S = PlF, , we find again ∅ ∞ + PlF,∞ + m ord ℓF,∅ m = ℓF,m, then ℓF,m = ℓF,m/cℓ( (xp ) p Pl ∞ )= ℓF,m. C C C C h i ∈ F, C S S (ix) cℓF : IF,T ℓF,m, canonical map which must be read as cℓF,m for suitable m and S, according−→C to the case of class group considered, when there is no ambiguity.

2.2. Class fields and corresponding class groups. We define the generalized Hilbert class fields as follows: + (i) HF is the Hilbert class field in the narrow sense (maximal Abelian extension of F unramified for prime ideals and possibly complexified at , which means that + ∞ the field HF may be non-real even if F is totally real); we have Gal(H+/F ) ℓ+ = I /P +; F ≃C F F F Pl∞ ord + (ii) HF = HF HF is the Hilbert class field in the ordinary sense (maximal Abelian extension of⊆F , unramified for prime ideals, and splitted at ); we have ∞ Gal(Hord/F ) ℓord = I /P ; F ≃C F F F S + (iii) HF HF is the S-split Hilbert class field (maximal Abelian extension of F unramified⊆ for prime ideals and splitted at S); we have S S + Gal(H /F ) ℓ = ℓ / cℓ (S) Z; F ≃C F C F h F i INVARIANTGENERALIZEDIDEALCLASSES 5

+ recall that the decomposition group of p S0 (resp. S ) is given, in ℓF , by the ∈ ∞ + S C cyclic group generated by the class of p (resp. (xp)); hence Gal(HF /HF ), generated by these decomposition groups, is isomorphic to cℓ (S) Z. h F i + (iv) HF,m is the m-ray class field in the narrow sense, ord HF,m is the m-ray class field in the ordinary sense, S S HF,m is the S-split m-ray class field of F (denoted F (m) in [Gr2]); we have S S + Gal(H /F ) ℓ = ℓ / cℓ (S) Z F,m ≃C F,m C F,m h F i (see (viii) and (ix) for the suitable definitions of cℓF depending on the class group S + considered). In other words, HF,m is the maximal subextension of HF,m in which the (finite and infinite) places of S are totally split. ord Plp For instance, for a prime p, the p-Sylow subgroups of ℓF and ℓF , for the set S = Pl := p, p p , have a significant meaning in someC dualityC theorems. p { | } G 3. Computation for the order of ℓ+ / C K,m H
Let K/k be any cyclic extension of number fields, of degree d, of Galois group G, and let σ be a fixed generator of G. We fix a modulus m of k with support T + which implies that HK,m/k is Galois (by abuse we keep the same notation for the extensions of m and T in K). Then let ℓ+ H⊆C K,m be an arbitrary sub-G-module of ℓ+ . C K,m + + Remarks 3.1. (i) The group G acts on ℓK,m, hence on Gal(HK,m/K) by conju- C+ HK,m/K + gation via the Artin isomorphism A A Gal(HK,m/K), for all A IK,T + 7→
∈ ∈ (modulo PK,m), for which + + HK,m/K HK,m/K 1 τ = τ τ − , for all τ G.  A  ·  A  · ∈ (ii) The sub-G-module fixes a field L H+ which is Galois over k and in the H ⊆ K,m same way, Gal(L/K) ℓ+ / is a G-module. ≃C K,m H + S (iii) Taking = cℓK (S) Z, S PlK (see (viii)) leads to ℓK,m/ = ℓK,m & L = S H h i ⊂ C H C HK,m (assuming that cℓK(S) is a sub-G-module). (iv) If we take, more generally, a modulus M of K “above m”, it must be invariant by G; so necessarily, M = (m) extended to K, except if some P M is ramified ep | since (p)= P p P . But in class field theory, it is always possible to work with a Q | + + multiple M′ of M (because H H ′ ), so that the case M = (m) is universal K,M ⊆ K,M for our purpose and is, in practice, any multiple of the conductor fL/K of L/K. We intend to compute #( ℓ+ / )G = #Gal(L/K)G, which is equivalent, from C K,m H exact sequences (1), to obtain the degree [Lab : K], where Lab is the maximal subextension of L, Abelian over k. Our method is straightforward and is based on the well-known “ambiguous class number formula” given by Chevalley [Ch1, (1933)], and used in any work on class field theory (e.g., [Ch2], [AT], [L], [Ja1, Chap. 3], [L3]), often in a hidden manner, since it is absolutely necessary for the interpretation, in the cyclic case, of the ab famous idelic index (Jk : k×NK/k(JK )) = [K : k], valid for any finite extension K/k and which gives the product formula between normic symbols in view of the Hasse norm theorem (in the cyclic case). This formula has also some importance for Greenberg’s conjectures [Gre2] on Iwa- sawa’s λ, µ invariants for the Zp-extensions of a totally real number field [Gr15]. 6 GEORGES GRAS

Chevalley’s formula in the cyclic case is based on (and roughly speaking equivalent rc (Ek:NK/k(EK )) 2 to) the nontrivial computation of the Herbrand quotient 1−σ = [K:k] of (NEK :EK ) the group of units EK , where NEK is the subgroup of units of norm 1 in K/k and where rc is the number of real places of k, complexified in K. Chevalley’s formula was established first by Takagi for cyclic extensions of prime degree p; the generalization to arbitrary cyclic case by Chevalley was possible due to the so called “Herbrand theorem on units” [H]. Many fixed point formulas where given in the same framwork for other notions of classes (e.g., logarithmic class groups, [Ja3], [So], p-ramification torsion groups, [Gr2, Theorem IV.3.3], [MoNg]).

3.1. The main exact sequence and the computation of #( ℓ+ / )G. C K,m H 3.1.1. Global computations. Recall that is a sub-G-module of ℓ+ =I /P + . H C K,m K,T K,m Put

+ 1 σ (2) = h ℓ , h − ; H { ∈C K,m ∈ H} e it is obvious that G (3) ℓ+ / = / . C K,m H
H H e We have the exact sequences

+ G 1 σ − 1 σ 1 ℓK,m ( ) − 1 (4) −→C −−−→ H −−−→ H −→ Ne e 1 K/k N ( ) 1, −→ NH −−−→ H −−−→ K/k H −→ 1 with N = Ker(NK/k), where NK/k denotes the arithmetical norm as opposed to H d 1 the algebraic norm defined in Z[G] by ν = 1+ σ + + σ − , and for which K/k ··· we have the relation νK/k = jK/k NK/k, where jK/k is the map of extension of ideals from k to K (it corresponds,◦ via the Artin map, to the transfer map fp for Galois groups); for a prime ideal P of K, jK/k NK/k(P) = jK/k(p ) = ep fp ◦ νK/k ( P′ p P′ ) (where ep is the ramification index), which is indeed P since G Q | [K:k] operates transitively on the P′ p with a decomposition group of order . | epfp By definition, for an ideal A of K, we have NK/k(cℓK (A)) = cℓk(NK/k(A)), and for any ideal a of k, we have jK/k(cℓk(a)) = cℓK(jK/k(a)), which makes sense since N (P + ) P + and j (P + ) P + , seeing the modulus m of k extended K/k K,m ⊆ k,m K/k k,m ⊆ K,m in K in some writings.

To simplify the formulas, we write N for NK/k. Recall that for ℓ prime, such that ℓ ∤ d = [K : k], the ℓ-Sylow subgroup ℓ+ Z C k,m ⊗ ℓ is isomorphic to ( ℓ+ Z )G since the map j : ℓ+ Z ℓ+ Z is C K,m ⊗ ℓ K/k C k,m ⊗ ℓ −→ C K,m ⊗ ℓ injective, and the map N : ℓ+ Z ℓ+ Z is surjective. K/k C K,m ⊗ ℓ −→C k,m ⊗ ℓ Let be any subgroup of I such that cℓ ( )= , i.e., I K,T K I H (5) P + P + = ; I· K,m  K,m H the group P + is unique and we then have I· K,m (6) N( ) = N( ) P + P + N( ) N( ) P + . H I · k,m  k,m ≃ I  I ∩ k,m

1 For K/k Galois, the arithmetical norm NK/k is defined multiplicatively on the group of ideals fp of K by NK/k(P) = p for prime ideals P of K, where p is the prime ideal of k under P and fp its residue degree in K/k. If A = (α) is principal in K, then NK/k(A)= (NK/k(α)) in k. INVARIANTGENERALIZEDIDEALCLASSES 7

