On modified circular units and annihilation of real classes

34 Pages·2017·0.7 MB·English

On modified circular units
and annihilation of real classes
Jean-Robert Belliard Th´ˆong Nguy˜ˆen-Quang-D˜ˆo
¯
january 24 2005
Abstract
For an abelian totally real number field F and an odd prime number
p which splits totally in F, we present a functorial approach to special
“p-units” previously built by D. Solomon using “wild” Euler systems.
This allows us to prove a conjecture of Solomon on the annihilation of
the p-class group of F (in the particular context here), as well as related
annihilation results and index formulae.
0 Introduction
Let F be an imaginary abelian field, G=Gal(F/Q), and let Cl± be the ”plus”
F
and ”minus” parts defined by complex conjugation acting on the class group
Cl of F. The classical Stickelberger theorem asserts that the Stickelberger
F
ideal, say Stick(F) ⊂ Z[G], is an annihilator of Cl−. Sinnott has shown that
F
the index (Z[G]− : Stick(F)−) is essentially ”equal” to the order h− of Cl−.
F F
Using the p-adic point of view, these results fit into the theory of Iwasawa and
of p-adic L-functions via what is still called the Main Conjecture, even though
it is now a theorem (Mazur-Wiles).
For Cl+, our knowledge is more fragmented. For simplicity take p6=2. The
F
Main Conjecture relates (in a “numerical” way) the p-part X+ of Cl+ to the
F F
p-partofU /C ,whereU (resp. C )isthegroupofunits(resp. circular
F+ F+ F+ F+
units) of the maximal real subfield F+ of F. In the semi-simple case, it also
gives an ideal MW(F) which annihilates a certain part of X+ (the maximal
F
subgroup acting trivially on the intersection of the Zp-cyclotomic extension of
F+ with the p-Hilbert class field of F+). In the papers [So1], [So2] Solomon
constructed what could be considered as real analogues of Gauß sums and of
Stickelberger’s element, from which he conjectured an annihilation of X+. Be-
F
fore sketching Solomon’s construction let us recall some main principles of the
demonstrationofStickelberger’stheoremusingGaußsums: forallprimeideals
℘ not dividing the conductor of F, one constructs the Gauß sum g(℘) (which
belongs to some cyclotomic extension of F), and if x is a denominator of the
0MSCnumberprimary: 11R23,11R18,11R20
1
0 INTRODUCTION 2
StickelbergerelementθF ∈Q[G],then(g(℘)x)=℘xθF. Westresstwoimportant
ingredients :
• one canalwayschooseinthe classof℘ primeidealsLwhichsplitinF/Q.
• the Gauß sum is an ℓ-unit (L | ℓ), and the computation of its L-adic
valuation plays a crucial role.
We return to Solomon’s construction. For simplicity, we now assume that F
is a (totally) real field, and p an odd prime number splitting in F. From a
fixednormcoherentsequenceofcyclotomicnumbers(intheextensionF(µp∞)),
Solomon constructs, using a process which could be seen as a ”wild” variant of
Kolyvagin-Rubin-Thaine’smethod, a special”(p)-unit” κ(F,γ) (which depends
on the choice of a topological generator of Gal(F(µp∞)/F(µp)), γ say). In fact
κ(F,γ)∈UF′ ⊗Zp where UF′ is the group of (p)-units of F. The main result of
[So1] describes the P-adic valuation of κ(F,γ). Let f be the conductor of F,
putζ =exp(2iπ/f),anddefineε =N (1−ζ ). ThenforP|pwehave
f F Q(ζf)/F f
the p-adic equivalence :
1
v (κ(F,γ))∼ log (ı (ε ))
P p p P F
where ıP is the embedding F ֒→Qp defined by P. From this computation and
fixing an embedding ı: F ֒→Qp, Solomon introduces the element
1
solF := p logp(ı(εgF))g−1 ∈Zp[G],
g∈G
X
and conjectures that this element annihilates the p-part X of the class group
F
of F ([So1], conjecture 4.1; actually Solomon states a more general conjecture
assuming only that p is unramified in F). Here some comments are in order :
• Solomon’sconstructionisnotafunctorialone,whichmeansthatheisdea-
lingwithelementsinsteadofmorphisms(thisobjectionmaybeaddressed
to most ”Euler system” constructions); this may explain the abundance
oftechnicalcomputationsrequiredtoshowthemainresultorsome(very)
special cases of the conjecture.
