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Local units and Gauss sums
ARTICLE IN PRESS
Journal of Number Theory 101 (2003) 270–293
http://www.elsevier.com/locate/jnt
Local units and Gauss sums Miho Aoki Department of Mathematics, Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachioji-shi, Tokyo-to, 192-0397, Japan Received 17 May 2002; revised 7 January 2003 Communicated by D. Goss
Abstract In this paper, we will determine the structure of a certain module which is related to the plus part of the ideal class groups in terms of the divisibility of Gauss sums in some local fields. This result is a generalization of a result of Iwasawa and the previous work of Ichimura and Hachimori. r 2003 Elsevier Science (USA). All rights reserved. MSC: 11R23 Keywords: Gauss sums; Euler system
1. Introduction Let F be an abelian number field. Fix an odd rational prime p and let FN denote the cyclotomic Zp -extension of F : We have a sequence of fields F ¼ F0 CF1 C?CFm C?CFN such that Fm =F is a cyclic extension of degree pm : For any number field K; let AK be the p-primary component of the ideal class group of K; and set X ¼ lim AFm : By the action of complex conjugation, X is ’
decomposed to X þ "X : Fix a topological generator g of G ¼ GalðFN =F Þ:
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0022-314X/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0022-314X(03)00056-8
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Suppose first F ¼ Qðmp Þ; where mp is the group of all pth roots of unity. Kolyvagin [6] and Rubin [9] determined the structure of ðX ÞG ¼ X =ðg 1ÞX CA F as an abelian group (Theorem 1). The structure is determined by certain elements obtained from Stickelberger elements. For the proof, they used the Euler system of Gauss sums. One of the keys of the proof is to know the maximum n such that certain n elements constructed from the Gauss sums belong to ðF Þp : On the other hand, the local properties of Gauss sums (namely, the properties of Gauss sums in the completion of F for a prime ideal above p) give information on the plus part X þ : Iwasawa [5] gave a condition for X þ ¼ 0 (this is equivalent to the conjecture of Vandiver) by using the local properties of Gauss sums. Let k : G-Z p be the cyclotomic character. In this paper, we will determine the structure of X þ =ðg kðgÞÞX þ by using Gauss sums for an abelian number field F satisfying certain conditions. Note that if F ¼ Qðmp Þ; then the p-rank of X þ =ðg kðgÞÞX þ (namely, dimZ=p ðX þ =ðg kðgÞÞX þ #Z Z=pÞ) is the same as that of X þ =ðg 1ÞX þ CAþ F : For the proof, we will use local properties of the Euler system of Gauss sums, and also use some properties on the Euler system of Gauss sums proved by Kolyvagin and Rubin. Let D be a finite abelian group and c : D-Qp be a character. Let Oc denote the extension ring of Zp which is generated by the values of c; and Oc be the D-module which is Oc as an additive group on which D acts via c: We define the idempotent ec P 1 1 by ec ¼ jDj sAD Traceðc ðsÞÞs; where Trace : Qp ðcðsÞjsADÞ-Qp is the trace map. For any Zp ½D -module Y ; we define the c-part Yc of Y by Yc ¼ Y #Zp ½D Oc : % Fix an odd character w of GalðQ=QÞ which is not the Teichmuller . character o: Here, we assume the following two assumptions on w: (1) the order of w is finite and prime-to-p: (2) wðpÞa1 and ow1 ðpÞa1: Let Fw be the fixed field of the kernel of w; and let F ¼ Fw ðmp Þ: Set D ¼ GalðF =QÞ: By assumption (1), the order of D is prime to p; and for any Zp ½D -module Y ; we have Yc Cec Y
and
Y ¼ " Yc ; c
where c runs over all representatives of Qp -conjugate classes of characters of D: For an element a in Y ; we denote the image of a in Yc by ac : Let M be a power of p such that MXjAF ;w j2 : For any integer iX0; let Si ¼ fnAZ40 j square-free; n ¼ c1 ?ci ðproduct of primesÞ; cj 1 ðmod MNF Þg
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S where NF is the conductor of F : Let S ¼ iX0 Si : As mentioned before, Rubin determined the structure of Xw =ðg 1ÞXw CAF ;w in terms of certain elements dðnÞ; nAS which are obtained from Stickelberger elements (see Section 3.2 for the definition). Especially d ¼ dð1Þ is given the largest power of p dividing the generalized Bernoulli number B1;w1 : Theorem 1 (Kolyvagin [6, Theorem 7], Rubin [9, Theorem 4.4]). Write t
Xw =ðg 1ÞXw ðCAF ;w Þ ¼ " Ow =pei ; i¼1
e1 X?Xet
as Ow -modules. Then we have peiþ1 þ?þet ¼ minfdðnÞ j nASi g; for any i; 0pipt 1: Our aim is to determine the structure of Xow1 =ðg kðgÞÞXow1 as an Ow -module. For each nAS; let TðnÞ denote the Ow -submodule of ðF #Z Qp =Zp Þw which is generated by elements ðtr #1=dÞew where tr is a Gauss sum of a prime ideal r of F dividing n: Let MN;n =FN be the Kummer extension given by m
MN;n ¼ FN ðfa1=p j a#1=pm ATðnÞgÞ: For any prime ideal r of F lying above a rational prime r 1 ðmod nNF Þ; we e% (Lemma 9), here, for any will show that ðtr w ÞdðnÞ=d is an element of MN;n element xAZp ½D ; x% denotes an element of Z½D satisfying x% x ðmod dÞ: Set br;n ¼ e%
ðtr w ÞdðnÞ=d and define the integer gr ðnÞ by the largest power of p which satisfies Þgr ðnÞ (MN;n;p is the completion of MN;n for a prime divisor above p). br;n AðMN;n;p Further let gðnÞ ¼ minfgr ðnÞ j r :prime ideals of F lying above a rational prime r 1 ðmod nNF Þg: Our main theorem is as follows. Theorem 2. Write t
Xow1 =ðg kðgÞÞXow1 ¼ " Ow =pei ; i¼1
e1 X?Xet
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as Ow -modules. Then we have peiþ1 þ?þet ¼ minfgðnÞ j nASi g; for any i; 0pipt 1: For any Ow -module Y ; we define the w-rank of Y by dimOw =p ðY #Ow Ow =pÞ: From the theorem, we get the following corollary immediately, because dimOw =p ðXow1 =ðg kðgÞÞXow1 #Ow Ow =pÞ ¼ dimOw =p ðXow1 =ðg 1ÞXow1 #Ow Ow =pÞ: Corollary 1. The w-rank of AF ;ow1 is smaller than or equal to i if and only if minfgðnÞ j nASi g ¼ 1: Corollary 2. Let F 0 =F be the maximal abelian pro-p extension unramified outside p; and T the Zp -torsion subgroup of GalðF 0 =F Þ: Then Tw C"ti¼1 Ow =pei with ei as in Theorem 2. Let Uð1Þ ¼
Q
pjp
Q
Uð1Þ p ; where p runs over all prime ideal of F over p: By the natural
ð1Þ injection F + pjp Fp ; we can regard bn1 r;1 as an element of U : Let G be the closure of the image of
fbn1 r;1 j r: prime ideal of F lying above a rational prime r 1 ðmod NF Þg in Uð1Þ : We get the following equation of the order of Xow1 =ðg kðgÞÞXow1 and that of Uð1Þ w =Gw : Corollary 3 (Ichimura and Hachimori [4, Theorem 1.2]). jUð1Þ w =Gw j ¼ jXow1 =ðg kðgÞÞXow1 j: Ichimura and Hachimori also studied the order of Uð1Þ w =Gw for the characters 1 wðpÞ ¼ 1 or ow ðpÞ ¼ 1: In the case F ¼ Qðmp Þ; Iwasawa [5] showed that Uð1Þ w ¼ Gw for every odd character w ðaoÞ of D implies Aþ ¼ 0 (the conjecture of Vandiver). F We will give the proofs of Corollaries 2 and 3 in Section 4.4 together with that of Theorem 2.