Remark 3.2. The generalized class groups being finite and since any ideal class can be represented by a finite or infinite place, we can find a finite set SK = SK,0 SK, + m ∪ +∞ of non-complex places such that the classes P PK,m (for P SK,0) and (xP) PK,m · ∈ · m (for P SK, ) generate , so that we can take = P Z (xP) Z ∈ ∞ H I P LSK,0h i · P LSK,∞h i ∈ ∈ as canonical subgroup of I defining . Thus ℓ+ / = ℓSK in the meaning K,T H C K,m H C K,m of § 2.1 (viii). But, to ease the forthcoming computations, we keep the writing with the subgroup . I Note that we do not assume that or SK are invariant under G contrary to and + I τ H PK,m; so, if for instance P SK,0, for any τ G we have, cℓK(P ) = cℓK (P′) I· τ ∈ + ∈ for some P′ S , whence P = P′ (x), x U . ∈ K ∈ K,m From the exact sequence, where ψ(u) = (u) for all u U + , ∈ k,m ψ (7) 1 E+ U + P + 1, −→ k,m −−−→ k,m −−−→ k,m −→ we then put 1 + + (8) Λ := ψ− N( ) P = x U , (x) N( ) ; I ∩ k,m
{ ∈ k,m ∈ I } we have the obvious inclusions E+ Λ U + . k,m ⊆ ⊆ k,m We can state (fundamental exact sequence): Theorem 3.3. Let K/k be any cyclic extension, of Galois group G =: σ . Let = P + P + be a sub-G-module of ℓ+ , where is a subgrouph ofiI , H I· K,m  K,m C K,m I K,T and let + 1 σ = h ℓ , h − . H { ∈C K,m ∈ H} 1 σ e We have ( ) − and the exact sequence (see (2) and (5) to (8)): H ⊆ NH e + + ϕ 1 σ (9) 1 E N(U ) Λ Λ ( ) − 1, −→ k,m K,m
∩ −−−→ −−−→ NH H −→ 1 σ e where, for all x Λ, ϕ(x)= cℓ (A) ( ) − , for any A such that N(A) = (x). ∈ K · H ∈ I e Proof. If x Λ, we have (x) P + and by definition (x) is of the form N(A), ∈ ∈ k,m A , and thus cℓK (A) N ; if (x) = N(B), B , there exists C IK such that ∈ I 1 1 σ ∈ H ∈ I ∈ B A− = C − . It is known that IK,T is a Z[G]-module (and a free Z-module) · 1 1 such that H (G, IK,T ) = 0; since B A− IK,T is of norm 1, it is of the required form with C I . Then · ∈ ∈ K,T 1 σ 1 σ 1 (cℓ (C)) − = cℓ (C − )= cℓ (B A− ) cℓ ( )= , K K K · ∈ K I H 1 σ 1 σ and by definition cℓ (C) , which implies (cℓ (C)) − ( ) − . Hence the fact K ∈ H K ∈ H that the map ϕ is well defined.e e If A is such that cℓ (A) , then N(A) = (x), x U + , thus x Λ and it ∈ I K ∈ NH ∈ k,m ∈ is a preimage; hence the surjectivity of ϕ. 1 σ We now compute Ker(ϕ): if x Λ, (x) = N(A), A , and if cℓK(A) ( ) − , ∈ ∈ I 1 σ∈ H there exists B IK,T such that cℓK(B) and cℓK (A) = cℓK(B) − ; soe there + ∈ 1 σ ∈ H exists u U such that A = B − (u), givinge (x) = N(A) = (N(u)), hence ∈ K,m · x = ε N(u), ε Eord; · ∈ k since x and N(u) are in U + , we get ε E+ and x E+ N(U + ). k,m ∈ k,m ∈ k,m K,m Reciprocally, if x Λ is of the form ε N(u), ε E+ and u U + , this yields ∈ · ∈ k,m ∈ K,m (x) = N(u) = N(A), A , ∈ I 1 σ which leads to the relation A = (u) B − where, as we know, we can choose 1 · + 1 σ B I since A (u)− I . Since (u) P , cℓ (B) − = cℓ (A) , ∈ K,T ∈ K,T ∈ K,m K K ∈ H 1 σ hence cℓ (B) , and we obtain cℓ (A) ( ) − .  K ∈ H K ∈ H e e 8 GEORGES GRAS

We deduce from (4), + G 1 σ + G # ℓ #( ) − # ℓ C K,m · H C K,m (10) ( : )= e = ; H H #N( ) #N #N( ) ( : ( )1 σ) e H · H H · NH H − thus from (3), (9) and (10), e # ℓ+ G + G C K,m # ℓK,m/ = + + C H
#N( ) (Λ : (Ek,m N(UK,m)) Λ) (11) H · ∩ # ℓ+ G = C K,m . #N( ) (Λ N(U + ): E+ N(U + )) H · K,m k,m K,m We first apply this formula to = P + /P + (U + /U + ) (E+ /E+ ) H0 K,T K,m ≃ K,T K,m  K K,m which is the sub-module of ℓ+ corresponding to the Hilbert class field H+ since, C K,m K using the id´elic Chinese remainder theorem (cf. [Gr2, Remark I.5.1.2]), or the well-known fact that any class contains a representative prime to T , we get the surjection I /P + I /P + giving an isomorphisme, whence K,T K,T → K K G (12) ℓ+ / (I /P + )G (I /P +)G ℓ+ G. C K,m H0
≃ K,T K,T ≃ K K ≃C K Take := P + ; then I0 K,T (13) N( ) = N(P + ) & N( ) = N(P + ) P + /P + , I0 K,T H0 K,T · k,m k,m and (14) Λ = x U + , (x) N(P + ) = (E+ N(U + )) U + . 0 { ∈ k,m ∈ K,T } k K,T ∩ k,m It follows, from (11) applied to , from (12), and N(U + ) Λ (see (14)), H0 K,m ⊆ 0 (15) # ℓ+ G = # ℓ+ G #N( ) (E+ N(U + )) U + : E+ N(U + ) . C K,m C K · H0 · k K,T ∩ k,m k,m K,m
Now, N( ) in (13) can be interpreted by means of the exact sequence H0 1 E+U + U + E+N(U + )U + U + −→ k k,m k,m −−−→ k K,T k,m k,m N( ) = N(P + ) P + P + 1, −−−→ H0 K,T · k,m k,m −→ giving (E+ N(U + ) : (E+ N(U + )) U + ) k K,T k K,T ∩ k,m (16) #N( 0)= + + ; H (Ek : Ek,m) thus from (15) and (16), + + + + + G + G (Ek N(UK,T ): Ek,m N(UK,m)) (17) # ℓK,m = # ℓK + + . C C · (Ek : Ek,m) The inclusions N(U + ) E+ N(U + ) E+ N(U + ) lead from (17) to K,m ⊆ k,m K,m ⊆ k K,T (E+ N(U + ) : N(U + )) # ℓ+ G = # ℓ+ G k K,T K,m , C K,m C K · (E+ : E+ ) (E+ N(U + ) : N(U + )) k k,m · k,m K,m K,m in other words (E+ N(U + ) : N(U + )) (N(U + ) : N(U + )) (18) # ℓ+ G = # ℓ+ G k K,T K,T · K,T K,m . C K,m C K · (E+ : E+ ) (E+ N(U + ) : N(U + )) k k,m · k,m K,m K,m Chevalley’s formula in the narrow sense ([Gr2, Lemma II.6.1.2], [Ja1, p. 177]) is + # ℓk p Pl 0 ep (19) # ℓ+ G = C · Q ∈ k, , C K [K : k] (E+ : E+ N(K )) · k k ∩ × where ep is the ramification index in K/k of the finite place p. INVARIANTGENERALIZEDIDEALCLASSES 9

Lemma 3.4. For any finite set T , we have the relation

+ + (20) U N(K×) = N(U ). k,T ∩ K,T Proof. Let x U + of the form N(z), z K ; put (z) = C where k,T × p Plk,0 p ω ∈ ∈ Q ∈ ω Cp = P , for a fixed P p and ω Z[G] depending on p. Since the N(Cp) = N(P ) 0 0 | ∈ 0 must be prime to T , we have ω (1 σ) ω′, ω′ Z[G], for all p T . Hence 1 σ ∈ − · ∈ ∈ + (z)= C A − with C IK,T and A IK . We can choose in the class modulo PK · ∈ ∈ + (narrow sense) of A an ideal B prime to T , hence B = A (y′), y′ K× , giving 1 σ · ∈ z′ := zy′ − prime to T ; then we can multiply y′ by y′′, prime to T , to obtain 1 σ y := y′ y′′ such that the signature of y − be suitable, which is possible because of the relation N(z) 0 (i.e., the signature of z is in the kernel of the norm, see [Gr3, ≫ 1 σ + Proposition 1.1]); then z′′ := zy − yields N(z′′)= x with z′′ U .  ∈ K,T + + + + + + So (Ek : Ek N(UK,T )) = (Ek : Ek N(K×)). More generally, if x Uk,T ∩ + ∩ ∈ must be in N(UK,T ) this is equivalent to say that x must be in N(K×) (i.e., a global norm without any supplementary condition) which is more convenient to use x , K/k normic criteria (with Hasse’s symbols p for instance; see Remark 4.7). Recall + +
+ + that for T = , U = K× and the lemma says that k× N(K×) = N(K× ). ∅ K,T ∩ The lemma is valid with a modulus m if its support T has no ramified places. From (18) , (19) and (20), we have obtained

+ + + # ℓk p Pl 0 ep (N(UK,T ) : N(UK,m)) # ℓ+ G = C · Q ∈ k, · , C K,m [K : k] (E+ : E+ ) (E+ N(U + ) : N(U + )) · k k,m · k,m K,m K,m hence using (11)

+ + + G # ℓk p Pl 0 ep (N(UK,T ) : N(UK,m)) (21) # ℓ+ / = C · Q ∈ k, · . C K,m H
[K : k] #N( ) (E+ : E+ ) (Λ N(U + ) : N(U + )) · H · k k,m · K,m K,m

+ + 3.1.2. Local study of (NK/k(UK,T ) : NK/k(UK,m)). For a finite place P of K, let KP be the P-completion of K at P. Then let K,P be the group of local units of U + KP and K,T := P T K,P P T KP×; we denote by K,m the closure of UK,m U Q ∈ U ⊂ Q ∈ U in (T and m seen in K). We have analogous notations for the field k. UK,T The arithmetical norm NK/k =: N can be extended by continuity on P T KP× and Q ∈ the groups N( K,T ) and N( K,m) are open compact subgroups of k,T . It follows U + θ U U that the map N(U ) N( )/N( m) is surjective. K,T −→ UK,T UK, + Consider its kernel. Let N(u), u UK,T , be such that N(u) = N(αm), αm K,m. 1 ∈ ∈U Since H (G, P T KP×) = 0 (Shapiro’s Lemma and Hilbert Theorem 90), there Q ∈ 1 σ exists β P T KP× such that u = αmβ − . ∈ Q ∈ + + We can approximate (over T ) β by v K× and αm by um UK,m; then u = 1 σ ∈ ∈ um v − ξ, with ξ near from 1 in P T KP× and totally positive; then let u′ = (1 σ)· Q+ ∈ + u v− − ; this leads to u′ = um ξ UK,m and N(u′) = N(u) N(UK,m). The kernel + ∈ ∈ of the map θ is N(UK,m). Thus