• In the semi-simple case (i.e. p∤[F :Q]), Solomon’s conjecture is true but
it gives nothing new since one can easily see that sol ∈MW(F) in that
F
case.
The object of this paper is to present Solomon’s construction in a more func-
torialway, making a more wholehearted use of Iwasawatheory. This functorial
approach is not gratuitous, since on the one hand it is an alternative to the
technical computations in [So1],[So2], and on the other hand it allows us to
prove a slightly modified version of Solomon’s conjecture and other related an-
nihilationresultsin the mostimportantspecialcase,namelywhen p splits inF
(see theorem 5.4).
0 INTRODUCTION 3
We now give an idea of our functorial construction and of its main deriva-
tives :
Assuming the abovementioned hypotheses (F is totally realand p splits in F),
following U and C , we introduce U′ , the group of (p)-units of F, together
F F F
′
with the p-adic completions U , C and U . The natural idea would be
F F F
′
to apply Iwasawa co-descent to the corresponding Λ-modules U , C , U ,
∞ ∞ ∞
obtainedbytakinginverselimitsupthecyclotomicZp-extensionofF,butsince
we are assuming that p splits in F, some natural homomorphisms, for example
(C ) −→ C , are trivial! The main point will be to construct other natural
∞ Γ F
(but non trivial) morphisms, which will allow us to compare the modules at
the levelofF withthe correspondingmodules arisingfromIwasawaco-descent.
To fix the ideas let us see how to compare C with (C ) . We construct
F ∞ Γ
successively two homomorphisms :
1) The first one, whose construction essentially uses an analog of ”Hilbert’s
′
theorem 90 in Iwasawatheory”, allows us to map (C ) to U after “dividing
∞ Γ F
by (γ −1)” (see theorem 2.7). One may even go to the quotient by U , the
F
natural projection being injective when restricted to the image of (C ) . So
∞ Γ
we have canonical maps :
(C ) α //U′ and (C ) β //U′ /U . Then the image under β of
∞ Γ F ∞ Γ F F
an ad hoc element in C is nothing else but Solomon’s element κ(F,γ) and
∞
functorial computations with β allow us to solve all index questions arising in
[So1],[So2].
2) The second homomorphism constructs a ”bridge” linking Sinnott’s exact se-
quence with the class-field exact sequence. More precisely, C′′ := α((C ) ) is
F ∞ Γ
′
consideredasa submodule ofU /U asin 1),while C is consideredas asub-
F F F
moduleofU′ /U′ ,whereU′ issomekernelconsistingoflocalecyclotomicnorms,
F F F
usuallycalledtheSinnott-Grosskernel. Aprioritheredoesn’texistanynatural
map from U′ /bU′ to U′ /bU , since U U′ = {1}, but when restricting to
F F F F F F
C and using the map β, one is able to construct a map ϕ: C −→ (C′′)
F T F Tr F
(the kernelof thbe algebraictrace ofZp[G] acbting on CF′′), which is injective and
whose cokernel may be computed (see theorem 4.5) . e
Once the morphisms α,β,ϕ have been constructeed one is able to compute
the Fitting ideals in Zp[G] of various codescent modules in terms of Solomon’s
elements. This yields annihilation results for various classes, e.g. ideal classes,
but also quotients of global units or semi-local units modulo circular units (see
theorem 5.7). Note that in the non semi-simple case, these differ from the ones
obtained by means of usual (tame) Euler systems.
Inrecentdevelopments([BB2],[BG],[RW2])aroundthe“liftedrootnumber
conjecture”andits“twisted”versions(or“equivariantTamagawanumber”con-
jecture forTatemotivesoverQ inthe languageofBurnsandFlach),Solomon’s
elements appear to play a role. In some special cases, the LRN conjecture is
equivalent to the existence of S-units satisfying a variety of explicit conditions;
in particular, it implies a refinement of Solomon’s main result (op. cit.) on
1 A FEW EXACT SEQUENCES 4
theP-adicvaluationofSolomon’selement([BB2],3.2and4.14). Moreover,the
proof of the ETN conjecture for abelian fields proposed recently by [BG] uses
Iwasawa co-descent in its final step and, in the splitting case, Solomon’s ele-
ments appear again ([BG], 8.7, 8.11, 8.12, 8.13), together with the usual heavy
calculations (see also [RW2]). Actually, all the technicalties seem to be related
to the phenomenon of “trivial zeroes” of p-adic L-functions (with an intended
vagueness, this means : zeroes of local Euler factors at p). This renders our
present approach all the more interesting, since it appears to give a functorial
(as opposed to technical) process to “bypass trivial zeroes”. We hope to come
back to this topic in a subsequent work.