2. Preliminary lemmas 2.1. For a field K; we denote the maximal abelian extension of K by K ab : Let F ; Fm and FN be as in Section 1, and fix a prime ideal p of F above p: We denote the
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decomposition group and the inertia group of the prime ideal above p in GalðFmab =Fm Þ by Dm and Im ; respectively. Let Fm;p be the completion at the prime ab =Fm;p : We have an exact ideal above p: We can regard Dm as the Galois group of Fm;p sequence 0-Im -Dm -GalðFab q =Fq Þ-0; where Fq is the residue field of Fm;p : Let D be the decomposition group of p in D ¼ GalðF =QÞ: The above sequence is exact as D-modules. For our fixed odd character w; we denote the restriction 1 1 of ow1 to D by ow1 D : Since we assumed ow ðpÞa1; we get owD is not the trivial character. Hence, we have the following isomorphism from the above exact sequence ðIm #Z Zp Þow1 CðDm #Z Zp Þow1 : D
ow1 D
Let Fm
ð2:1:1Þ
D
be the subfield of Fmab =Fm such that ow1 D
GalðFm
=Fm ÞCðGalðFmab =Fm Þ#Z Zp Þow1 : D
Isomorphism (2.1.1) implies that the decomposition group of the prime ideal above p ow1 D
in GalðFm
=Fm Þ coincides with the inertia group. ow1
We define FN D in the same way. By taking the projective limit, we know that the ow1
decomposition group of the prime divisor above p in GalðFN D =FN Þ coincides with the inertia group. 2.2. Let LN denote the maximal unramified abelian pro-p extension over FN : In this section, we will consider the Kummer extension LN =FN : By the Kummer theory, there is a subgroup V DFN #Z Qp =Zp ;
such that m
LN ¼ FN ðfa1=p j a#1=pm AV gÞ; and there exists a Kummer pairing X V -mpN ; m
ðx; a#1=pm Þ//x; a#1=pm S ¼ ða1=p Þx1 ;
ð2:2:1Þ
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where mpN denotes the group consisting of all pm th root of unities for all mX0: This is a non-degenerate bilinear pairing, and for any t in GalðFN =QÞ; we have /x; a#1=pm St ¼ /xt ; at #1=pm S: Recall D ¼ GalðF =QÞ;
G ¼ GalðFN =F Þ:
By our assumption p[½F : Q ; we have GalðFN =QÞCD G: Let ðV G Þ> denote the annihilator of V G in X with respect to the pairing (2.2.1), namely ðV G Þ> ¼ fxAX j /x; vS ¼ 1; for every vAV G g: Lemma 1. ðV G Þ> ¼ ðg kðrÞÞX : Proof. The proof is standard, so we will sketch it. First, let ðg kðgÞÞx be an element of ðg kðgÞÞX : For every vAV G ; we have/ðg kðrÞÞx; vS ¼ 1: Hence we get ðV G Þ> *ðg kðgÞÞX : Next, let us prove the inclusion ðV G Þ> Cðg kðrÞÞX :
ð2:2:2Þ
Let fðg kðgÞÞX g> ¼ fvAV j /s; vS ¼ 1; for every xAðg kðgÞÞX g denote the annihilator of ðg kðgÞÞX in V with respect to pairing (2.2.1). We will show the inclusion V G *fðg kðgÞÞX g> ; because this implies inclusion (2.2.2). Let v be an element of fðg kðgÞÞX g> : For 1 every xAX ; we have /x; vg 1 S ¼ 1: Since pairing (2.2.1) is non-degenerate, we 1 have vg 1 ¼ 1: We conclude that v is an element of V G : & By Lemma 1, we have the following non-degenerate bilinear pairing. X =ðg kðgÞÞX V G -mpN :
ð2:2:3Þ
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Further, from the same arguments of [10, Section 10.2] we have the following nondegenerate bilinear pairing: Xow1 =ðg kðgÞÞXow1 VwG -mpN #Zp Ow :
ð2:2:4Þ
2.3. Let K=Q be a Galois extension, not necessarily finite with Galois group G ¼ GalðK=QÞ: Let IK denote the ideal group of K; ClK the ideal class group of K; and AK the Sylow p-subgroup of ClK : For an infinite number field K; these groups are defined as follows. IK ¼ lim IKi ; -
ClK ¼ lim ClKi ;
AK ¼ lim AKi ;
-
-
where Ki runs over all subfields QCKi CK such that Ki is finite over Q: We define a Zp ½G -submodule WK of K #Z Qp =Zp : WK ¼ fa#1=pm AK #Z Qp =Zp j ðaÞApm IK g: We can easily show the following sequence: j
0-EK #Z Qp =Zp -WK ! AK -0
ð2:3:1Þ
is exact where EK denotes the group of units in K; and the map j is defined by jða#1=pm Þ ¼ class of a;
m
where ðaÞ ¼ ap ; aAIK :
From now on, we assume that K is a CM-field. Let J denote the complex conjugation in G ¼ GalðK=QÞ: For any Zp ½G -module M; we define the submodule M þ and M by M 7 ¼ fx j AM; Jx ¼ 7xg: Since we assumed p42; we have M ¼ M þ "M : Lemma 2. WK CA K: Proof. Since p42; from the exact sequence (2.3.1), we have 0-ðEK #Z Qp =Zp Þ -WL -A L -0: Since ðEK #Z Qp =Zp Þ ¼ 0 (cf. [10, Theorem 4.12]), we obtain the assertion.
&
2.4. Let K be an algebraic extension of Q (resp. Qp ), not necessarily finite over Q (resp. Qp ). We consider F satisfying the conditions in Section 1 with Galois group D ¼ GalðF =QÞ; and an odd character wðaoÞ:
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Lemma 3. Assume that K=Q is an abelian extension with Galois group G ¼ GalðK=QÞ and there exists a subfield K 0 of K such that K 0 -F ¼ Q and K ¼ FK 0 : Then we can consider D ¼ GalðF =QÞ to be a subgroup of G ¼ GalðK=QÞ: For every intermediate field K0 of F and K; we have isomorphisms of Ow modules: m
m
p H p (1) ðK =ðK Þ Þw CðK0 =ðK0 Þ Þw ; for every m40; H (2) ðK #Z Qp =Zp Þw CðK0 #Z Qp =Zp Þw ;
where H ¼ GalðK=K0 Þ: Proof. See [9, Lemma 2.2] for (1). By taking the direct limit of (1), we get (2).
&
Lemma 4. (1) If K satisfies the conditions of Lemma 3, then for every element m a#1=pm AðK #Z Qp =Zp Þw ; a#1=pm ¼ 0 if and only if a ¼ bp for some bAK : (2) If K contains mpN ; then for every element a#1=pm AK #Z Qp =Zp ; a#1=pm ¼ m 0 if and only if a ¼ bp for some bAK : m
Proof. It is obvious in both cases that a ¼ bp implies a#1=pm ¼ 0: From n nþm for some bAK and the definition, a#1=pm ¼ 0 if and only if ap ¼ bp pm because the map some nX0: In case (1), this implies a ¼ b n n nþ1 ðK #Z 1=p Z=ZÞw -ðK #Z 1=p Z=ZÞw is injective for any n40: In case (2), ap ¼ m
implies a ¼ bp z for some zAmpn and we have a ¼ bp zAðK Þp because K bp contains mpN : & nþm
m
m
We get the following lemma as an immediate corollary of Lemma 4. Lemma 5. (1) Assume that K satisfies the conditions of Lemma 3. Let a#1=pm be an element of ðK #Z Qp =Zp Þw such that aeðK Þp : Then we have /a#1=pm SOw COw =pm as Ow -modules. Hence, we see j/a#1=pm SOw j ¼ qm with q ¼ jOw =pj: (2) Assume that K contains mpN : Let a#1=pm be an element of K #Z Qp =Zp such that aeðK Þp : Then we have /a#1=pm SZ CZ=pm as abelian groups.
3. Gauss sums 3.1. We will review the definition of Stickelberger elements and Gauss sums. We will also prove a small lemma which states the Zp ½D -module WF in Section 2.3 can be written as a set of Gauss sums.