+ + ( k,T : N( K,m)) (N(UK,T ) : N(UK,m)) = (N( K,T ) : N( K,m)) = U U U U ( k,T : N( K,T )) (22) U U ( : m) ( m : N( m)) = Uk,T Uk, · Uk, UK, . ( : N( )) Uk,T UK,T By local class field theory we know that ( k,T : N( K,T )) = p T ep. where ep is U U Q ∈ the ramification index of p in K/k. 10 GEORGES GRAS

Remark 3.5. The index ( k,m : N( K,m)) may be computed from higher ramifica- tion groups in K/k (cf. [Se1,U ChapitreU V]) by introduction of the usual filtration of λp λp the groups k,p and K,P. If m = p T p , λp 1, then k,m = p T (1+p p) U U ∈ ≥ U ∈ O λpeQp Q and K,m = p T P p(1 + P P), where p and P are the local rings of U Q ∈ Q | O O O integers. This local index only depends on the given extension K/k. To go back to ℓ+ , we have the following formula (cf. [Gr2, Corollary I.4.5.6 (i)]) C k,m + + + + (Uk,T : Uk,m) + ( k,T : k,m) (23) # ℓk,m = # ℓk + + = # ℓk U + U+ , C C · (Ek : Ek,m) C · (Ek : Ek,m) where the integer ( : m) is given by the generalized Euler function of m. Uk,T Uk, Then using (21), (22), (23), we obtain the main result: Theorem 3.6. Let K/k be a cyclic extension of Galois group G; let m be a nonzero integer ideal of k and let T be the support of m. Let ep be the ramification index in K/k of any finite place p of k. Then for any sub-G-module of ℓ+ and any H C K,m subgroup of I such that P + /P + = , we have I K,T I· K,m K,m H + + G # ℓk,m p/T ep ( k,m : N( K,m)) (24) # ℓ / = C · Q ∈ · U U . C K,m H
[K : k] #N( ) (Λ:Λ N(U + )) · H · ∩ K,m where N = N is the arithmetical norm and Λ := x U + , (x) N( ) . K/k { ∈ k,m ∈ I } Using, where appropriate, Lemma 3.4, we get the following corollaries: Corollary 3.7. [Gr3, Th´eor`eme 4.3, p. 41]. Taking T = , we obtain: ∅ + G # ℓk p Pl 0 ep (25) # ℓ+ / = C · Q ∈ k, , C K H
[K : k] #N( ) (Λ:Λ N(K )) · H · ∩ × + where Λ := x k× , (x) N( ) . { ∈ ∈ I } Corollary 3.8. [HL, (1990)]. If T does not contain any prime ideal ramified in K/k, we obtain, since in the unramified case ( m : N( m))=1 regardless m: Uk, UK, + G # ℓk,m p Pl 0 ep (26) # ℓ+ / = C · Q ∈ k, . C K,m H
[K : k] #N( ) (Λ:Λ N(K )) · H · ∩ × Corollary 3.9. If T = and if = cℓK (SK ), where SK is any finite set of places ∅ H NSK NSK of K, we obtain Λ/Λ N(K×) E /E N(K×) (see Remark 3.2), and: ∩ ≃ k k ∩ + # ℓk p Pl 0 ep (27) # ℓSK G = C · Q ∈ k, , C K [K : k] #cℓ ( NS ) (ENSK : ENSK N(K )) · k h K i · k k ∩ × NSK where the group Ek of “NSK -units” is defined by: NSK Sk E = x E , vp(x) 0 (mod fp) p Sk & vp(x)=0 p Plk, Sk, , k { ∈ k ≡ ∀ ∈ ∀ ∈ ∞ \ ∞} Sk being the set of places of k under SK and fp the residue degree of p. + Proof. We have Λ = x k× , (x) N( ) , where = P P S 0 (yP) P S ∞ . { ∈ ∈ I } I h i ∈ K, · h i ∈ K, If x Λ, then (x) = N(A) N(A), A P P S 0 , A (yP) P S ∞ ; hence, ∈ · ∈ h i ∈ K, ∈ h i ∈ K, up to NK×, x is represented by a NSK -unit ε. One verifies that the map which NSK NSK associates x with the image of ε in E /E N(K×) is well-defined and leads k k ∩ to the isomorphism. Note that E+ ENSK .  k ⊆ k Remark 3.10. We have in [Ja1, p. 177 (1986)] another writing of this formula: # ℓSk e d S G k p/Sk p p Sk p # ℓ K = C · Q ∈ · Q ∈ , C K [K : k] (ESk : ESk N(K )) · k k ∩ × where dp = ep fp is the local degree of K/k at p with ep = 1 for infinite places: use Sk NSK the relation E N(K×)= E N(K×) and the exact sequence k ∩ k ∩ INVARIANTGENERALIZEDIDEALCLASSES 11

Sk NSK 1 E /E S Z/ NS Z cℓ ( S Z)/cℓ ( NS Z) 1 −→ k k −−−→h ki h K i −−−→ k h ki k h K i −→ for the comparison. Taking SK = PlK, in the two formulas, we get ∞ ord # ℓk p Pl 0 ep p Pl ∞ fp (28) # ℓord G = C · Q ∈ k, · Q ∈ k, , C K [K : k] (Eord : Eord N(K )) · k k ∩ × which is the true original Chevalley’s formula (in the ordinary sense), where fp =2 (resp. 1) if p Plk, is complexified (resp. is not). ∈ ∞ 3.2. Genera theory and heuristic aspects. The usual case (S = T = ), in the cyclic extension K/k, can be interpreted by means of the following diagram∅ of finite extensions: ℓ+ C K + + K KHk HK

K H+ H+ ∩ k k N ℓ+ C K ℓ+ k C k

+ Here K Hk /k is the maximal subextension of K/k, unramified at finite places, ∩ + + + and the norm map NK/k : ℓK ℓk is surjective if and only if K Hk = k. So formula (25) can be interpretedC −→ Cas follows (which will be very impor∩ tant for numerical computations); using the relations [K : k] = [K : K H+] [K H+ : k] & # ℓ+ = #N( ℓ+ ) [K H+ : k], ∩ k · ∩ k C k C K · ∩ k we shall get a product of two integers

+ G #N( ℓ ) p Pl 0 ep (29) # ℓ+ / = C K Q ∈ k, . C K H
#N( ) · [K : K H+] (Λ:Λ N(K )) H ∩ k · ∩ × Thus in the computations using a filtration Mi (see Section 4), the G-modules = cℓK ( ) are denoted Mi = cℓK( i); the Mi and N(Mi) will be increasing H I + + I + subgroups of ℓ and ℓ , respectively, so that M = ℓ for some n. C K C k n C K + Then we know that Λi = xi k× , (xi) N( i) , which means that xi, being the norm of an ideal and totally{ positive,∈ is a local∈ I norm} at each unramified finite place and at each infinite place (from Remark 4.7, (α), (β)); so it remains to consider the local norms at ramified prime ideals since by the Hasse norm theorem, x N(K×) if and only if x is a local norm everywhere (apart from one place). This∈ can be done by means of norm residue symbols computations of Remark 4.7, (γ), in the context of “genera theory” (see the abundant literature on the subject, for instance from the bibliographies of [Fr], [Fu], [Gr2], [L4]), so that the integers:

p Pl 0 ep Q ∈ k, , i 0, [K : K H+] (Λ :Λ N(K )) ≥ ∩ k · i i ∩ × are decreasing because of the injective maps + + ֒ (×E /E N(K×) ֒ ֒ Λ /Λ N(K×) ֒ Λ /Λ N(K k k ∩ → · · · → i i ∩ → i+1 i+1 ∩ → · · · giving increasing indices (Λ :Λ N(K×)). i i ∩ Let Ip(K/k) be the inertia groups (of orders ep) of the prime ideals p and put

(30) Ω(K/k)= n(τp)p Ip(K/k), τp =1o; ∈ pLPl0 p QPl0 ∈ ∈ 12 GEORGES GRAS we have the genera exact sequence of class field theory (interpreting the product formula of Hasse symbols, [Gr2, Proposition IV.4.5]) + + ω π + + 1 Ek /Ek N(K×) Ip(K/k) Gal(HK/k/Hk ) 1, −→ ∩ −−−→ pLPl0 −−−→ −→ ∈ + + ab where HK/k := HK is the genera field defined as the maximal subextension of + + HK , Abelian over k, where ω associates with x Ek the family of Hasse symbols x, K/k ∈ p in Ip(K/k) (hence in Ω(K/k)), and where π associates with  p Pl0 pLPl0 ∈ ∈ (τp)p Ip(K/k) the product p τp′ of the lifts τp′ of the τp, in the inertia groups ∈ pLPl0 Q + ∈ + + + of HK/k/Hk (these inertia groups generate the group Gal(HK/k/Hk ) which is the image of π); from the product formula, if (τp)p is in the image of ω, then this product + + + + τ ′ fixes both H and K, hence KH . Thus π(Ω(K/k)) = Gal(H /KH ) Qp p k k K/k k with π ω(E+) = 1, giving the isomorphisms ◦ k + + + + + + Ω(K/k) ω(E ) Gal(H /KH )& ω(E ) E E N(K×).  k ≃ K/k k k ≃ k  k ∩ ep p Plk,0 + + 1 σ We have #Ω(K/k)= Q ∈ and H being fixed by ( ℓ ) − , we get [K : K H+] K/k C K ∩ k p Pl 0 ep + + + Q ∈ k, + G [HK/k : K] = [Hk : K Hk ] + + + = # ℓK ∩ · [K : K Hk ] (Ek : Ek N(K×)) C as expected. ∩ · ∩ Since Λ contains E+, we have π ω(Λ /E+) Gal(H+ /KH+). Therefore we i k ◦ i k ⊆ K/k k have at the final step i = n, using (29) for = M = ℓ+ , H n C K e p Plk,0 p (Λn :Λn N(K×)) = Q ∈ = #Ω(K/k), ∩ [K : K H+] ∩ k + + + whence ωn(Λn) = Ω(K/k) and πn ωn(Λn/Ek ) = Gal(HK/k/KHk ), which ex- ◦ + plains that an obvious heuristic is that # ℓK has no theoretical limitation about the integer n (but its structure may have someC constraints, see Section 4). An interesting case leading to significant simplifications is when there is a single ramified place p0 in K/k; indeed, the product formula (from Ω(K/k) = 1) implies ep0 (Λ : Λ N(K×)) = 1 and + = 1, so that formula (29) reduces to ∩ [K : K Hk ] + ∩ G #N( ℓ ) # ℓ+ / = C K , where #N( ℓ+ ) = [H+ : K H+] is known. If p is C K H
#N( ) C K k ∩ k 0 H + G # ℓ totally ramified, then # ℓ+ / = C k . C K H
#N( ) From the above formulas (e.g., Formula (27)),H we get some practical applications:

Theorem 3.11. Let K/k be a cyclic p-extension of Galois group G. Let SK be a finite set of non-complex places of K such that cℓ ( S ) is a sub-G-module. K h K i Consider the p-class group ℓSK , for which we have the formula C K + #N( ℓ ) p Pl 0 ep SK G K Q ∈ k, # ℓK = C + NS NS . C #cℓk( NSK ) · [K : K H ] (E K : E K N(K )) h i k k k × + ∩ · ∩ Then we have cℓK (SK ) Z = ℓK (i.e., SK generates the p-class group of K) if and only if the twoh followingi conditionsC are satisfied: (i) N( ℓ+ )= cℓ ( NS ), C K k h K i p Pl 0 ep NSK NSK Q ∈ k, (ii) (Ek : Ek N(K×)) = + = #Ω(K/k) (see (30)). ∩ [K : K Hk ] + ∩ If K Hk = k and if all places P SK are unramified of residue degree 1 in K/k, the two∩ conditions become: ∈ INVARIANTGENERALIZEDIDEALCLASSES 13

+ (i′) ℓ = cℓ ( S ), where S is the set of places p under P S , C k k h ki k ∈ K ep Sk Sk p Plk,0 (ii′) (E : E N(K×)) = Q ∈ = #Ω(K/k). k k ∩ [K : k] So, if the p-class group ℓ+ is numerically known, to characterize a set S of C k K generators for ℓ+ needs only local normic computations with the group ESk of C K k Sk-units of k which are known. Moreover, we can restrict ourselves to the case of p-class groups in a cyclic extension of degree p. Example 3.12. Consider K = Q(√82), k = Q and p = 2 (the fundamental unit is of norm 1, hence ordinary and narrow senses coincide). We shall use the primes − 3 and 23 which split in K, and prime ideals P3 and P23 above. It is clear that the 2-rank of the class group of K is 1 (usual Chevalley’s formula (28)). The conditions SQ SQ of the theorem are equivalent to (E : E N(K×)) = 2 since the product of Q Q ∩ ramification indices is equal to 4; for instance, ESQ = 3 for S = P . Q h i K { 3} + We have to compute, for some x Q× (norm of an ideal, thus local norm at each ∈ x,K/Q unramified place), the Hasse symbol which is equal to 1 if and only if x  41  is local norm at 41 (which is equivalent to be global norm in K/Q because of the x,K/Q x,K/Q product formula = 1 and the Hasse norm theorem).  41  ·  2  But from the method recalled in Remark 4.7, we have to find an “associate number” x′ such that x′ 1 (mod 8) & x′ x (mod 41), then to compute the Kronecker 82 ≡ ≡ symbol   (we have used the fact that the conductor of K is 8 41). x′ · We compute that x = 3 is not norm of an element of K×, whence P3 generates the 82 2-class group of K (for x = 3, x′ = 249, and = 1). We can verify that P is 249
− 3 of order 4 since the equation u2 82 v2 =3e (with gcd(u, v) = 1) has no solution − · 4 with e = 1 or e = 2, but N(73 + 8 √82) = 3 ; however, the knowledge of # ℓK is not required to generate the class group. C 82 Now we consider x = 23 for which x′ = 105 and 105 = 1. We compute that 65+7 √82
indeed 3 is of norm 23; this is given by the PARI instruction (cf. [P]): bnfisnorm(bnfinit(x2 82), 23)). − Then we can verify that 23 is not the norm of an integer; so we deduce that the class of P23 does not generate the 2-class group of K and is of order 2 (indeed, 2 2 N(761 + 84 √82) = 23 giving P23 = (761+84 √82)). Remark 3.13. Another important fact is the relation ν = j N when K/k K/k ◦ K/k some classes of k capitulate in K (i.e., jK/k non-injective). It is obvious that the classes of order prime to the degree d of K/k never capitulate; this explains that we shall restrict ourselves to p-class groups in p-extensions. The generalizations of Chevalley’s formula do not take into account this phenomena since they consider only groups of the form N ( ) without mystery (when ℓ+ K/k H C k is well known), contrary to νK/k . H This property of NK/k is valid if K/k is any Galois extension; if K/k has no unramified Abelian subextension L/K (what is immediately noticeable !) then NK/k is surjective, but possibly not νK/k. We have given in [Gr5], [Gr5′], numerical setting of this to disprove some statements concerning the propagation of p-ranks of p-class groups in p-ramified p-extensions K/k. These local normic calculations deduced from Theorem 3.6 have been extensively studied in concrete cases from the pioneer work of Inaba [I, (1940)], in quadratic, cubic extensions, etc. and applied to non-cyclic extensions (dihedral ones, etc): 14 GEORGES GRAS

see, e.g., [Fr], [Re], [Gr3], [Gr3′], [Gr4], [HL], [L1] (in the semi-simple case of G- modules), [Bol], [L2], [L3], [L4], [Kol], [KMS], [Ge1], [Ge2], [Ge3], [Kl], [Gr5], [Y1], [Y2], and the corresponding references of all these papers ! These techniques may give information on some class field towers problems, capitu- lation problems, often with the use of quadratic fields ([ATZ1], [ATZ2], [Go], [GW1], [GW2], [GW3], [Su], [SW], [Ter], [Mai], [Ma1], [Ma2], [Miy1], [Miy2], [MoMo], some examples in [Gr5] and numerical computations in [Gr5′], [Gr13], [Ku] for capitu- lation in Abelian extensions, then many results of N. Boston, F. Hajir and Ch. Maire, and many others as these matters are too broad to be exposed here).

4. Structure of p-class groups in p-extensions

4.1. Recalls about the filtration of a Zp[G]-module M, with G Z/p Z. Let K/k be a cyclic extension of prime degree p, of Galois group G = σ ≃. + + h i Let ℓK, ℓk be the class groups in the narrow sense (same theory with the ordinary C C + + sense for any data). We shall look at the p-class groups ℓK Zp, ℓk Zp, still + + C ⊗ C ⊗ denoted ℓK , ℓk thereafter, by abuse of notation. C C + We consider the Zp[G]-module M := ℓK for which we define the filtration evocated in Section 1: C G Mi+1/Mi := (M/Mi) ,M0 = 1; we denote by n the least integer i such that M = M. For all i 0 we have i ≥ 1 σ (1 σ)i n 1 M − Mi, Mi = h M, h − =1 , and #M = − #(Mi+1/Mi). i+1 ⊆  ∈ Qi=0 1 σ For all i 1, the maps Mi+1/Mi − Mi/Mi 1 are injective, giving a decreasing sequence≥ for the orders #(M /M−−−→) as i grows,− whence #(M /M ) #M . i+1 i i+1 i ≤ 1 If for instance #M1 = p, then #(Mi+1/Mi)= p for 0 i n 1. + ≤ ≤ − Remark that ℓk has no obvious G-module definition from M (it is not isomorphic GC ν p 1 to M = M , nor to M K/k for ν := 1 + σ + + σ − ); this is explained by 1 K/k ··· the difference of nature between νK/k and the arithmetical norm NK/k of class field theory.

ν ν p 1 4.2. Case M = 1. When M = 1 for ν := ν =1+ σ + + σ − , M is a K/k ··· Zp[G]/(ν)-module and we have p 1 Z [G]/(ν) Z [X]/(1 + X + + X − ) Z [ζ], p ≃ p ··· ≃ p where ζ is a primitive pth root of unity; then we know that m nj M Zp[ζ]/(1 ζ) , 1 n1 n2 nm, m 0, ≃ jL=1 − ≤ ≤ ≤···≤ ≥ whose p-rank can be arbitrary. The exact sequence

G 1 σ 1 σ 1 M = M M − M − 1 −→ 1 −−−→ −−−→ −→ becomes in the Zp[ζ]-structure: m nj 1 nj 1 (1 ζ) − Zp[ζ]/(1 ζ) −→ jL=1 − − −−−→ (31) m m 1 ζ nj nj M = Zp[ζ]/(1 ζ) − (1 ζ) Zp[ζ]/(1 ζ) 1, jL=1 − −−−→ jL=1 − − −→

nj where the submodules Mi are given by Mi = Zp[ζ]/(1 ζ) (for 0 i n, j,L nj i − ≤ ≤ ≤ where n = nm). m nj Each factor Nj := Zp[ζ]/(1 ζ) (such that M = Nj , not to be confused with − jL=1 the Mi = Nj) has a structure of group given by the following result: j,L n i j ≤ INVARIANTGENERALIZEDIDEALCLASSES 15