1 A few exact sequences
In this section (which can be skipped at first reading), we collect for the con-
venience of the reader a few “well known” exact sequences which come from
class-field theory and Iwasawa theory, and will be used freely in the rest of the
paper. Let F be a number field, p anodd prime, S =S(F) the set of primes in
F which divide p. We ’ll adopt once and for all the following notations :
U (resp. U′ ) = the group of units (resp. S-units) of F.
F F
X (resp. X′ ) = the p-groupof ideal classes (resp. S-idealclasses)of F.
F F
X =theGaloisgroupoverF ofthemaximalS-ramified(i.e. unramified
F
outside S) abelian pro-p-extensionof F.
For any abelian group, we’ll denote by A¯= lim A/Apm the p-completion
←
m
of A. If A is of finite type, then A¯=A⊗Zp.
1.1 Exact sequences from class-field theory
By class-field theory, X (resp. X′ ) is canonically isomorphic to the Galois
F F
group over F of the maximal unramified (resp. unramified and S-decomposed)
abelian extension H (resp. H′ ) of F. We have two exact sequences :
F F
– one relative to inertia :
UF diag //U := v∈SUv1 Artin//XF //XF //0 (1)
Here U1 = U is the gLroup of principal local units at v, and “diag” is
v Fv
induced by the diagonal map. Assuming Leopoldt’s conjecture for F and
p (which holds e.g. when F is abelian), it is well known that “diag” is
injective.
– one relative to decomposition :
U′F diag //F := v∈SFv× Artin// XF //XF′ //0 (2)
L
1 A FEW EXACT SEQUENCES 5
Here again,the map“diag”isinjective modulo Leopoldt’s conjecture. By
putting together (1) and (2) (or by direct computation), we get a third
exact sequence :
0 //U′F/UF µ //F/U Artin//XF //XF′ // 0 (3)
whereµisinducedbythevaluationmapsatallv ∈S. Bylocalclassfieldtheory
F×/U1 is canonicallyisomorphic to the Galois groupoverF of the unramified
v v v
Zp-extensionofFv. IfF isGaloisoverQ,thenF/U ≃Zp[S]asGaloismodules,
and we can rewrite (3) as :
0 // U′F/UF µ //Zp[S] Artin//XF //XF′ // 0 (3 µ)
1.2 Sinnott’s exact sequence
For any v ∈ S, let F× be the subgroup of F× which corresponds by class-field
v v
theory to the cyclotomic Zp-extensionFv,∞ of Fv i.e. Fv×/Fv× ≃Gal(Fv,∞/Fv)
andF× isusuallycablledthe groupofuniversal (cyclotomic) normsofF /F .
v v,∞ v
Let δ: U′ −→ ⊕ F×/F× be the homomorphism inducbed by the diagonal
F v∈S v v
map.bItskernel,U′ say,usuallycalledtheGross kernel([FGS];seealso[Kuz1]),
F
consists of elements of U′bwhich are everywhere universal cyclotomic norms.
F
Its cokernel is debscribed by the Sinnott exact sequence (see the appendix to
[FGS]), for which we need additional notations.