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Let N40 be an integer. We identify GalðQðmN Þ=QÞ with ðZ=NÞ in the usual way, and denote the isomorphism by GalðQðmN Þ=QÞCðZ=NÞ ; sa 2a mod N: Define the Stickelberger element yN by N X a 1 sa AQ½GalðQðmN Þ=QÞ : N a¼1
yN ¼
ða;NÞ¼1
Let F be as in Section 1 with Galois group D ¼ GalðF =QÞ: Let NF denotes the conductor of F : By the definition of F ; NF is the least common multiple of the conductor of w and p: For each prime ideal R of QðmNF Þ which splits completely in QðmNF Þ=Q; we define the character wR : ðZ=rÞ -mNF (r is the rational prime below R) which is given by wR ðaÞ aðr1Þ=NF mod R: We define the Gauss sum tR by tR ¼
r1 X
wR ðaÞzar AQðmrNF Þ ;
a¼1
where zr is a fixed primitive rth root of unity. For the prime ideal r of F below R; we define tr ¼ NormðtR ÞAF ðmr Þ ; where Norm : QðmrNF Þ -F ðmr Þ is the norm map. tr is well defined because of tRs ¼ tsR for any sA GalðQðmrNF Þ=F ðmr ÞÞ: We consider the fixed odd character wðaoÞ of D as a character of GalðQðmrNF Þ=QÞ: We choose sa AGalðQðmNF Þ=QÞ such that wðsa Þ a is invertible in Ow : By Stickelberger’s theorem, we have ðwðsa ÞaÞew
ðtr
Þ ¼ ðwðsa Þ aÞyNF ;w rw
in ðIF #Z Zp Þw : Since wðsa Þ aAO w ; we have e
ðtr w Þ ¼ yNF ;w rw ; in ðIF #Z Zp Þw : We see yNF ;w ¼ ð1 w1 ðpÞÞB1;w1 ;
ð3:1:1Þ
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where B1;w1 denotes the generalized Bernoulli number. By the assumption wðpÞa1; we have yNF ;w BB1;w1 ; here, B means that both sides have the same p-adic valuation. Let f ¼ ½Ow : Zp and the integer d40 such that d f ¼ jAF ;w j: By the theorem of Mazur and Wiles [7], we have B1;w1 Bd: Hence we can write yNF ;w ¼ ud; for some uAO w : Define the element Z of Zp ½D by Ow COw ew CZp ½D ew ; u1 /u1 ew /Zew : From (3.1.1), we have Ze
ðtr w Þ ¼ drw
ð3:1:2Þ
in ðIF #Z Zp Þw : Let e%w and Z% be elements of Z½D such that e%w ew ðmod dÞ and Z% Z ðmod dÞ: Fix an integer k40; and define a subset W ð1Þ of ðF #Z Qp =Zp Þw by e%
W ð1Þ ¼ ftr w #1=d ¼ ðtr #1=dÞew j r: prime ideals of F lying above a rational prime r 1 ðmod kNF Þg: Lemma 6. Let WF be as in Section 2.3. For any integer k40; we have WF ;w ¼ W ð1Þ: Especially, the set W ð1Þ is an Ow -module and it is independent of an integer k: Proof. Put e% Z% W 0 ð1Þ ¼ ftr w #1=d ¼ ðtr #1=dÞew Z j r: prime ideals of F lying above a rational prime r 1 ðmod kNF Þg: If we show an equality WF ;w ¼ W 0 ð1Þ; then W 0 ð1Þ is an Ow -module, so we have WF ;w ¼ W 0 ð1Þ ¼ uW 0 ð1Þ ¼ W ð1Þ: Hence we will show the equality WF ;w ¼ W 0 ð1Þ: By (3.1.2), it is sufficient to show the inclusion WF ;w CW 0 ð1Þ: Recall WF ;w ¼ fa#1=pm AðF #Z Qp =Zp Þw j ðaÞApm IF g: Let a#1=pm be an element of WF ;w : Suppose pm 4d: From the isomorphism of Ow modules WF ;w CAF ;w (Lemma 2), we see a#d=pm ¼ 0: By Lemma 4(1), we can write m a ¼ bp =d ; for some bAF : Hence we have m
a#1=pm ¼ bp
=d
#1=pm ¼ b#1=d:
Thus we may assume pm pd: Since a#1=pm is an element of WF ;w ; we can write ðaÞ ¼ pm a; for some aAIF : Let cAAF ;w be the ideal class of a: By the Chebotarev density theorem, we can choose a prime ideal r of F lying above a rational prime r 1 ðmod kNF Þ; and whose class
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is c: Write r ¼ aðbÞ; for some bAF : Then m
ðad=p bd Þew ¼ daw þ ðbew d Þ ¼ drw ; Ze
in ðIF #Z Zp Þw : From this and (3.1.2), we have ðtr w Þ ¼ ðad=p bd Þew in ðIF #Z Zp Þw : Zew
Since ðEF #Z Zp Þw ¼ 0; we have tr
m
¼ ðad=p bd Þew in ðF #Z Zp Þw : Hence m
m
a#1=pm ¼ ad=p bd #1=d ¼ ðtr #1=dÞew Z in ðF #Z Qp =Zp Þw :
&
3.2. In this section, we will review a result of Rubin [9] which is obtained from arguments of the Euler system of Gauss sums. Let M be a power of p such that M4jAðF Þw j2 : For any integer iX0; let Si denote the set of positive squarefree integers S which are divisible by exactly i primes c which satisfy c 1 ðmod MNF Þ: Let S ¼ iX0 Si : For each prime cAS; fix a generator sc of GalðF ðmc Þ=F ÞCGalðQðmc =QÞ; and let Dc ¼
c2 X
isic :
i¼0
For each nAS; let Dn ¼
Y cjn
Dc ;
Nn ¼
X
t;
tAGn
where Gn ¼ GalðF ðmn Þ=F ÞCGalðQðmn Þ=QÞ: We define dðnÞAðZ=MÞ½D to be the element satisfying Dn ðsa aÞynNF ¼ Nn dðnÞ (see [1,9] for the precise and see [2] for the explicit formula for dðnÞ). Let dðnÞ be the largest power of p which divides dðnÞw AðOw =MÞew : Note that dð1Þ ¼ dBB1;w1 : For each nAS; let TðnÞ denote the Ow -submodule of ðF #Z Qp =Zp Þw : which is generated by the elements ðtr #1=dÞew where r divides n: By Lemma 6, TðnÞ is also an Ow -submodule of WF ;w : If nASi ; then TðnÞ is generated by i elements, because ðtrs #1=dÞew ¼ ðtsr #1=dÞew ¼ ðtr #1=dÞwðsÞew ; for any sAD: By the definition, for n; mAS such that n divides m; we have 0 ¼ Tð1ÞCTðnÞCTðmÞ: For each nAS; let BF ðnÞ denote the Zp ½D - submodule of AF which is generated by the classes of prime ideals of F dividing n: By the isomorphism of Lemma 2 and
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(3.1.2), we have WF ;w
C
AF ;w
, TðnÞ , /ðtr #1=dÞew SOw
, C
BF ðnÞw
C
, /½rw SOw :
ð3:2:3Þ
Hence WF ;w =TðnÞCAF ;w =BF ðnÞw :
ð3:2:4Þ
Rubin [9] showed the following inequality for F ¼ Qðmp Þ by using arguments of the Euler system of Gauss sums. As he mentioned the remark at the end of [9], the result is equally shown for our F ; because we assume p[½F : Q :
Proposition 1 (Rubin [9, Corollary 4.2]). For each nAS; we have jAF ;w =BF ðnÞw jpdðnÞ f ; where f ¼ ½Ow : Zp : Hence from (3.2.4), we have jWF ;w =TðnÞjpdðnÞ f ;
ð3:2:5Þ
for each nAS:
4. Proofs 4.1. In Section 4, we will give the proofs of our main results in Section 1. Recall that our aim is to determine the structure of Ow -module Xow1 =ðg kðgÞÞXow1 in Section 2.2. We will begin with the following lemma. Lemma 7. Let V G be as in Section 2.2, and WF be as in Section 2.3. Then VwG is an Ow submodule of WF ;w : Proof. By the definition, we have VwG CðFN #Z Qp =Zp ÞGw : By Lemma 3, we have an #Z Qp =Zp ÞGw CðF #Z Qp =Zp Þw : Hence we can isomorphism of Ow -modules ðFN regard VwG as an Ow -submodule of ðF #Z Qp =Zp Þw :
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Let a#1=pm ðaAF Þ be an element of VwG : Since a#1=pm gives an unramified extension, it follows that m
ðaÞ ¼ bp
in IFn ;
for some nXm; and bAIFn : Since Fn =F is unramified outside p; we have m
ðaÞ ¼ ap a0
in IF ;
where a; a0 AIF and a0 is a product of primes above p: It is sufficient to show ðaÞ ¼ ðaew Þ ¼ ew a0 ¼ 0
in IF =pm IF :
Let IF ;p be subgroup of IF which is generated by primes above p: Since our assumption wðpÞa1; we get ðIF ;p #Z Zp Þw CZp ½D=D w ¼ 0; where D is the decomposition group of p: The above assertion follows from this. & By Lemma 7, VwG is finite because WF ;w CAF ;w is a finite group. Hence from pairing (2.2.4), we have a noncanonical isomorphism of Ow -modules: Xow1 =ðg kðgÞÞXow1 CVwG
ð4:1:1Þ
(cf. [10, Lemmma 3.1]). We will study the structure of VwG instead of Xow1 =ðg kðgÞÞXow1 : By Lemma 3, we can regard WF ;w ; VwG ; TðnÞ ðnASÞ as Ow -submodules of ðFN #Z Qp =Zp ÞGw : Let m
MN ¼ FN ðfa1=p j a#1=pm AWF ;w gÞ; m
LN;0 ¼ FN ðfa1=p j a#1=pm AVwG gÞ; and for any nAS; let m