Theorem 4.1. Under the assumption M νK/k = 1 in the cyclic extension K/k of degree p, put nj = aj (p 1)+ bj , aj 0 and 0 bj p 2, in the decomposition of M in elementary components− as above.≥ Then≤ ≤ − n a +1 b a p 1 b N := Z [ζ]/(1 ζ) j (Z/p j Z) j (Z/p j Z) − − j , j =1, . . . , m. j p − ≃ L ∀ (32) m a +1 b a p 1 b M (Z/p j Z) j (Z/p j Z) − − j . ≃ jL=1 h L i Proof. We have N := Z [ζ]/(1 ζ)nj Z [ζ]/paj (1 ζ)bj . So, to have the j p − ≃ p − structure of group, it is sufficient to compute the pk-ranks for all k 1 (i.e., the k−1 k p p k ≥ dimensions over Fp of Nj /Nj ), which is immediate since this p -rank is p 1 for k a , b for k = a +1, and 0 for k>a + 1. − ≤ j j j j This implies that the p-rank of N is p 1 if a 1 and b if a = 0 (i.e., b = j − j ≥ j j j nj p 2). So the parameters aj and bj will be important in a theoretical and ≤ − k numerical point of view. Put M (k) := h M, hp =1 , k 0.  ∈ ≥ ν (k) k Lemma 4.2. If M =1, then M = Mk (p 1), k 0, and the p -rank Rk of M · − ∀ ≥k(p 1) 1 (k 1) (k) Rk − − is the Fp-dimension of M − /M . Then p = i=(k 1)(p 1) #(Mi+1/Mi). Q − − Proof. Immediate from the Zp[ζ]-structure and properties of Abelian p-groups.  4.3. Case M ν = 1. We have, in the same framwork, the following result in the case M ν = 1, but6 #(M /M )= p [Gr2, Proposition 4.3, pp. 31–32]: 6 i+1 i Theorem 4.3. Let K/k be a cyclic extension of prime degree p, of Galois group G = σ and let M be a finite Z [G]-module such that M νK/k = 1. Let n be the h i p 6 least integer i such that Mi = M. We assume that #M1 = p. Put n = a (p 1) + b, with a 0 and 0 b p 2. Then we have necessarily n 2 and the· following− possibilities:≥ ≤ ≤ − ≥ n 2 (i) Case np. Then M Z/pa+1Z Z/paZ − − . ≃
L
Proof. The proof needs two lemmas (in which we keep the notation Mn for M). Lemma 4.4. For all k 1 we have the exact sequence − ≥ − −1 pk 1 pk pk 1 pk 1 σ pk pk (33) 1 M1 Mn /M1 Mn Mn /Mn − Mn 1 /Mn 1 1. −→ ∩ ∩ −−−→ −−−→ − − −→ Proof. Under the assumption #M1 = p, we know fron § 4.1 that #(Mi+1/Mi)= p, 1 σ − 1 σ 0 i n 1; we have the exacte sequence 1 M1 Mi+1 Mi+1− 1, ≤ ≤ − 1 σ 1→σ −→ −→1 σ → which shows that #(M /M − )= p, hence M − = M since M − M . i+1 i+1 i+1 i i+1 ⊆ i − pk 1 1 σ pk Let x Mn such that x − = y , y Mn 1. There exists z Mn such that − −1 1∈σ 1 σ pk (1 σ) ∈pk 1 σ p∈k pk y = z − and x − = z · − ; thus (x z− ) − = 1 so that x z− M1 Mn , giving · · ∈ ∩ k−1 k Ker(1 σ) M M p /M M p , − ⊆ 1 ∩ n 1 ∩ n the opposite inclusion being obvious as well as the surjectivity. 

Lemma 4.5. If n = p then the p-rank of Mn is equal to the p-rank of Mn 1. 6 − p 1 Proof. From the relation (1 ζ) − = p A(ζ), where A(ζ) 1 (mod (1 ζ)), we p 1 − · ≡− − have ν = (1 σ) − p A(σ), A(σ) 1 (mod (1 σ)) (i.e., A(σ) invertible in − − · ≡− − Zp[G]).

(a) Case n>p. Let x Mn 1 Mn 2 (this makes sense since n p +1 3) and −2 − − (1 σ)n ∈ \ ≥ ≥ let y = x − ; then y M1, y = 1 because of the choice of x. There exists ∈ n 2 6 p 1 B(σ) B(σ) Z [G] such that (1 σ) − = B(σ) (1 σ) − and with z = x one ∈ p − · − 16 GEORGES GRAS

− (1 σ)p 1 1 σ ν ν obtains y = z − . Since Mn 1 = Mn− one gets Mn 1 = 1, so that z = 1 and − − (1 σ)p 1 p A(σ) p − z − = z · which shows that y Mn ; the assumption #M1 = p implies p p ∈ the inclusion M1 Mn (in fact y Mn 1). The exact sequence (33) applied with ⊆ ∈ p− p k = 1 leads to the isomorphism Mn/Mn Mn 1/Mn 1. ≃ − − n 1 1 σ (b) Case np. We note that, with obvious notation, (Mi)j = Mj for j i; so we k ≤ can apply Theorem 4.1 to Mn 1. Lemma 4.4 shows that the p -rank of Mn is larger − k than (or equal to) that of Mn 1; as the p -rank of a group is a decreasing function − n 1 k of k, Lemma 4.5 and the above remark show that for k p−1 , the p -ranks of ≤  −  Mn and Mn 1 are equal to p 1. − − n 1 Put n 1= a′ (p 1)+ b′, 0 b′ p 2 (in fact a′ = p−1 ). − − ≤ ≤ −  −  The exact sequence of Lemma 4.4 shows a priori three possibilities:

(α) Case b′ = 0. Necessarily, Ra′+1(Mn) = 1 and Ra′+1(Mn 1) = 0. − (β) Case b′ > 0 and Ra′+1(Mn)= Ra′+1(Mn 1) + 1. − (γ) Case b′ > 0 and Ra′+1(Mn)= Ra′+1(Mn 1) and Ra′+2(Mn) = 1. − So it remains to prove that the case (γ) is not possible. Let x Mn, x / Mn 1; −1 −1 − ν ν (1 σ)p p A(σ) (1∈ σ)p ∈ we have x M1 & x = x − x− · ; put x′ := x − and x′′ = p A(σ) ∈ · x− · ; we have x′ Mn (p 1) = M(a′ 1)(p 1)+b′+1 Ma′(p 1); but Ma′(p 1) = ∈ ′− − − − ⊂ ′ − ′− (a ) pa +1 pa (Mn 1)a′(p 1) = (Mn 1) . As x / Mn 1, we have x = 1, hence x′′ = 1. − − − ′ − ′ ∈(a ) (a 6 ) ν 6 Thus we have obtained x′ (Mn 1) and x′′ / (Mn 1) ; since x M1 and ′ ∈ − ν (a∈) − ∈ν 1 a′ = 0 (we have n p+1), one has x (Mn 1) , in other words x′′ = x x′− ′ 6 (a ) ≥ ∈ − · ∈ (Mn 1) (absurd).  − This finishes a particular case of structure when M νK/k is not specified. Of course, ν ν we have M K/k M1 and when #M1 = p, we have #M K/k =1 or p. It would be interesting to have⊆ more general structure theorems.

4.4. Numerical computations for p-class groups. Now we apply these results to the p-class group M = ℓ+ in K/k cyclic of degree p. Many cases are possible: C K If the transfer map j is injective then ( ℓ+ )νK/k N ( ℓ+ ). K/k C K ≃ K/k C K The map N is surjective except if K/k is unramified (i.e., K H+, the p-Hilbert K/k ⊂ k class field of k); if K/k is ramified we get N ( ℓ+ )= ℓ+. K/k C K C k The transfer map may be non-injective while NK/k is surjective, which causes more intricate theoretical calculations. But as we know, if NK/k is not surjective (un- ramified case), then jK/k is never injective (Hilbert’s Theorem 94, [GW1], [GW2], [GW3], [Su], [Ter]). To simplify, we suppose K/k cyclic of degree p and not unramified (otherwise, we + + # ℓk # ℓk get #M1 = C and more generally #(Mi+1/Mi) = C , which can be p p #N(Mi) carried out in the same way). · INVARIANTGENERALIZEDIDEALCLASSES 17

We suppose that K/k is ramified at some prime ideals p1,..., pt of k (t 1). We ν ≥ make no assumptions about #M1 and M K/k . With the previous notations and definitions, we then have the simplified formulas (25) for which the submodule is an element M =: cℓ ( ) of the filtration of M: H i K Ii + G # ℓk p Pl 0 ep # ℓ+ /M = #(M /M )= C · Q ∈ k, , C K i
i+1 i [K : k] #N(M ) (Λ :Λ N(K )) · i · i i ∩ × + where Λi = xi k× , (xi) N( i) . If p> 2 one can use the ordinary sense and remove the mention{ ∈ + in all the∈ forthcomingI } expressions. G (i) Computation of M1 = M from M0 = 1, which means that 0 = 1, hence + + I N(M0) = 1 and Λ0 = x0 k× , (x0) N(1) = Ek , giving the following { ∈ + + ∈ } δ0 expression where we have put (E : E N(K×)) =: p : k k ∩ + t 1 + G # ℓk p − #(M1/M0) = # ℓK = C · (34) C (Λ0 :Λ0 N(K )) ∩ × + t 1 δ0 =: # ℓ p − − . C k · First we remark that we have the isomorphism: + G G + + ℓ cℓ (I ) E N(K×)/N(E ), C K  K K ≃ k ∩ K + G G which shows how to obtain M1 = ℓK from cℓK (IK ) (called the group of strongly ambiguous classes) and global normicC computations with units of k. But the group + N(EK ) is not effective and we must proceed otherwise. In other words, the group G + G of strongly ambiguous classes cℓK(IK ) is not a “local” invariant, contrary to ℓK . + C So in the first step (which is a bit particular since 0 =1 and Λ0 = Ek ), we shall I + look at the x0 Λ0 which are norms of some y1 K× . ∈ + ∈ 1 σ So (x ) = N(y ) = (1), y K× , which yields (y ) A − = 1, where A is 0 1 1 ∈ 1 · 1 1 defined up to an invariant ideal, so that 1 contains at least such non-invariants 1 r1 I 1 s ideals A1,..., A1 , and invariant ideals (in which are ideals a ,..., a , generating + 1 t ℓk , extended to K, and ramified prime ideals P ,..., P ). C + 1 σ Reciprocally, if cℓK(A1′ ) M1, there exists y1 K× such that (y1) A1′ − = (1), giving N(y )= x Λ . ∈ ∈ · 1 0 ∈ 0 Thus, it is not difficult to see that the classes of these ideals generate M1, whence = A1,..., Ar1 ; (a1),..., (as); P1,..., Pt . I1 { 1 1 } + This gives N(M1) by means of the computation, in ℓk , of N( 1) (M1 does not need + C I + to be computed as a subgroup of ℓ ), then, with Λ = x k× , (x ) N( ) : C K 1 { 1 ∈ 1 ∈ I1 } + t 1 # ℓk p − #(M2/M1)= C · #N(M1) (Λ1 :Λ1 N(K×)) (35) + · ∩ # ℓk t 1 δ1 =: C p − − . #N(M1) · + Remark 4.6. The p-class group ℓK is equal to the group of ambiguous classes + C + if and only if cℓk(N 1) = ℓk & δ1 = t 1. If ℓk = 1, the group Λ1 is easily I + C − C obtained from N 1 Pk , whence the computation of δ1; since 1 only depends on + I ⊂ I Ek NK× and the ramification in K/k, we can hope to characterize the fields K fulfilling∩ these conditions.