Let F∞ = n≥0Fn be the cyclotomic Zp-extension of F, L′∞ the maximal
unramified abelian pro-p-extension of F which is totally decomposed at all
∞
places dividingSp (hence at all places), L′ the maximal abelian extension of F
0
contained in L′ . Then Sinnott’s exact sequence reads :
∞
0 // U′ /U′ δ // F×/F× Artin// Gal(L′/F) // X′ // 0 (4)
F F v∈S v v 0 F
By class-field theory, Gal(L′ /F ) ≃ X′ := lim X′ and Gal(L′/F ) ≃
b L ∞ b∞ ∞ ← Fn 0 ∞
(X′ ) (the co-invariantsof X′ by Γ=Gal(F /F)). By the product formula,
∞ Γ ∞ ∞
the image of δ is contained readily in ⊕ F×/F× (:= the kernel of the map
v∈S v v
“sum of components”) and (4) can be rewritten as :
0 //U′F/UF′ δ // v∈SFv×e/Fv× Artinb// (X∞′ )Γ //XF′ (5)
The image ofthe naturalmap(X′ ) −→X′ is nothing butGal(H′ /H′ ∩
L ∞ Γ F F F
F ). Gross’ conjectubre assertsfthat (X′ )b is finite. It holds (which is the case
∞ ∞ Γ
if F is abelian) if and only if UF′ has Zp-rank equal to r1 +r2. Let us look
more closely at the Galois structure of ⊕ F×/F×. For simplicity, suppose
v∈S v v
that the completions of F are libnearly disjoint from those of Q∞ (this property
should be called “local linear disjointness”) : thisbhappens e.g. if p is totally
split in F. Then the local norm map gives an isomorphism Fv×/Fv× ≃Q×p/Q×p;
besides Q×p/Q×p ≃1+pZp is isomorphic to pZp via the p-adic logdarithm,hebnce
⊕v∈SFv×/Fv× ≃ pZp[S] in this situation. Let us denote by I(S) the kernel of
the sum in Zbp[S]. Assuming that p is totally split in F, we can rewrite (5) as :
b
2 HILBERT’S THEOREM 90 IN IWASAWA THEORY 6
0 // U′F/UF′ λ //pI(S) // (X∞′ )Γ // XF′ //0 (5 λ)
Oncomparingthe exactsequences(3 µ)and(5 λ), itis tempting to tryand
“drawabridge”betbweenthem, butunfortunately,there seemstobe nonatural
linkbetweenU′ /U′ andU′ /U becauseU′ andU are“independent”inthe
F F F F F F
following sense :
b b
Lemma 1.1 Suppose that F satisfies Leopoldt’s conjecture and p is totally split
in F. Then
U′ ∩U ={1}
F F
Proof. Assuming Leopoldt’s conjecture, we can embed U and U′ in F =
b F F
⊕v∈SFv×. If p is totally split, then for any v ∈ S, Fv× = Q×p = pZp and
Uv1 =1+pZp, hence an element u∈UF belongs to UF′ if and onlybif u=1.
(cid:3) b b
Nevertheless, one of our main results will be thebconstruction of a natural
map (§ 4) between two appropriate submodules of U′ /U′ and U′ /U .
F F F F
b
2 Hilbert’s theorem 90 in Iwasawa theory
2.1 Some freeness results
We keep the notations of § 1. Let Λ = Zp[[Γ]] be the Iwasawa algebra. If
M =limM isaΛ-module,α =(α ,...,α ,...)willdenoteatypicalelement
∞ n ∞ 0 n
←
of M , the index zero referring to F =F . The canonicalmap (M ) −→M
∞ 0 ∞ Γ 0
sendsα +I M (whereI istheaugmentationideal)tothecomponentα of
∞ Γ ∞ Γ 0
α . LetusdenotebyM(0) thesubmoduleofM consistingoftheelementsα
∞ ∞ ∞ ∞
suchthatα =0(inadditivenotation). Wewanttostudytherelationsbetween
0
′ ′
theΛ-modulesU :=limU andU :=limU . Obviously,theyhavethesame
∞ n ∞ n
← ←
Λ-torsion, which is Zp(1) or (0) according as F contains or not a primitive pth
root of unity (see e.g. [Kuz1] or [Wi]). Let Fr (U ) := U /Tor (U ) and
Λ ∞ ∞ Λ ∞
′ ′ ′
Fr (U ):=U /Tor (U ).
Λ ∞ ∞ Λ ∞
Proposition 2.1 Let F be any number field, [F : Q] = r1 +2r2. Then the
′
Λ-modules Fr (U ) and Fr (U ) are free, with Λ-rank equal to r +r .
Λ ∞ Λ ∞ 1 2
Proof. TheassertionconcerningFr (U′ )isatheoremofKuz′min([Kuz1],7.2).
Λ ∞
′
As for Fr (U ), it is enoughto notice that the quotient Fr (U )/Fr (U )≃
Λ ∞ Λ ∞ Λ ∞
′
U∞/U∞ has no Zp-torsion, hence has no non-trivial finite submodule. This is
′
equivalenttosayingthatFr (U )/Fr (U )isofprojectivedimensionatmost
Λ ∞ Λ ∞
1 over Λ (see e.g. [N1]). Since Λ is local, the Λ-freeness of Fr (U ) follows by
Λ ∞
Schanuel’s lemma.