MN;n ¼ FN ðfa1=p j a#1=pm ATðnÞgÞ: MN =FN is a finite abelian p-extension unramified outside p: Every intermediate field K of FN and MN which corresponds to an Ow -submodule Y of WF ;w is a Galois extension over Q; because D acts on Y via w and G acts on Y trivially. Recall that LN is the maximal unramified abelian p-extension over FN : Since V -WF ;w ¼ VwG ; we have LN -MN ¼ LN;0 : Hence LN;0 (resp. MN;n LN;0 ) is the
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maximal unramified extension of FN (resp. MN;n ) contained in MN : MN;n
u:r:
j u:r:
FN
MN;n -LN;0
MN;n LN;0
MN
j u:r:
ð4:1:2Þ
LN;0
Here, u.r. means an unramified extension. The rest of Section 4.1 is devoted to the study of the Kummer extension MN =MN;n with nAS: Lemma 8. For any nAS; MN;n is a CM-field. Proof. Since MN;n =FN is a Galois extension and FN is a totally imaginary field, MN;n is also totally imaginary field. Let J denote the complex conjugation and /JS the Galois group which is generated by J: We have
þ /JS where FN is the maximal real subfield of FN ; and MN;n is the fixed field of J contained in MN;n : Since J acts on GalðMN;n =FN Þ via ow1 and w is an odd character, we have sJ ¼ JsJ 1 ¼ s; for every sAGalðMN;n =FN Þ: Hence J and every element of /JS þ =FN is a Galois GalðMN;n =FN Þ are commutative. It follows from this that MN;n þ /JS þ /JS is extension. Since FN is a totally real field and 2 does not divide ½MN;n : FN ; MN;n a totally real field. &
For each nAS; let cn : FN #Z Qp =Zp -MN;n #Z Qp =Zp
be the natural map. Define W ðnÞ ¼ cn ðWF ;w ÞCWM N;n
(cf. Section 2.3 for the definition of WM ). We have the following diagram of two N;n
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Kummer pairings: GalðMN =FN Þ m GalðMN =MN;n Þ
WF ;w
-mpN #Zp Ow
k jj W ðnÞ -mpN #Zp Ow
and by the definition of MN;n and (4.1.2), we have an isomorphism of Ow modules: W ðnÞCWF ;w =TðnÞ: We know from (3.2.5), ½MN : MN;n ¼ jW ðnÞjpdðnÞ f ; with f ¼ ½Ow : Zp : Next, we will define an element br;n of MN;n for each nAS and each prime ideal of r of F lying above a rational prime r 1 ðmod nNF Þ: Lemma 9. Let nAS and r be a prime ideal of F lying above a rational prime r e% 1 ðmod nNF Þ: Then ðtr w ÞdðnÞ=d is an element of MN;n : Proof. We may assume dðnÞod: By the argument of Euler system of Gauss sums (cf. [1, Proposition 3.1; 9, Proposition 2.3]), we have dðnÞrw ¼ a þ ðaÞ in ðIF #Z Zp Þw ; where a is a product of prime ideals dividing n and aAðF #Z Zp Þw : For the element uAO w defined in Section 3.1, we have dðnÞruw ¼ ða þ ðaÞÞu : By the definition of MN;n ; a is principal in MN;n : Hence we see dðnÞruw ¼ ðbÞ in IMN;n #Z Zp ; #Z Zp : For the complex conjugation J; we have for some bAMN;n
dðnÞruw ¼ ðbð1JÞ=2 Þ in IMN;n #Z Zp : ; here b% is an element of MN;n such Hence b%1J #1=ð2dðnÞÞ is an element of WM N;n e % dðnÞ w that b b% ðmod ðMN;n Þ Þ: On the other hand, by Lemma 6, tr #1=d is an element
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of WF ;w : Considering the commutative diagram
e% From (3.1.2), the image of b%1J #1=ð2dðnÞÞ and that of tr w #1=d in A MN;n are both u ½rw : Hence it follows from the isomorphism WMN;n CAMN;n that e%
tr w #1=d ¼ b%ð1JÞud=dðnÞ #1=d
in MN;n #Z Qp =Zp ; e%
Þd=dðnÞ : This completes the for some integer u40: By Lemma 4(2), we get tr w AðMN;n proof. &
For each nAS and each prime ideal r of F lying above a rational prime r 1 ðmod nNF Þ; define by the previous lemma, e%
br;n ¼ ðtr w ÞdðnÞ=d AMN;n :
Remark. If dðnÞod; then br;n is not uniquely determined. Namely, there is a difference of d=dðnÞth root of unity by the definition. But ðEMN;n #Z Qp =Zp Þ ¼ 0 implies that br;n #1=dðnÞAWM is uniquely determined. N;n Recall the Kummer pairing GalðMN =MN;n Þ W ðnÞ-mpN #Zp Ow and ; W ðnÞ ¼ cn ðWF ;w ÞCWM N;n
for nAS: The next lemma follows from Lemma 6 and the definition of br;n : Lemma 10. For any nAS; we have W ðnÞ ¼ fbr;n #1=dðnÞ j r : prime ideal of F lying above a rational prime r 1 ðmod nNF Þg: 4.2. In this subsection, we will prove the key proposition (Proposition 2) for the proof of Theorem 2. We will study the Ow -module VwG CXow1 =ðg kðgÞÞXow1 : Write t
VwG ¼ " Ow =pei ; i¼1
Ow =pei C/ai #1=pmi SOw
e1 X?Xet ;
ðai #1=pmi AðF #Z Qp =Zp Þw Þ:
ð4:2:1Þ
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By Lemmas 6, 7 and the definition of TðnÞ in Section 3.2, there exists nASi such that TðnÞ ¼ "ij¼1 /aj #1=pmj SOw ; (note that Tð1Þ ¼ 0). Our aim in this subsection is the following proposition. Proposition 2. For any i; 0pipt; there exists ni ASi such that i
Tðni Þ ¼ " /aj #1=pmj SOw j¼1
and jW ðni Þj ¼ jWF ;w =Tðni Þj ¼ dðni Þ f : Proof. We will choose ni ASi inductively. There is nothing to do for i ¼ 0: Suppose that there exists ni ASi which satisfies the above conditions. By Lemma 6, we can write aiþ1 #1=pmiþ1 ¼ ðtr #1=dÞew ; for some prime ideal r of F lying above a rational prime c 1 ðmod ni NF MÞ: By the same arguments in [9, the proof of Theorem 4.1], there exists a prime ideal r of F lying above a rational prime r 1 ðmod ni NF MÞ with dðni ÞXdðni rÞ such that /½rw SOw ¼ /½rw SOw CAF ;w
ð4:2:2Þ
and in ðIF #Z Zp Þw ;
dðni Þ=dðni rÞrw ¼ a þ ðaÞ
where a is a product of prime ideals dividing ni and a is an element of ðF #Z Zp Þw : Let BF ðni Þ be the Zp ½D -submodule of AF which is defined in Section 3.2. From (3.2.3), we have WF ;w
C
, Tðni Þ"/ðtr #1=dÞew SOw
AF ;w ,
C
BF ðni Þw "/½rw SOw :
From this and (4.2.2), we have dðni Þ=dðni rÞrw ¼ ðbÞ
in ðIF #Z Zp Þw ;
for some bAðF #Z Zp Þw : For the element uAO w defined in Section 3.1, we have dðni Þ=dðni rÞruw ¼ ðbu Þ
in ðIF #Z Zp Þw :
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Hence we get bu #dðni rÞ=dðni ÞAWF ;w : The image of bu #dðni rÞ=dðni Þ and that of ðtr #1=dÞew in AF ;w are both ½ruw ; so it follows that ter%w #1=d ¼ ðtr #1=dÞew ¼ bu #dðni rÞ=dðni Þ ¼ buðddðni rÞÞ=dðni Þ #1=d: e%
By Lemma 4(1), we have trw AðF Þðddðni rÞÞ=dðni Þ : e%
k
Let kX0 be the largest integer such that trw AðF Þp : Then we have k p Xðddðni rÞÞ=dðni Þ: It follows from Lemma 5(1) that j/ter%w #1=dSOw j ¼ ðd=pk Þ f pðdðni Þ=dðni rÞÞ f ;
ð4:2:3Þ
where f ¼ ½Ow : Zp : Let niþ1 ¼ ni rASiþ1 : From (4.2.2) and (4.2.3) we have Tðni Þ"/aiþ1 #1=pmiþ1 SOw ¼ Tðni Þ"/ðtr #1=dÞew SOw ¼ Tðniþ1 Þ and jW ðniþ1 Þj ¼ jWF ;w =Tðniþ1 Þj ¼ jW ðni Þj=j/ðtr #1=dÞew SOw j X dðni Þ f ðdðniþ1 Þ=dðni ÞÞ f ¼ dðniþ1 Þ f : Using (3.2.5), we get the conclusion. & 4.3. Let MN and LN;0 be as in Section 4.1. Recall that LN;0 is the maximal unramified extension of FN contained in MN (see (4.1.2)). In addition, from Section 2.1 we know that LN;0 is the maximal extension of FN contained in MN in which every prime divisor of FN lying above p is completely decomposed. Since MN =FN is an extension unramified outside p; the group GalðMN =LN;0 Þ is generated by decomposition groups in GalðMN =FN Þ of prime divisors above p: Fix a prime ideal p of F lying above p; and let MN;p =FN;p denote the completion of the extension at a prime divisor lying above p: Let D denote the decomposition group of p in D ¼ GalðF =QÞ; and for any character c of D; let cD denote the restriction of c to D: GalðMN;p =FN;p Þ is an OwD -module because D acts on it via ow1 From the above arguments, we have GalðMN =LN;0 Þ ¼ D : /GalðMN;p =FN;p ÞSOw :