(ii) For the computation of 2, we process from the elements of Λ1 which are + I norms of some y2 K× and the analogous fact that if x1 Λ1 is norm, then ∈ + ∈ (x1) = N(y2) = N(B1), B1 1, y2 K× , hence there exists A2 IK such that 1 σ ∈ I ∈ ∈ B1 = (y2) A2− . · 1 σ Reciprocally, let h2 = cℓK(A2′ ) M2 for some A2′ IK ; since h2− M1, there + ∈ 1 σ ∈ ∈ exists y K× such that (y ) A′ − = A′ , hence N(A′ ) = N(y ) =: (x ), 2 ∈ 2 · 2 1 ∈ I1 1 2 1 18 GEORGES GRAS x Λ (since for all i, E+ Λ and invariant ideals are in , the choices of x 1 ∈ 1 k ⊆ i Ii 2 and A2′ do not matter). Then these ideals of the form A1,..., Ar2 must be added to to create : 2 2 I1 I2 = A1,..., Ar1 ; A1,..., Ar2 ; (a1),..., (as); P1,..., Pt , I2 { 1 1 2 2 } whence N(M2) and + Λ = x k× , (x ) N( ) , 2 { 2 ∈ 2 ∈ I2 } and so on. Hence, the algorithm is very systematic and the use of normic symbols to find the subgroups Λi N(K×) is effective: indeed, for the most general case of computation of Hasse symbols,∩ see the Remark 4.7 below; otherwise use Hilbert symbols (xi, α)p by adjunction to k of a primitive pth roots of unity ζp to obtain the Kummer extension

p K′ := K(ζ ) =: k′(√α ), α k′×, p ∈ over k′ := k(ζp), and use the obvious Galois structure in K′/k′/k for the radical α and the decomposition of ramified prime ideals, i.e., the duality of characters given by the reflection principle [Gr2, §§ II.1.6.8, II.5.4.2, II.5.4.3, II.7.1.5, II.7.5]; this leads to generalizations of R´edei’s matrices over Fp; the rank of the matrices, denoted δi, may be introduced in the general formula to give: + # ℓk t 1 δi (36) #(Mi+1/Mi)= C p − − , #N(Mi) · with increasing δ up to the value i = n giving δ = t 1 and #N(M ) = # ℓ+. i i − i C k This was done in [Gr3′, (1973)] essentially for p =2, 3, and in [KMS, Theorem 5.16 (2015)], for p = 5, when the base field contains ζp and for particular α (essentially 5 k = Q(ζ5) and K = k(√q) where q N is for instance a prime satisfying some conditions, so that the 5-rank can be bounded∈ explicitly by a precise computation of the filtration); this approach by [KMS] applies to the arithmetic of elliptic curves in the Z5-extension of k. Remark 4.7. For convenience, recall (from [Gr2, II.4.4.3]) the hand computation of normic Hasse symbols x , K/k , by global means, in any Abelian extension K/k. p
Let m be a multiple of the conductor f of K/k (it does not matter if the support T of m strictly contains the set of (finite) places ramified in K/k, which will be the mp case if the conductor is not precisely known). Set m =: p with mp > 0. pQT ∈ Let x k× and let p be a place of k (x is not assumed to be prime to p); let ∈ us consider several cases, where K/k denotes the Frobenius automorphism of p p
in K/k (for an unramified p; for an infinite complexified place, the Frobenius is a complex conjugation), and let vp be the p-adic valuation: x , K/k K/k vp(x) (α) p Pl (real infinite place). We have = , where vp(x)=0 ∈ ∞ p
p
(resp. 1) if σp(x) > 0 (resp. σp(x) < 0). v (x) (β) p Pl T . Similarly, since p is unramified, we have x , K/k = K/k p . ∈ 0 \ p
p
(γ) p T . Let x′ k× (called a p-associate of x) be such that (using the multiplicative∈ Chinese∈ remainder theorem):

1 mp (i) x′x− 1 (mod p ), ≡ mp′ (ii) x′ 1 (mod p′ )), for each place p′ T, p′ = p, ≡ ∈ 6 (iii) σp′ (x′) > 0 for each infinite place p′ Pl , complexified in K/k. ∈′ ∞ ′ x , K/k x , K/k 1 − Then, by the product formula, we have p = p′ Pl,p′=p p′ , and ′
∈ 6
x , K/k x , K/k Q since p = p by (i) and the definition of the local p-conductor of K/k,

′ x , K/k x , K/k 1 − we have p = p′ Pl,p′=p p′ ; let us compute the symbols occurring
Q ∈ 6
in the right hand side: INVARIANTGENERALIZEDIDEALCLASSES 19

′ mp′ x , K/k if p′ T p , x′ 1 (mod p′ ) (by (ii)) and we have p′ = 1, • ∈ \{ } ′ ≡
x , K/k K/k if p′ Pl , ′ = 1 since either ′ = 1 if p′ is complex or non- • ∈ ∞ p
p
complexified real, or vp′(x′)=0 for p′ complexified real (by (iii)), ′ ′ x , K/k K/k vp′ (x ) if p′ Pl0 T , p′ is unramified and we know that p′ = p′ ; • ∈ \ ′

x , K/k K/k vp′ (x ) ′ ′ − finally, we have obtained p = p Pl0 T p . It follows that since
Q ∈ \
vp(x′)= vp(x) by (i), we can write: ′ vp(x ) vp(x) (x′) =: p a = p a, (a is prime to T by (ii)), x, K/k K/k 1 and we have obtained (for p T ), = − , where the Artin symbol ∈  p   a  K/k is by definition built multiplicatively from the Frobenius automorphisms  a  x, K/k K/k vp(x) of the prime divisors of a. Recall that if p / T , we have = . ∈  p   p  When we find that x is a global norm in K/k, bnfisnorm(bnfinit (P ), x) of PARI [P] (for k = Q and K given via the polynomial P ), gives a solution y; if x = N(y) and (x) = N(A) for an ideal A of K, then it is immediate to get numerically B such 1 σ that (y) B − = A. This was used for the Example 3.12. · One can find numerical computations, densities results, notions of “governing fields” and heuristic principles in many papers like [Gr3′], [Mo1], [Mo2], [St2], [Wi], [Y2], [Ge4], etc. We think that the local framwork given by the algorithm may confirm these heuristic results since normic symbols are independant (up to the product formula) and take uniformly all values with standard probabilities. 4.5. p-triviality criterion for p-class groups in a p-extension. When K/k is + + G cyclic of p-power degree, the triviality of ℓK, equivalent to ℓK = 1, is easily characterized from the Chevalley’s formulaC (28) and gives: C + # ℓk p Pl 0 ep p Pl 0 ep C · Q ∈ k, = #N( ℓ+ ) Q ∈ k, =1, [K : k] (E+ : E+ N(K )) C K · [K : K H+] (E+ : E+ N(K )) k k ∩ × ∩ k k k ∩ × + + + which leads to the two conditions Hk K & (Ek : Ek N(K×)) = #ΩK/k, which ⊆ + ∩ is coherent with the fact that the genera field HK/k is K (see (29) and (30), § 3.2). Any generalization (S-class groups with modulus, quotients by a sub-module ) is left to the reader. H The following result gives, when the p-group G is not cyclic, a charcterisation of S the condition ℓK = 1 despite the fact that the usual Chevalley’s formula does not exist in the non-cyclicC case; so this involves more deep invariants as the knot group κ S S and the p-central class field CK/k (i.e., the largest subextension of HK /K, Galois S S over k, such that Gal(CK/k/K) is contained in the center of Gal(CK/k/k)). Theorem 4.8. Let K/k be a p-extension with Galois group G (not necessarily Abelian), let S be a finite set of non-complex places of k and let ℓS be the p-Sylow C K subgroup of the S-class group of K. Then ℓS =1 if and only if the following three C K conditions are satisfied, where JK is the id`ele group of K: S (i) Hk K, ⊆ ab ab ab p/S ep p S ep fp S S Q ∈ × Q ∈ ab ab (ii) (Ek : Ek NK/k(JK )) = ab S , where ep (resp. fp ) is ∩ [K : Hk ] the ramification index (resp. the residue degree) of the place p of k in the maximal subextension Kab of K, Abelian over k, S S (iii) #κ = (E N (J ): E N (K×)), where the knot group κ is by k ∩ K/k K k ∩ K/k definition k× N (J ) N (K×). ∩ K/k K  K/k 20 GEORGES GRAS

The knot group, which may be nontrivial in the non-cyclic case, measures the “defect” of the Hasse principle, i.e., of local norms compared to global norms. The S S S proof is based on the fact that ℓK = 1 if and only if ℓK = IG ℓK, where IG is the augmentation ideal of G, becauseC when G is a p-groupC there·C exists a power of S 0 S IG which is contained in p Z[G]. Since by duality, H0(G, ℓK ) and H (G, ℓK∗) have S S C S G C same order, we obtain the relation ( ℓK : IG ℓK) = #( ℓK∗) , which means that S S G C ·C C [CK/k : K] = #( ℓK∗) ; thus we recover the condition by using the classical fixed C S point theorem for finite p-groups. From the formula giving [CK/k : K] (cf. [Gr2, Theorem IV.4.7]), we deduce the three conditions of the theorem. For a detailed proof, see [Gr2, § IV.4.7.4] giving a historic of the genera and central classes theories from works of Scholz, Fr¨ohlich, Furuta, Gold, Garbanati, Jehne, Miyake, Razar, Shirai, and many others; see [L4] for an history of genus theory and related results.