(cid:3)
2 HILBERT’S THEOREM 90 IN IWASAWA THEORY 7
2.2 Cyclotomic submodules
If the field F is abelian over Q, we have at our disposal the group CF (resp.
C′ ) of Sinnott’s circular units (resp. circular S-units), which is defined as
F
being the intersection of U (resp. U′ ) with the group of circular numbers
F F
of F ([Si1], [Si2]; see also §4.3 below). Sinnott’s index formula states that
(U : C ) = c h+, where h+ is the class number of the maximal real subfield
F F F F F
of F, and c is a rational constant whose explicit definition does not involve
F
the class group. Going up the cyclotomic Zp-extension F∞ = nFn, it is
known that the constants c remain bounded. It is a classical result (see e.g.
Fn S
[Gr1]) that C and U havethe same Λ-rank. It is anobviousconsequence of
∞ ∞
′
the definition that C and C have the same Λ-rank. If the base field F is a
∞ ∞
cyclotomicfield,itisalsoknown,byresultsondistributions“a`laKubert-Lang”,
′
thatFr (C )andFr (C )areΛ-free(seee.g. [Kuz2]). Lifewouldbetooeasy
Λ ∞ Λ ∞
if such results could be extended to any abelian field. By proposition 2.1 and
the exact sequences 0 //Fr (C ) //Fr (U ) // U /C //0,
Λ ∞ Λ ∞ ∞ ∞
and 0 // Fr (C′ ) //Fr (U′ ) //U′ /C′ //0 , we see that the
Λ ∞ Λ ∞ ∞ ∞
′
Λ-freeness of Fr (C ) (resp Fr (C )) is equivalent to the triviality of the
Λ ∞ Λ ∞
′ ′
maximalfinitesubmoduleofU /C (resp. U /C ). LetuscallMF (resp.
∞ ∞ ∞ ∞ ∞
MF′ ) these maximal finite submodules. In order to get hold of MF and
∞ ∞
MF′ , let us define :
∞
Definition 2.2 The Kuˇcera-Nekov´aˇrkernel is defined by
KN := Keri
F,n n,m
m≥n
[
where for m≥n∈N, in,m: Un/Cn −→Um/Cm is the natural map.
The orders of the kernels KN are bounded independently of n : this is
F,n
an immediate consequence of the main result of [GK] and the one of [KN].
Let us denote by KN the projective limit (relatively to norm maps and n)
F
KN := limKN . This kernel KN is the obstruction to the Λ-freeness of
F F,n F
←
′
Fr(C ) and Fr(C ), in the following sense :
∞ ∞
Proposition 2.3
(i) (U /C )Γ is finite.
∞ ∞
(ii) We have equalities KN = MF = MF′ , and canonical isomorphisms
F ∞ ∞
(KN )Γ ≃ Tor (Fr(C )) ≃ Tor (Fr(C′ )) . In particular, Fr (C )
F Zp ∞ Γ Zp ∞ Γ Λ ∞
′
(resp. Fr (C )) is free if and only if KN =0.
Λ ∞ F
Proof. (i) The Main Conjecture (or Mazur-Wiles’ theorem) applied to the ma-
ximal real subfield F+ of F implies that the Λ-torsion modules U+/C+ =
∞ ∞
U /C and X+ have the same characteristic series (where the + denotes
∞ ∞ ∞
2 HILBERT’S THEOREM 90 IN IWASAWA THEORY 8
the objects related to F+, i.e. the submodule on which complex conjugation
acts trivially because F is abelian and p 6= 2). Because Leopoldt’s conjecture
holds all along the cyclotomic tower, it is well known that X+ and (X′ )+ are
∞ ∞
pseudo-isomorphic (see e.g. [Wi]). By Gross ’ conjecture (which holds because
F is abelian), ((X′ )+)Γ is finite, and so is (U /C )Γ.
∞ ∞ ∞
(ii) We have an exact sequence :
0 // U∞/C∞ //U′∞/C′∞ //U′∞/(U∞+C′∞) // 0
It givesaninclusionMF ⊂MF′ . To provethe inverseinclusionit is enough
∞ ∞
′ ′
toshowthatU∞/(U∞+C∞)iswithoutZp-torsion. LetS be the (finite) setof
′
places above p of F . Clearly U /U is isomorphic to a submodule of finite
∞ ∞ ∞
index of Zp[S] with the natural action of Gal(F∞/Q) on both sides. Call M
′ ′ ′
this submodule. It follows that U /(U +C ) is isomorphic to M/µ(C ),
∞ ∞ ∞ ∞
where µ is obtained from the valuations at finite levels, i.e. is the limit of the
homomorphisms µ of the exact sequences (3).