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Lemma 11. GalðMN;p =FN;p Þ is a finite cyclic OwD -module. t:r: Proof. The finiteness follows from GalðMN =FN ÞoN: Let FN;p denote the maximal totally ramified abelian p-extension of FN;p : By local class field theory, we get an ð1Þ ð1Þ t:r: isomorphism GalðFN;p =FN;p ÞCUð1Þ p;N ¼ lim Up;m ; where Up;m denotes the group of ’
ow1
principal units of the completion of Fm for the prime ideal above p: Let FN D be as in ow1
Section 2.1. Since D acts on GalðMN =FN Þ via ow1 ; we see FN CMN CFN D : By the arguments in Section 2.1, the decomposition group of prime divisor above p in GalðMN =FN Þ coincides with the inertia group. Hence, MN;p =FN;p is a totally ramified extension and D acts on GalðMN;p =FN;p Þ via ow1 D : We have a surjection of Lw ¼ Ow ½½G
-modules ðUð1Þ p;N Þow1 #OwD Ow -GalðMN;p =FN;p Þ#OwD Ow : D
ð4:3:1Þ
#OwD Ow is isomorphic to Lw From our assumption wðpÞa1; we know that ðUð1Þ p;N Þow1 D [3, Proposition 1]. On the other hand, G acts on GalðMN =FN Þ via the cyclotomic character k; because of the Kummer pairing GalðMN =FN Þ WF ;w -mpN #Zp Ow ; and WF ;w CðF #Z Qp =Zp Þw : Hence from (4.3.1), we have the surjection of Lw modules: Lw =ðg kðgÞÞLw -GalðMN;p =FN;p Þ#OwD Ow : By the isomorphism Lw =ðg kðgÞÞLw COw ; GalðMN;p =FN;p Þ#OwD Ow is a cyclic Ow module, and hence GalðMN;p =FN;p Þ is a cyclic OwD -module. & For each nAS and each prime ideal r of F lying above a rational prime r 1 ðmod nNF Þ; let br;n be the element of MN;n which is defined in Section 4.1. Fix a prime ideal p of F lying above p; and let MN;n;p denote the completion of MN;n for a prime divisor lying above p: Let gr ðnÞAZ be the largest power of p which satisfies br;n AðMN;n;p Þgr ðnÞ ; and define gðnÞ ¼ minfgr ðnÞ j r: prime ideals of F lying above a rational prime r 1 ðmod nNF Þg: Remark. We can easily see that the integer gðnÞ does not depend on a prime divisor of MN;n lying above p: Proposition 3. Let nAS: dðnÞ f =½MN : MN;n LN;0 ¼ minfgðnÞ; dðnÞg f :
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Proof. Recall the Kummer pairing in Section 4.1: GalðMN =MN;n Þ W ðnÞ-mpN #Zp Ow : By Lemma 10, we can write k
W ðnÞ ¼ " OwD =pmi ; i¼1
OwD =pmi C/bri ;n #1=dðnÞSOwD ; with prime ideals ri of F lying above a rational prime ri 1 ðmod nNF Þ: m For every i with 1pipk; let Ki ¼ MN;n ðfa1=p j a#1=pm A/bri ;n #1=dðnÞSOwD gÞ: Then MN is the composite field of all Ki ; 1pipk: For a fixed prime ideal p of F ; let MN;n;p ; Ki;p and MN;p be the completions of the extensions at a prime divisor lying above p: Clearly, Ki;p ¼ m
MN;n;p ðfa1=p j a#1=pm A/bri ;n #1=dðnÞSOwD gÞ and MN;p is the composite of all Ki;p ; 1pipk: For any prime ideal r of F lying above a rational prime r 1 ðmod nNF Þ; we have br;n #1=dðnÞAW ðnÞ by Lemma 10. Since W ðnÞ is generated by all bri ;n #1=dðnÞ with 1pipk; we get gr ðnÞXminfgri ðnÞ; dðnÞ j 1pipkg by Lemma 4(2). Hence we get minfgðnÞ; dðnÞg ¼ minfgri ðnÞ; dðnÞ j 1pipkg: By Lemma 11, GalðMN;p =MN;n;p Þ is a cyclic OwD -module. Hence we get ½MN;p : MN;n;p ¼ maxf½ðKi Þp :ðMN;n Þp j 1pipkg: (1) Case dðnÞpgri ðnÞ for all i such that 0pipk: In this case, we have Ki;p ¼ MN;n;p for all i: Hence we get MN;p ¼ MN;n;p : Since GalðMN =MN;n LN;0 Þ is generated by decomposition groups in GalðMN =MN;n Þ of prime divisors above p; we have ½MN : MN;n LN;0 ¼ 1; and then we complete the proof in this case. (2) Case dðnÞ4gri ðnÞ for some i: Let i0 be such that gri0 ðnÞ ¼ minfgri ðnÞ j 1pipkg: We have by Lemma 4(2) GalððMN Þp =ðMN;n Þp Þ ¼ GalððKi0 Þp =ðMN;n Þp C OwD =ðdðnÞ=gri0 ðnÞÞ and GalðMN =MN;n LN;0 ÞCOw =ðdðnÞ=gri0 ðnÞÞ: Hence it follows that ½MN : MN;n LN;0 ¼ ðdðnÞ=gri0 ðnÞÞ f ; and the proof if complete. &
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Q Q 4.4. Let JF be the ide`le group of F ; and F v[p Uv be the closure of F v[p Uv in JF where Uv is the group of units of Fv (for an infinite prime divisor, Uv is defined to Q be the multiplicative group of Fv ). Let YF ¼ JF =F v[p Uv and YF fpg denotes the p-primary component of YF : By class field theory, we have the following exact sequence. 0-Uð1Þ =Uð1Þ -EF -YF fpg-AF -0; Q ð1Þ where Uð1Þ ¼ pjp Uð1Þ p ; Up is the group of principal units of Fp ; and EF is the group of units of F : By taking the w-part of the above exact sequence, we get the f isomorphism AF ;w CYF fpgw =Uð1Þ w : Let dAZ40 be as before, that is, d ¼ jAF ;w j and e%w AZ½D such that e%w ew ðmod dÞ: Since dAF ;w ¼ 0; we have d ð1Þ ðUð1Þ w Þ CdðYF fpgw ÞCUw :
By the assumption ow1 ðpÞa1; we have Uð1Þ w COw [3, Proposition 1]. Let n be the order of the residue field of a prime ideal of F lying above p; and r a prime ideal Q of F lying above a rational prime r 1 ðmod NF Þ: By the natural injection F + pjp Fp ; Q e% we can regard ðtr w Þn1 as an element of Uð1Þ ¼ pjp Uð1Þ p : For any integer k40; we see e%
dðYF fpgw Þ ¼ /fðtr w Þn1 j r: prime ideals of F lying above a rational prime r 1 ðmod kNF ÞgSZp ;
ð4:4:1Þ
(cf. [4, Lemma 4.4]). Lemma 12. For any integer k40; there exists a prime ideal r of F lying above a rational prime r 1 ðmod kNF Þ such that gr ð1Þpd: Proof. For mX0 we put Gm ¼ GalðFm =F Þ: By the spectral sequence, we have an exact sequence m
m
p Gm =ðFm;p Þ Þ 0- H 1 ðGm ; " mpm ðFm;p ÞÞ- " Fp =ðFp Þp -ð" Fm;p pjp
pjp
pjp
- H ðGm ; " mpm ðFm;p ÞÞ-?: 2
pjp
By taking the w-part of the above exact sequence, we get ! 8 !Gm 9 < = m m p C " Fm;p =ðFm;p Þ ; " Fp =ðFp Þp : pjp ; pjp w
w
ð4:4:2Þ
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because of ð"pjp mpm ðFm;p ÞÞw ¼ 0 from ow1 ðpÞa1 [3, Proposition 1]. The assertion follows this and (4.4.1). & We write as in Section 4.2, t
i
VwG ¼ " Ow =pe ; i¼1
Ow =pei C/ai #1=pmi SOw ;
e1 X?Xet ;
ðai #1=pmi AðF #Z Qp =Zp Þw Þ:
Proof of Theorem 2. Let nASi ; 0pipt: We can easily see from (4.1.2) that jVwG =TðnÞ-VwG j ¼ ½LN;0 : MN;n -LN;0
¼ ½MN;n LN;0 : MN;n
¼ ½MN : MN;n =½MN : MN;n LN;0
¼ jW ðnÞj=½MN : MN;n LN;0 : From this and Proposition 3, we get jVwG =TðnÞ-VwG j dðnÞ f =jW ðnÞj ¼ dðnÞ f =½MN : MN;n LN;0
¼ minfgðnÞ; dðnÞg f : Note that dðnÞ f =jW ðnÞjX1 from (3.2.5). We can choose ni ASi from Proposition 2 which satisfies i
Tðni Þ ¼ " /aj #1=pmj SOw j¼1
and jW ðni Þj ¼ dðni Þ f : Hence we have minfjVwG =TðnÞ-VwG j j nASi g ¼ jVwG =Tðni Þj ¼ pðeiþ1 þ?þet Þf : Further, since jW ðni Þj ¼ dðni Þ f we have minfjVwG =TðnÞ-VwG j dðnÞ f =jW ðnÞj j nASi g ¼ pðeiþ1 þ?þet Þf :
ð4:4:3Þ
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From (4.4.3), we get minfgðnÞ; dðnÞ j nASi g ¼ peiþ1 þ?þet :
ð4:4:4Þ
On the other hand, we have FN;p ¼ MN;ni ;p for every prime ideal p of F lying above p because FN CMN;ni CLN;0 and p is totally decomposed in LN;0 =FN : By Lemma 12, there exists a prime ideal r of F lying above a rational prime r 1 ðmod ni NF Þ dðn Þ=d such that gr ð1Þpd: By the definition, br;ni ¼ br;1 i is an element of MN;n and gr ðni Þ i Þgr ðni Þ ¼ ðFN;p Þgr ðni Þ : Hence we get is the largest power of p satisfying br;ni AðMN;n;p from gr ð1Þpd that
gr ðni Þ ¼ gr ð1Þ dðni Þ=d p dðni Þ: It concludes from this that minfgðnÞ; dðnÞ j nASi g ¼ minfgðni Þ; dðni Þg ¼ gðni Þ ¼ minfgðnÞ j nASi g; and using (4.4.4), we complete the proof.