Remark 4.9. Condition (iii) is empty when G is cyclic (Hasse principle), or when κ = 1. The condition κ = 1 can be checked in the Abelian case via Razar’s criterion, see [Ra]; on the contrary it becomes nontrivial in the other cases so that, in practice, there does not exist any easy numerical criterion for the triviality of the p-class group in a non-cyclic p-extension. In the particular case k = Q, S = , condition (i) is empty, condition (ii), equivalent ab ab ∅ κ to ep = [K : Q], is easy to check, and condition (iii) is equivalent to = 1; p QPl0 this∈ implies that for k = Q with the narrow sense, the above problem is essentially reduced to that of the Hasse principle.

5. Relative p-class group of an Abelian field of prime to p degree We fix a prime number p. To simplfy, we suppose p> 2. We shall apply the above results of Sections 3 and 4 to study the Galois structure of the relative p-class group of an imaginary Abelian extension k/Q, of prime to p degree, using both the genera theory with characters in a suitable extension K/k, cyclic of degree p, and the “principal theorem” of Thaine–Ribet–Mazur–Wiles– Kolyvagin in k [MW]. This section, based on [Gr11, (1993)], emphasizes an interesting phenomena which is, roughly speaking, that when one grows up in suitable p-extensions K/k, the p-class group of K becomes “more regular” and gives informations on the p-class group of the base field k; the most spectacular case being Iwasawa theory in Zp- extensions [Iw] giving for instance (under the nullity of the µ-invariant) Kida’s formula for the λ−-invariants in finite p-extensions K/k of CM-fields, which is nothing else than a “genera theory” comparison of p-ranks of relative class groups “at infinity”, i.e., in K /k where k and K are the cyclotomic Zp-extensions of k and K, respectively∞ (see∞ various∞ approaches∞ in [Iw], [Ki], [Sin]). For instance, when K/k is cyclic of degree p one gets for the whole λ-invariants, assuming K k = k ([Iw, Theorem 6 (1981)]): ∩ ∞

λ(K) 1= p (λ(k) 1)+(p 1) (χ(G, EK∞ )+1)+ ew(K /k ) 1 , − · − − · Pw ∞ ∞ −
where w ranges over all non-p-places of K , where pχ(G,EK∞ ) is the Herbrand 2 ∞ H (G,EK∞ ) quotient 1 of the group EK∞ of units of K (similar situation as for H (G,EK∞ ) ∞ Chevalley’s formula which needs the knowledge of the Herbrand quotient of EK ) and where ew(K /k ) is the ramification index of w in K /k . ∞ ∞ ∞ ∞ This aspect, in p-extensions different from Zp-extensions, is probably not sufficiently thorough. INVARIANTGENERALIZEDIDEALCLASSES 21

5.1. Abelian extensions of Q and characters. Now we fix a prime number ab p > 2. Let Q , seen in Cp (the completion of an algebraic closure of Qp), be the maximal Abelian extension of Q (as we know, it is the compositum of all cyclotomic extensions of Q), and let Gab := Gal(Qab/Q). ab Let Ψ be the group of C -irreducible characters ψ : G C× of finite order, and p −→ p let be the set of Qp-irreducible characters χ (such a character χ is the sum of theXQ -conjugates ψ of a character ψ Ψ; then we say that these conjugates ψ p i ∈ i divide χ, denoted ψi χ). | ab We denote by kχ (cyclic over Q) the subfield of Q fixed by the kernel Ker(χ) of ψ and by Rχ the ring of values of ψ over Zp (kχ, Ker(χ), Rχ do not depend on the choice of the conjugate of ψ, whence the notation); furthermore, these objects only depend on the Q-irreducible character ρ above ψ or χ (ρ is the sum of all Q-conjugates of ψ then a sum of some χ). The degree of kχ/Q is equal to the order of ψ χ. | The ring Rχ is a cyclotomic local ring whose maximal ideal is denoted Mχ; more n precisely, if ψ χ is of order dp , p ∤ d, n 0, then Rχ = Zp[ξd pn ] = Zp[ξd][ξpn ], | n ≥ where ξd and ξpn are primitive dth and p th roots of unity, respectively; the prime p n 1 is unramified in Qp(ξd)/Qp and totally ramified in Qp(ξpn )/Qp of degree (p 1) p − , so that we get − − (p 1)pn 1 M − = p R . χ · χ Let

0 := χ , ψ χ is of order prime to p and X { ∈ X | } := χ , ψ χ is of p-power order . Xp { ∈ X | } We verify that = 0 p since for any χ and ψ χ, we have the unique factorization ψ =X ψ Xψ ·where X ψ is of order prime∈ X to p and| ψ is of p-power order, 0 · p 0 p then χ = χ χ , where ψ χ and ψ χ , since Q (ξ )/Q and Q (ξ n )/Q are 0 · p 0 | 0 p | p p d p p p p linearly disjoint over Qp. Note that χp is also the Q-irreducible character deduced from ψp since Q-conjugates and Qp-conjugates of ψp coincide. The local degree fχ0 := [Qp(ξd): Qp] is the residue degree of p in Q(ξd)/Q. We say that χ is even (resp. odd) if ψ(s 1) = 1 (resp. ψ(s 1)= 1), where s 1 is − − − − the complex conjugation. We denote by ±, ± and ± the corresponding sets X X0 Xp of even or odd characters (note that since p = 2, = +). 6 Xp Xp For any subfield K of Qab we denote by (then , ) the set of characters XK XK,0 XK,p of K (i.e., such that Gal(Qab/K) Ker(χ) or k K). ⊆ χ ⊆ 5.2. The universal χ-class groups (χ , p> 2). Let ℓF denotes the p-class group of any field F Qab (since p > 2,∈ we X have implicitelyC the ordinary sense). Let χ . ⊂ ∈ X 1 1 (i) If χ = χ 0, let eχ0 = χ0(s− ) s be the idempotent of 0 [kχ0 :Q] ∈ X s Gal(Pkχ /Q) ∈ 0 Z [Gal(k /Q)] associated with χ = χ ; so we have Z [Gal(k /Q)] e R . p χ0 0 p χ0 · χ0 ≃ χ0 Then we define the χ0-class group as the corresponding semi-simple component of

ℓkχ0 defined by C eχ0 ℓχ0 := ℓ . C C kχ0 (ii) If χ = χ χ with χ and χ , χ = 1, let k′ be the unique subfield 0 · p 0 ∈ X0 p ∈ Xp p 6 of kχ such that [kχ : k′] = p (we have k′ = kχ0 only if χp is of order p); thus ′ the arithmetical norm Nkχ/k induces the following exact sequence of Rχ0 -modules defining ℓχ: C ′ Nk /k eχ0 χ eχ0 1 ℓχ ℓ ℓ ′ 1, −→C −−−→C kχ −−−→C k −→ the surjectivity being obvious because kχ is the direct compositum over Q of kχ0 and kχp which is a cyclic p-extension of Q, thus totally ramified at least for a prime 22 GEORGES GRAS

′ number, whence kχ/k′ ramified. Since ℓχ is anihilated by eχ0 and by Nkχ/k which p 1 C corresponds to 1 + σ + + σ − in the group algebra of Gal(k /k′) =: σ , ℓ is ··· χ h i C χ canonically a Rχ-module (and not only a Rχ0 -module).

This defines, by an obvious induction in kχ/kχ0 , the universal family of components ℓχ for all χ for which we have the following formulas for any cyclic extension K/C Q of degree∈ Xd pn, p ∤ d, n 0: · ≥ e χ0 # ℓK = # ℓK , C χ QK,0 C 0∈X (37) n e χ0 # ℓK = # ℓχi , χ0 K,0, C iQ=0 C ∀ ∈ X n−i where, for each χ , χ = χ χ , where χ is above ψp for ψ χ . 0 ∈ XK,0 i 0 · p,i p,i p p | p We denote by ω, of order p 1, the Teichm¨uller character for p> 2; we have kω = Q(ζ ) where ζ is a primitive− pth root of unity and by definition ω(ζ ζa) a p p p → p ≡ (mod p) for a =1,...,p 1. − With these definitions, we can give the statement of the “principal theorem” of Thaine–Ribet–Mazur–Wiles–Kolyvagin [MW] in the particular context of imagi- nary fields K for the relative class groups ℓ− , hence with odd characters. C K Theorem 5.1. Let p = 2 and let χ = χ χ −. We assume that χ = ω 6 0 · p ∈ X 0 6 when kχ is the cyclotomic field Q(ζpn ) (otherwise ℓχ = 1). For ψ χ, let bχ be 1 1 C | the ideal B1(ψ− ) Rχ where B1(ψ− ) is the generalized Bernoulli number of the character ψ. Then· we have # ℓ = #(R /b ). C χ χ χ But as it is well known, this result does not give the structure of ℓχ as Rχ-module; t C indeed, if bχ = Mχ, we may have the general structure: e e ti ℓχ Rχ/Mχ , 1 t1 te, e 0, ti = t. C ≃ iL=1 ≤ ≤···≤ ≥ iP=1 For instance, ℓ is R -monogenic if and only if e = 1. C χ χ 5.3. Definition of admissible sets of prime numbers. Still for p = 2 and e χ6 0 χ0 0−, χ0 = ω, consider the cyclic field k := kχ0 for which ℓχ0 = ℓk , where ∈ X 1 6 1 C C eχ0 = [k :Q] χ0(s− ) s. We intend to apply the previous sections of χ0 s Gal(Pk /Q) ∈ χ0 this paper on genera theory to obtain informations on the structure of ℓ . C χ0 Definitions 5.2. (i) For any t 1, let be the familly of sets ℓ ,...,ℓ of t ≥ St { 1 t} prime numbers fulfilling the following conditions (for given χ − and ψ χ ): 0 ∈ X0 0 | 0 ℓ 1 (mod p), for i =1,...,t (i.e., p [Q(ζ ): Q]); i ≡ | ℓi