Let B∞ be the cyclotomic Zp-extension of Q, and Bn its nth step. Let
ε denotes the normcoherentsequenceofnumbersε =(p,N (1−
Q,∞ Q,∞ Q(ζp2)/B1
ζp2),...,NQ(ζpn+1)/Bn(1 − ζpn+1),...). Since F∞ is abelian over Q it is at
most tamely ramified over B∞, with ramification index e say. It follows that
µ(εQ,∞)=e v∈Sv,withe∈Ndividing(p−1)andthereforeaunitinZp. Each
′
generatorsofC eitherisaunitorbehaveslikeε . Thisshowsthatµ(C )=
P∞ Q,∞ ∞
′ ′
Zp v∈Sv. Consequently we have U∞/(U∞ + C∞) ≃ M/(Zp v∈Sv) ⊂
Zp[PSFo]/r(Zthpe fivr∈stSevq)uaanlidtyt,htisheprreoveexsistthaensenconsducehqutahlaittyfoorf palrlopno≥sitinoPn,2γ.p3n(iaic)t.s
triviallyonPMF . Puttingω =γpn−1as0usual,andtakingn≥n 0andm−n
∞ n 0
large enough in the commutative triangle :
U /C ωm/ωn //U /C
m m m m
JNJmJJ,nJJJJJ%% tttttintt,mtt99
U /C
n n
where N is the obvious norm map, we see immediately that MF ⊂KN ,
m,n ∞ F
and the finiteness of KN shows the equality. Moreover,the exact sequence of
F
Λ-modules
0 //Fr (C ) //Fr (U ) //U /C //0
Λ ∞ Λ ∞ ∞ ∞
gives by descent an exact sequence of Zp-modules :
0 //(U /C )Γ //Fr (C ) //Fr (U ) //(U /C ) //0
∞ ∞ Λ ∞ Γ Λ ∞ Γ ∞ ∞ Γ
By (i) (U /C )Γ is finite, hence the equality (U /C )Γ = (MF )Γ. It fol-
∞ ∞ ∞ ∞ ∞
lowsthenfrom2.1that(MF )Γ =Tor (Fr (C ) ). Theremainingassertions
∞ Zp Λ ∞ Γ
in 2.3 are straightforward.
(cid:3)
In what follows the finite order #(KN )Γ will play for descent modules a
F
role analogousto the one playedby Sinnott’s constantc at finite level. Let us
F
put the :
2 HILBERT’S THEOREM 90 IN IWASAWA THEORY 9
Notation 2.4 κ :=#(KN )Γ
F F
Remark : We takethis opportunity to correcta few mistakesin§3 and§4
of [BN] (due to the eventual non triviality of the constant κ ). In short :
F
every statement concerning only pseudo-isomorphisms or characteristic
series remains true; every index formula should be corrected if necessary
by a factor involving κ ; in every monomorphism or isomorphism state-
F
ment related to (C ) , this module should be replaced by its image in
∞ Γ
U . Another,quickersolutionwouldbetorestrictgeneralityandsuppose
F
everywhere that κ =1.
F
In many situations (e.g.when p ∤ [F : Q], or when F is a cyclotomic field),
it is known that KN = 0, but note that this is indirect evidence, coming
F
fromthe Λ-freenessofFr (C ) ([Kuz2], [B1]...), not fromthe definition ofthe
Λ ∞
obstruction kernels. Let us describe examples of non trivial KN (again by
F
indirect evidence, this time finding a (C∞)Γ containing non trivial Zp-torsion).