&
Proof of Corollary 2. It is sufficient to show the isomorphism Xow1 =ðg kðgÞXow1 CTw : Since we assumed ow1 ðpÞa1; this follows from the isomorphism Tw CHomOw ðXow1 =ðg kðgÞÞXow1 ; mpN #Zp Ow Þ (cf. [8]).
&
Proof of Corollary 3. Let D be the decomposition group of p in D ¼ GalðF =QÞ; and ð1Þ wD the restriction of w to D: Since Uð1Þ w CðUp ÞwD #OwD Ow and (4.4.2), we get the conclusion. &
Acknowledgments I thank the thesis advisor Professor M. Kurihara for valuable discussions and advice.
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References [1] M. Aoki, The Iwasawa main conjecture and Gauss sums, J. Number Theory 89 (2001) 151–164. [2] M. Aoki, Notes on the structure of the ideal class groups of abelian number fields, submitted. [3] R. Gillard, Unite´s cyclotomiques, unite´s semi-locales et Zc -extensions II, Ann. Inst. Fourier 29 (4) (1979) 1–15. [4] Y. Hachimori, H. Ichimura, Semi-local units modulo Gauss sums, Manuscripta Math. 95 (1998) 377–395. [5] K. Iwasawa, A note on Jacobi sums, Symp. Math. XV (1975) 447–459. [6] V. Kolyvagin, Euler systems, in the Grothendieck Festschrift II, Progr. Math. 87 (1990) 435–483. [7] B. Mazur, A. Wiles, Class fields of abelian extensions of Q; Invent. Math. 76 (1984) 179–330. [8] T. Nguyen Quang Do, Sur la Zp -torsion de certains modules galoisiens, Ann. Inst. Fourier 36 (1986) 27–46. [9] K. Rubin, Kolyvagin’s system of Gauss sums, in Arithmetic Algebraic Geometry, Progr. Math. 89 (1991) 309–324. [10] L. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, 2nd Edition, Vol. 83, Springer, Berlin, New York, 1997.
Journal of Number Theory 101 (2003) 270–293
http://www.elsevier.com/locate/jnt
Local units and Gauss sums Miho Aoki Department of Mathematics, Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachioji-shi, Tokyo-to, 192-0397, Japan Received 17 May 2002; revised 7 January 2003 Communicated by D. Goss
Abstract In this paper, we will determine the structure of a certain module which is related to the plus part of the ideal class groups in terms of the divisibility of Gauss sums in some local fields. This result is a generalization of a result of Iwasawa and the previous work of Ichimura and Hachimori. r 2003 Elsevier Science (USA). All rights reserved. MSC: 11R23 Keywords: Gauss sums; Euler system
1. Introduction Let F be an abelian number field. Fix an odd rational prime p and let FN denote the cyclotomic Zp -extension of F : We have a sequence of fields F ¼ F0 CF1 C?CFm C?CFN such that Fm =F is a cyclic extension of degree pm : For any number field K; let AK be the p-primary component of the ideal class group of K; and set X ¼ lim AFm : By the action of complex conjugation, X is ’
decomposed to X þ "X : Fix a topological generator g of G ¼ GalðFN =F Þ:
Fax: +81-426-77-2472. E-mail address: [email protected].
0022-314X/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0022-314X(03)00056-8
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Suppose first F ¼ Qðmp Þ; where mp is the group of all pth roots of unity. Kolyvagin [6] and Rubin [9] determined the structure of ðX ÞG ¼ X =ðg 1ÞX CA F as an abelian group (Theorem 1). The structure is determined by certain elements obtained from Stickelberger elements. For the proof, they used the Euler system of Gauss sums. One of the keys of the proof is to know the maximum n such that certain n elements constructed from the Gauss sums belong to ðF Þp : On the other hand, the local properties of Gauss sums (namely, the properties of Gauss sums in the completion of F for a prime ideal above p) give information on the plus part X þ : Iwasawa [5] gave a condition for X þ ¼ 0 (this is equivalent to the conjecture of Vandiver) by using the local properties of Gauss sums. Let k : G-Z p be the cyclotomic character. In this paper, we will determine the structure of X þ =ðg kðgÞÞX þ by using Gauss sums for an abelian number field F satisfying certain conditions. Note that if F ¼ Qðmp Þ; then the p-rank of X þ =ðg kðgÞÞX þ (namely, dimZ=p ðX þ =ðg kðgÞÞX þ #Z Z=pÞ) is the same as that of X þ =ðg 1ÞX þ CAþ F : For the proof, we will use local properties of the Euler system of Gauss sums, and also use some properties on the Euler system of Gauss sums proved by Kolyvagin and Rubin. Let D be a finite abelian group and c : D-Qp be a character. Let Oc denote the extension ring of Zp which is generated by the values of c; and Oc be the D-module which is Oc as an additive group on which D acts via c: We define the idempotent ec P 1 1 by ec ¼ jDj sAD Traceðc ðsÞÞs; where Trace : Qp ðcðsÞjsADÞ-Qp is the trace map. For any Zp ½D -module Y ; we define the c-part Yc of Y by Yc ¼ Y #Zp ½D Oc : % Fix an odd character w of GalðQ=QÞ which is not the Teichmuller . character o: Here, we assume the following two assumptions on w: (1) the order of w is finite and prime-to-p: (2) wðpÞa1 and ow1 ðpÞa1: Let Fw be the fixed field of the kernel of w; and let F ¼ Fw ðmp Þ: Set D ¼ GalðF =QÞ: By assumption (1), the order of D is prime to p; and for any Zp ½D -module Y ; we have Yc Cec Y
and
Y ¼ " Yc ; c
where c runs over all representatives of Qp -conjugate classes of characters of D: For an element a in Y ; we denote the image of a in Yc by ac : Let M be a power of p such that MXjAF ;w j2 : For any integer iX0; let Si ¼ fnAZ40 j square-free; n ¼ c1 ?ci ðproduct of primesÞ; cj 1 ðmod MNF Þg
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S where NF is the conductor of F : Let S ¼ iX0 Si : As mentioned before, Rubin determined the structure of Xw =ðg 1ÞXw CAF ;w in terms of certain elements dðnÞ; nAS which are obtained from Stickelberger elements (see Section 3.2 for the definition). Especially d ¼ dð1Þ is given the largest power of p dividing the generalized Bernoulli number B1;w1 : Theorem 1 (Kolyvagin [6, Theorem 7], Rubin [9, Theorem 4.4]). Write t
Xw =ðg 1ÞXw ðCAF ;w Þ ¼ " Ow =pei ; i¼1
e1 X?Xet
as Ow -modules. Then we have peiþ1 þ?þet ¼ minfdðnÞ j nASi g; for any i; 0pipt 1: Our aim is to determine the structure of Xow1 =ðg kðgÞÞXow1 as an Ow -module. For each nAS; let TðnÞ denote the Ow -submodule of ðF #Z Qp =Zp Þw which is generated by elements ðtr #1=dÞew where tr is a Gauss sum of a prime ideal r of F dividing n: Let MN;n =FN be the Kummer extension given by m
MN;n ¼ FN ðfa1=p j a#1=pm ATðnÞgÞ: For any prime ideal r of F lying above a rational prime r 1 ðmod nNF Þ; we e% (Lemma 9), here, for any will show that ðtr w ÞdðnÞ=d is an element of MN;n element xAZp ½D ; x% denotes an element of Z½D satisfying x% x ðmod dÞ: Set br;n ¼ e%
ðtr w ÞdðnÞ=d and define the integer gr ðnÞ by the largest power of p which satisfies Þgr ðnÞ (MN;n;p is the completion of MN;n for a prime divisor above p). br;n AðMN;n;p Further let gðnÞ ¼ minfgr ðnÞ j r :prime ideals of F lying above a rational prime r 1 ðmod nNF Þg: Our main theorem is as follows. Theorem 2. Write t
Xow1 =ðg kðgÞÞXow1 ¼ " Ow =pei ; i¼1
e1 X?Xet
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as Ow -modules. Then we have peiþ1 þ?þet ¼ minfgðnÞ j nASi g; for any i; 0pipt 1: For any Ow -module Y ; we define the w-rank of Y by dimOw =p ðY #Ow Ow =pÞ: From the theorem, we get the following corollary immediately, because dimOw =p ðXow1 =ðg kðgÞÞXow1 #Ow Ow =pÞ ¼ dimOw =p ðXow1 =ðg 1ÞXow1 #Ow Ow =pÞ: Corollary 1. The w-rank of AF ;ow1 is smaller than or equal to i if and only if minfgðnÞ j nASi g ¼ 1: Corollary 2. Let F 0 =F be the maximal abelian pro-p extension unramified outside p; and T the Zp -torsion subgroup of GalðF 0 =F Þ: Then Tw C"ti¼1 Ow =pei with ei as in Theorem 2. Let Uð1Þ ¼
Q
pjp
Q
Uð1Þ p ; where p runs over all prime ideal of F over p: By the natural
ð1Þ injection F + pjp Fp ; we can regard bn1 r;1 as an element of U : Let G be the closure of the image of
fbn1 r;1 j r: prime ideal of F lying above a rational prime r 1 ðmod NF Þg in Uð1Þ : We get the following equation of the order of Xow1 =ðg kðgÞÞXow1 and that of Uð1Þ w =Gw : Corollary 3 (Ichimura and Hachimori [4, Theorem 1.2]). jUð1Þ w =Gw j ¼ jXow1 =ðg kðgÞÞXow1 j: Ichimura and Hachimori also studied the order of Uð1Þ w =Gw for the characters 1 wðpÞ ¼ 1 or ow ðpÞ ¼ 1: In the case F ¼ Qðmp Þ; Iwasawa [5] showed that Uð1Þ w ¼ Gw for every odd character w ðaoÞ of D implies Aþ ¼ 0 (the conjecture of Vandiver). F We will give the proofs of Corollaries 2 and 3 in Section 4.4 together with that of Theorem 2.