ψ0(ℓi)= 1, for i =1,...,t (i.e., ℓi totally splits in k = kχ0 ). (ii) For S , let Φ be the set of characters ϕ, of order p, with conductor ∈ St S ⊂ Xp ℓ1 ℓt (that is to say, kϕ Q(ζℓ1 ℓ ) is of conductor ℓ1 ℓt, whence if ki is ··· ⊆ ····· t ··· the unique subfield of Q(ζℓi ) of degree p, then kϕ is a subfield of degree p of the compositum k k and k is not in a compositum of less than t fields k ). 1 ··· t ϕ i t (iii) The character ϕ ΦS is said to be χ0-admissible if bχ0 ϕ = Mχ ϕ (see Theorem ∈ · 0· 5.1 for the definition of bχ0 ϕ). By extension we say that S t is χ0-admissible if there exists at least a χ -admissible· character ϕ Φ . ∈ S 0 ∈ S p (iv) Let rχ be the Rχ /p Rχ -dimension of ℓχ / ℓ . 0 0 0 C 0 C χ0 So the number t is known from the computation of a Bernoulli number depending on ϕ and it is not difficult to find χ0-admissible characters ϕ. Then we have proved in [Gr11] the following effective result: INVARIANTGENERALIZEDIDEALCLASSES 23

Theorem 5.3. Let p =2 and let χ −, χ = ω, and let k = k . 6 0 ∈ X0 0 6 χ0 Let S = ℓ1,...,ℓt t be a χ0-admissible set; then for i = 1,...,t, let li be a { } ∈ S e eχ prime ideal of k above ℓ and let h := cℓ (l ) χ0 be the image of cℓ (l ) in ℓ 0 . i i k i k i C k Then ℓ is the R -module generated by the h , i =1,...,t, and we have r t. C χ0 χ0 i χ0 ≤ Taking the minimal value of t yields rχ0 . The principle of the proof is an application of the computations of invariant classes of the Section 4 in K/k where K = kϕ k = kχ0 ϕ and where ϕ is the χ0-admissible character of order p. · ·

kϕ K = kχ ϕ 0· G Z/pZ ≃

Q k = kχ d, p∤d 0 e We consider the G-module M = ℓ χ0 as a component of the relative class group C K ℓK− ; in other words, a semi-simple component of the p-class group of K, since from C e ′ χ0 ℓ− = ℓ we have selected χ − and the associated filtration with K − K 0 k C χ′ L C ∈ X 0∈Xk eχ e 0 G G χ0 characters of M = ℓK for which M1 = M = ( ℓK) , G := Gal(K/k) Z/pZ (see [Gr12, (1978)]).C C ≃

We denote by Li the ideal of K above li (indeed, li is totally ramified in K/k) and eχ by Hi := cℓK(Li) 0 . Then the proof consists in proving the following lemmas (see [Gr11, Lemmes (1.2), (1.3), Corollaire (2.4)]: e e Lemma 5.4. The extension j : ℓ χ0 ℓ χ0 is injective. K/k C k −→C K This comes easily from the fact that χ0 is odd (the χ0-components of units are trivial for χ = ω, thus there is no capitulation of relative classes). 0 6 e e χ0 χ0 Lemma 5.5. We have M1 = j ( ℓ ) H1,...,Ht R and M1 j ( ℓ ) K/k k χ0 K/k k t C · h i  C ≃ (Rχ0 /p Rχ0 ) . e χ0 t f This expression giving #M = # ℓ p · χ0 , where f is the residue degree of 1 C k · χ0 p in Q(ξd)/Q, is nothing else than the χ0-Chevalley’s formula in K/k for an odd character χ0 (cf. [Gr12]). Lemma 5.6. The character ϕ Φ is χ -admissible if and only if M = M (in ∈ S 0 1 other words, if and only if there are no exceptional χ0-classes).

eχ0 eχ0 Thus, since ℓ = ℓe , we get #M := ℓ = # ℓχ # ℓχ ϕ from formula (37) k χ0 K 0 0· C C C C · C t with n = 1. From Theorem 5.1, we have M = M1 if and only if bχ0 ϕ = Mχ ϕ · 0· (χ0-admissibility). From the lemmas we get NK/k(M) = NK/k(M1), hence ℓχ0 = p C ℓχ h1,...,ht Rχ , whence ℓχ = h1,...,ht Rχ . C 0 · h i 0 C 0 h i 0 So, for practical use, we are reduced to the known algorithm which must stop at the first step. The ideals bχ0 ϕ generated by Bernoulli numbers are easily obtained from the Stickelberger element· of the field K: m K/Q 1 a 1 St(K) := − Gal(K/Q), aP=1  a  m − 2 ∈ where m is the conductor of K and K/Q the Artin symbol (for gcd (a,m) = 1). a
For more details see [Gr11] where it is also proved that admissible sets have a nontrivial Chebotarev density leading to the effectivness of the detemination of the structure and where relations with some results of Schoof [Sch1] are discussed (cf. [Gr11, §§ 4, 5]). 24 GEORGES GRAS

One can then find many numerical examples in the Appendix [Gr11, (A)] by e Berthier, showing some cases of non-monogenic ℓ χ0 as R -modules. For instance, C K χ0 let k = Q 541(37+ 6 √37) (quartic cyclic over Q) and p = 5; there exist two q−  eχ0 5-adic characters χ0 and χ0′ for which ℓk Rχ0 /(2 i)Rχ0 Rχ0 /(2 i)Rχ0 ′ L eχ C ≃ e − − and ℓ 0 = 1 (a rare example of non-monogenic ℓ χ0 ). See [Ber] for numerical C k C k tables where the case of even characters χ0 is also illustrated.

6. Conclusion and perspectives To conclude, we can say that the p-class group is perhaps not the only object for the class field theory setting of a number field k. Indeed, we prefer the very similar finite p-group, denoted k,p, and defined as the p-torsion subgroup of the Galois group of the maximal p-ramifiedT (i.e., unramified outside p), non-complexified, Abelian pra pro-p-extension of k denoted Hk,p in the following schema: Tk,p ord pra k kHk,p Hk,p e e

k Hord Hord ∩ k,p k,p e ℓord k C k,p

ord where k is the compositum of the Zp-extensions of k, Hk,p the p-Hilbert class field, and ℓorde is the p-class group of k (ordinary sense). C k,p This finite group k,p, connected with the Leopoldt conjecture at p and the residue of the p-adic zetaT function, has been studied by many authors by means of algebraic and analytic viewpoints (e.g., K. Iwasawa [Iw], J. Coates [Co, Appendix], H. Koch [Ko], J-P. Serre [Se2], etc.), and we have done extensive practical studies in [Gr2] from earlier publications [Gr7], [Gr8], [Gr9], and recently in a historical overview of the Bertrandias-Payan module (a quotient of k,p) by means of three different approaches by J-F. Jaulent, T. Nguyen Quang DoT and us (see the details in [Gr6] and its bibliography).

The functorial properties of these modules k,p are more canonical (especially in T G any p-extensions K/k of Galois group G) with an explicit formula for # K,p under the sole Leopoldt conjecture, so that a “Chevalley’s formula” does existT for any p-extension K/k, see [Gr2, Theorem IV.3.3] and [MoNg]; k,p contains any deep information on class groups and units (using, for instance,T reflection theorems to connect and ℓPlp when k contains the pth roots of unity, [Gr2, Proposition Tk,p C k,p III.4.2.2]); furthermore, it is a fundamental invariant concerning the structure of the Galois group of the maximal p-ramified pro-p-extension of k, saying that this pro-p-group is free if and only if k,p = 1 (fundamental notion called p-rationality of k; see [Gr2, Theorem III.4.2.5]).T

Moreover the properties of the k,p in a p-extension are in relation with the notion of p-primitive ramification introducedT in [Gr9, (1986)] and largely developed in many papers on the subject (e.g., [Ja2], [MoNg]). In a similar context, in connection with Gross’s conjecture [FG], mention the logarithmic class group introduced by J-F. Jaulent ([Ja3], [So]) governing the p-Hilbert kernel and the p-regular kernel.

The main property concerning these groups k,p is that, under the Leopoldt conjec- ture for p in K/k (even if K/k is not Galois),T the transfer map j : K/k Tk,p −→ TK,p (corresponding as usual to extension of ideals in a broad sense) is injective [Gr2, Theorem IV.2.1] contrary to the case of p-class groups. Furthermore, the property of p-rationality we have mentionned above, has important consequences as is shown INVARIANTGENERALIZEDIDEALCLASSES 25 by Galois representations theory (e.g., [Gre1, (2016)]) or conjectural and heuristic aspects (e.g., [Gr10, (2016)]). So we intend to make much advertise for these since the corresponding filtration Tk,p (Mi)i 0 in a finite cyclic p-extension K/k has not been studied to our knowledge. ≥ Acknowledgments. I thank Pr. Balasubramanian Sury for his kind interest and his valuable help for the submission of this paper. I am very grateful to the Referee for the careful reading and the suggestions for improvements of the paper.

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Villa la Gardette, chemin Chateauˆ Gagniere,` F–38520 Le Bourg d’Oisans. Email address: [email protected] url : http://www.researchgate.net/profile/Georges Gras