Note that another example for p = 3 has just been announced by R. Kuˇcera
([Kuˇc]). Suchexamplesmaybe consideredas exceptional: let usrecallthatwe
have to avoid Hypoth`ese B of [B1], which holds true in most cases. In order to
prove the non-triviality of the constant κ we need to add some very peculiar
F
decompositionhypotheses suchas those usedin [Gr2]. Then the Galois module
structureofcircularunitsislessdifficulttocontrol. Thisjustifiestheterminolo-
gy “gu¨nstige (p+1)-tuple” used in [Gr2]. We state these conditions :
1– the conductor f of F is of the form f = i=p+1l , where l are prime
i=1 i i
numbers, l ≡ 1[p], and for all j 6= i, there exist some x such that
i i,j
l ≡xp [l ]. Q
i i,j j
2– G:=Gal(F/Q)≃(Z/pZ)2
3– All the (p+1) subfields of absolute degree p of F (F1,F2,...,Fp+1 say)
have conductors cond(Fj)= i=p+1 l
i=1,i6=j i
Remark : Following GreiQther, and using Cˇebotarev density theorem,
it is not difficult to prove that there exist infinitely many (p+1)-
uplesofprimesl suchthatpandanysubfieldofthecyclotomicfield
i
Q(ζQli) satisfy condition 1. Such (p+1)-uples were called gu¨nstige
(p+1)-tuplein[Gr2]. We canthendeducethatforeachpthereexist
(infinitely) many fields F such that F and p satisfy condition 1–, 2–
and 3–.
Proposition 2.5 If the fixed field F together with the fixed prime p6=2 satisfy
condition 1–, 2– and 3–, then Fr(C ) is not Λ-free.
∞
Proof. Details are given in [B2].
(cid:3)
2 HILBERT’S THEOREM 90 IN IWASAWA THEORY 10
2.3 Dividing by T
Going back to the general (not necessarily abelian) situation, let us record a
useful result of co-descent, due to Kuz′min :
′ ′
Proposition 2.6 ([Kuz1], 7.3) The natural map (U ) −→U is injective.
∞ Γ F
Remark : For simple reasons of Zp-ranks, the analogousproperty for U∞
′
doesnothold. Inotherwordsthenaturalmap(U ) −→(U ) induced
∞ Γ ∞ Γ
′
by the inclusion U ⊂U , is not injective.
∞ ∞
Weaimtoreplacethislastinclusionbyamoreelaborateinjection. Tothisend,
we first prove an analog of Hilbert’s theorem 90 in Iwasawa theory :
Theorem 2.7 Recall that (U′ )(0) is defined as the kernel of the naive descent
∞
′ ′
map U −→U . Let us fix a topological generator γ of Γ=Gal(F /F). Then
∞ 0 ∞
multiplication by (γ−1) gives an isomorphism of Λ-modules :
U′ −∼→(U′ )(0).
∞ ∞
′ ′
Proof. Since Γ acts trivially on U , the image (γ−1)U is obviously con-
0 ∞
tainedin(U′ )(0). Moreover,thekernelof(γ−1)isΛ-torsion,buttheΛ-torsion
∞
′ ′
submodule of U∞ is (0) or Zp(1), hence (γ−1) is injective on U∞. It remains
to show that (γ −1)U′ = (U′ )(0). To this end, consider the composite of
∞ ∞
natural injective maps :
(U′ )(0)/(γ−1)U′ (cid:31)(cid:127) //U′ /(γ−1)U′ =(U′ ) (cid:31)(cid:127) //U′
∞ ∞ ∞ ∞ ∞ Γ F
(the injectivity on the left is obvious; on the right it follows from 2.6). By
definition (see the beginning of this section), this composite map is null, which
shows what we want.
(cid:3)
Corollary 2.8 Suppose that F satisfies Leopoldt’s conjecture and p is totally
split in F. Then multiplication by (γ−1) gives an isomorphism of Λ-modules
′ ∼
U −→U
∞ ∞
Proof. Using 2.7, it remains only to show the equality (U′ )(0) = U . Let us
∞ ∞
show successively the mutual inclusions :
(i) If α ∈(U′ )(0), then by 2.7,α may be written as α =(γ−1)β , with
∞ ∞ ∞ ∞ ∞
′
β ∈U . Since γ acts trivially on the primes of F above p, it is obvious that
∞ ∞
(γ−1)β ∈U . Hence (U′ )(0) ⊂U .
∞ ∞ ∞ ∞
(ii) Let u ∈ U . For all primes v above p and all integers n, we have
∞ ∞
u = N (u ) = N (u ), i.e. u is actually an element of U′ . But
0 Fn/F n Fv,n/Fv n 0 F
U U′ ={1} by 1.1. Hence U ⊂(U′ )(0).
F F ∞ ∞
(cid:3) b
T
Webare now in a position to construct the map we were looking for.
Definition 2.9