2. Preliminary lemmas 2.1. For a field K; we denote the maximal abelian extension of K by K ab : Let F ; Fm and FN be as in Section 1, and fix a prime ideal p of F above p: We denote the
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decomposition group and the inertia group of the prime ideal above p in GalðFmab =Fm Þ by Dm and Im ; respectively. Let Fm;p be the completion at the prime ab =Fm;p : We have an exact ideal above p: We can regard Dm as the Galois group of Fm;p sequence 0-Im -Dm -GalðFab q =Fq Þ-0; where Fq is the residue field of Fm;p : Let D be the decomposition group of p in D ¼ GalðF =QÞ: The above sequence is exact as D-modules. For our fixed odd character w; we denote the restriction 1 1 of ow1 to D by ow1 D : Since we assumed ow ðpÞa1; we get owD is not the trivial character. Hence, we have the following isomorphism from the above exact sequence ðIm #Z Zp Þow1 CðDm #Z Zp Þow1 : D
ow1 D
Let Fm
ð2:1:1Þ
D
be the subfield of Fmab =Fm such that ow1 D
GalðFm
=Fm ÞCðGalðFmab =Fm Þ#Z Zp Þow1 : D
Isomorphism (2.1.1) implies that the decomposition group of the prime ideal above p ow1 D
in GalðFm
=Fm Þ coincides with the inertia group. ow1
We define FN D in the same way. By taking the projective limit, we know that the ow1
decomposition group of the prime divisor above p in GalðFN D =FN Þ coincides with the inertia group. 2.2. Let LN denote the maximal unramified abelian pro-p extension over FN : In this section, we will consider the Kummer extension LN =FN : By the Kummer theory, there is a subgroup V DFN #Z Qp =Zp ;
such that m
LN ¼ FN ðfa1=p j a#1=pm AV gÞ; and there exists a Kummer pairing X V -mpN ; m
ðx; a#1=pm Þ//x; a#1=pm S ¼ ða1=p Þx1 ;
ð2:2:1Þ
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where mpN denotes the group consisting of all pm th root of unities for all mX0: This is a non-degenerate bilinear pairing, and for any t in GalðFN =QÞ; we have /x; a#1=pm St ¼ /xt ; at #1=pm S: Recall D ¼ GalðF =QÞ;
G ¼ GalðFN =F Þ:
By our assumption p[½F : Q ; we have GalðFN =QÞCD G: Let ðV G Þ> denote the annihilator of V G in X with respect to the pairing (2.2.1), namely ðV G Þ> ¼ fxAX j /x; vS ¼ 1; for every vAV G g: Lemma 1. ðV G Þ> ¼ ðg kðrÞÞX : Proof. The proof is standard, so we will sketch it. First, let ðg kðgÞÞx be an element of ðg kðgÞÞX : For every vAV G ; we have/ðg kðrÞÞx; vS ¼ 1: Hence we get ðV G Þ> *ðg kðgÞÞX : Next, let us prove the inclusion ðV G Þ> Cðg kðrÞÞX :
ð2:2:2Þ
Let fðg kðgÞÞX g> ¼ fvAV j /s; vS ¼ 1; for every xAðg kðgÞÞX g denote the annihilator of ðg kðgÞÞX in V with respect to pairing (2.2.1). We will show the inclusion V G *fðg kðgÞÞX g> ; because this implies inclusion (2.2.2). Let v be an element of fðg kðgÞÞX g> : For 1 every xAX ; we have /x; vg 1 S ¼ 1: Since pairing (2.2.1) is non-degenerate, we 1 have vg 1 ¼ 1: We conclude that v is an element of V G : & By Lemma 1, we have the following non-degenerate bilinear pairing. X =ðg kðgÞÞX V G -mpN :
ð2:2:3Þ
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Further, from the same arguments of [10, Section 10.2] we have the following nondegenerate bilinear pairing: Xow1 =ðg kðgÞÞXow1 VwG -mpN #Zp Ow :
ð2:2:4Þ
2.3. Let K=Q be a Galois extension, not necessarily finite with Galois group G ¼ GalðK=QÞ: Let IK denote the ideal group of K; ClK the ideal class group of K; and AK the Sylow p-subgroup of ClK : For an infinite number field K; these groups are defined as follows. IK ¼ lim IKi ; -
ClK ¼ lim ClKi ;
AK ¼ lim AKi ;
-
-
where Ki runs over all subfields QCKi CK such that Ki is finite over Q: We define a Zp ½G -submodule WK of K #Z Qp =Zp : WK ¼ fa#1=pm AK #Z Qp =Zp j ðaÞApm IK g: We can easily show the following sequence: j
0-EK #Z Qp =Zp -WK ! AK -0
ð2:3:1Þ
is exact where EK denotes the group of units in K; and the map j is defined by jða#1=pm Þ ¼ class of a;
m
where ðaÞ ¼ ap ; aAIK :
From now on, we assume that K is a CM-field. Let J denote the complex conjugation in G ¼ GalðK=QÞ: For any Zp ½G -module M; we define the submodule M þ and M by M 7 ¼ fx j AM; Jx ¼ 7xg: Since we assumed p42; we have M ¼ M þ "M : Lemma 2. WK CA K: Proof. Since p42; from the exact sequence (2.3.1), we have 0-ðEK #Z Qp =Zp Þ -WL -A L -0: Since ðEK #Z Qp =Zp Þ ¼ 0 (cf. [10, Theorem 4.12]), we obtain the assertion.
&
2.4. Let K be an algebraic extension of Q (resp. Qp ), not necessarily finite over Q (resp. Qp ). We consider F satisfying the conditions in Section 1 with Galois group D ¼ GalðF =QÞ; and an odd character wðaoÞ:
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Lemma 3. Assume that K=Q is an abelian extension with Galois group G ¼ GalðK=QÞ and there exists a subfield K 0 of K such that K 0 -F ¼ Q and K ¼ FK 0 : Then we can consider D ¼ GalðF =QÞ to be a subgroup of G ¼ GalðK=QÞ: For every intermediate field K0 of F and K; we have isomorphisms of Ow modules: m
m
p H p (1) ðK =ðK Þ Þw CðK0 =ðK0 Þ Þw ; for every m40; H (2) ðK #Z Qp =Zp Þw CðK0 #Z Qp =Zp Þw ;
where H ¼ GalðK=K0 Þ: Proof. See [9, Lemma 2.2] for (1). By taking the direct limit of (1), we get (2).
&
Lemma 4. (1) If K satisfies the conditions of Lemma 3, then for every element m a#1=pm AðK #Z Qp =Zp Þw ; a#1=pm ¼ 0 if and only if a ¼ bp for some bAK : (2) If K contains mpN ; then for every element a#1=pm AK #Z Qp =Zp ; a#1=pm ¼ m 0 if and only if a ¼ bp for some bAK : m
Proof. It is obvious in both cases that a ¼ bp implies a#1=pm ¼ 0: From n nþm for some bAK and the definition, a#1=pm ¼ 0 if and only if ap ¼ bp pm because the map some nX0: In case (1), this implies a ¼ b n n nþ1 ðK #Z 1=p Z=ZÞw -ðK #Z 1=p Z=ZÞw is injective for any n40: In case (2), ap ¼ m
implies a ¼ bp z for some zAmpn and we have a ¼ bp zAðK Þp because K bp contains mpN : & nþm
m
m
We get the following lemma as an immediate corollary of Lemma 4. Lemma 5. (1) Assume that K satisfies the conditions of Lemma 3. Let a#1=pm be an element of ðK #Z Qp =Zp Þw such that aeðK Þp : Then we have /a#1=pm SOw COw =pm as Ow -modules. Hence, we see j/a#1=pm SOw j ¼ qm with q ¼ jOw =pj: (2) Assume that K contains mpN : Let a#1=pm be an element of K #Z Qp =Zp such that aeðK Þp : Then we have /a#1=pm SZ CZ=pm as abelian groups.
3. Gauss sums 3.1. We will review the definition of Stickelberger elements and Gauss sums. We will also prove a small lemma which states the Zp ½D -module WF in Section 2.3 can be written as a set of Gauss sums.
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Let N40 be an integer. We identify GalðQðmN Þ=QÞ with ðZ=NÞ in the usual way, and denote the isomorphism by GalðQðmN Þ=QÞCðZ=NÞ ; sa 2a mod N: Define the Stickelberger element yN by N X a 1 sa AQ½GalðQðmN Þ=QÞ : N a¼1
yN ¼
ða;NÞ¼1
Let F be as in Section 1 with Galois group D ¼ GalðF =QÞ: Let NF denotes the conductor of F : By the definition of F ; NF is the least common multiple of the conductor of w and p: For each prime ideal R of QðmNF Þ which splits completely in QðmNF Þ=Q; we define the character wR : ðZ=rÞ -mNF (r is the rational prime below R) which is given by wR ðaÞ aðr1Þ=NF mod R: We define the Gauss sum tR by tR ¼
r1 X
wR ðaÞzar AQðmrNF Þ ;
a¼1
where zr is a fixed primitive rth root of unity. For the prime ideal r of F below R; we define tr ¼ NormðtR ÞAF ðmr Þ ; where Norm : QðmrNF Þ -F ðmr Þ is the norm map. tr is well defined because of tRs ¼ tsR for any sA GalðQðmrNF Þ=F ðmr ÞÞ: We consider the fixed odd character wðaoÞ of D as a character of GalðQðmrNF Þ=QÞ: We choose sa AGalðQðmNF Þ=QÞ such that wðsa Þ a is invertible in Ow : By Stickelberger’s theorem, we have ðwðsa ÞaÞew
ðtr
Þ ¼ ðwðsa Þ aÞyNF ;w rw
in ðIF #Z Zp Þw : Since wðsa Þ aAO w ; we have e
ðtr w Þ ¼ yNF ;w rw ; in ðIF #Z Zp Þw : We see yNF ;w ¼ ð1 w1 ðpÞÞB1;w1 ;
ð3:1:1Þ
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where B1;w1 denotes the generalized Bernoulli number. By the assumption wðpÞa1; we have yNF ;w BB1;w1 ; here, B means that both sides have the same p-adic valuation. Let f ¼ ½Ow : Zp and the integer d40 such that d f ¼ jAF ;w j: By the theorem of Mazur and Wiles [7], we have B1;w1 Bd: Hence we can write yNF ;w ¼ ud; for some uAO w : Define the element Z of Zp ½D by Ow COw ew CZp ½D ew ; u1 /u1 ew /Zew : From (3.1.1), we have Ze
ðtr w Þ ¼ drw
ð3:1:2Þ
in ðIF #Z Zp Þw : Let e%w and Z% be elements of Z½D such that e%w ew ðmod dÞ and Z% Z ðmod dÞ: Fix an integer k40; and define a subset W ð1Þ of ðF #Z Qp =Zp Þw by e%
W ð1Þ ¼ ftr w #1=d ¼ ðtr #1=dÞew j r: prime ideals of F lying above a rational prime r 1 ðmod kNF Þg: Lemma 6. Let WF be as in Section 2.3. For any integer k40; we have WF ;w ¼ W ð1Þ: Especially, the set W ð1Þ is an Ow -module and it is independent of an integer k: Proof. Put e% Z% W 0 ð1Þ ¼ ftr w #1=d ¼ ðtr #1=dÞew Z j r: prime ideals of F lying above a rational prime r 1 ðmod kNF Þg: If we show an equality WF ;w ¼ W 0 ð1Þ; then W 0 ð1Þ is an Ow -module, so we have WF ;w ¼ W 0 ð1Þ ¼ uW 0 ð1Þ ¼ W ð1Þ: Hence we will show the equality WF ;w ¼ W 0 ð1Þ: By (3.1.2), it is sufficient to show the inclusion WF ;w CW 0 ð1Þ: Recall WF ;w ¼ fa#1=pm AðF #Z Qp =Zp Þw j ðaÞApm IF g: Let a#1=pm be an element of WF ;w : Suppose pm 4d: From the isomorphism of Ow modules WF ;w CAF ;w (Lemma 2), we see a#d=pm ¼ 0: By Lemma 4(1), we can write m a ¼ bp =d ; for some bAF : Hence we have m
a#1=pm ¼ bp
=d
#1=pm ¼ b#1=d:
Thus we may assume pm pd: Since a#1=pm is an element of WF ;w ; we can write ðaÞ ¼ pm a; for some aAIF : Let cAAF ;w be the ideal class of a: By the Chebotarev density theorem, we can choose a prime ideal r of F lying above a rational prime r 1 ðmod kNF Þ; and whose class
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is c: Write r ¼ aðbÞ; for some bAF : Then m
ðad=p bd Þew ¼ daw þ ðbew d Þ ¼ drw ; Ze
in ðIF #Z Zp Þw : From this and (3.1.2), we have ðtr w Þ ¼ ðad=p bd Þew in ðIF #Z Zp Þw : Zew
Since ðEF #Z Zp Þw ¼ 0; we have tr
m
¼ ðad=p bd Þew in ðF #Z Zp Þw : Hence m
m
a#1=pm ¼ ad=p bd #1=d ¼ ðtr #1=dÞew Z in ðF #Z Qp =Zp Þw :
&
3.2. In this section, we will review a result of Rubin [9] which is obtained from arguments of the Euler system of Gauss sums. Let M be a power of p such that M4jAðF Þw j2 : For any integer iX0; let Si denote the set of positive squarefree integers S which are divisible by exactly i primes c which satisfy c 1 ðmod MNF Þ: Let S ¼ iX0 Si : For each prime cAS; fix a generator sc of GalðF ðmc Þ=F ÞCGalðQðmc =QÞ; and let Dc ¼
c2 X
isic :
i¼0
For each nAS; let Dn ¼
Y cjn
Dc ;
Nn ¼
X
t;
tAGn
where Gn ¼ GalðF ðmn Þ=F ÞCGalðQðmn Þ=QÞ: We define dðnÞAðZ=MÞ½D to be the element satisfying Dn ðsa aÞynNF ¼ Nn dðnÞ (see [1,9] for the precise and see [2] for the explicit formula for dðnÞ). Let dðnÞ be the largest power of p which divides dðnÞw AðOw =MÞew : Note that dð1Þ ¼ dBB1;w1 : For each nAS; let TðnÞ denote the Ow -submodule of ðF #Z Qp =Zp Þw : which is generated by the elements ðtr #1=dÞew where r divides n: By Lemma 6, TðnÞ is also an Ow -submodule of WF ;w : If nASi ; then TðnÞ is generated by i elements, because ðtrs #1=dÞew ¼ ðtsr #1=dÞew ¼ ðtr #1=dÞwðsÞew ; for any sAD: By the definition, for n; mAS such that n divides m; we have 0 ¼ Tð1ÞCTðnÞCTðmÞ: For each nAS; let BF ðnÞ denote the Zp ½D - submodule of AF which is generated by the classes of prime ideals of F dividing n: By the isomorphism of Lemma 2 and
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(3.1.2), we have WF ;w
C
AF ;w
, TðnÞ , /ðtr #1=dÞew SOw
, C
BF ðnÞw
C
, /½rw SOw :
ð3:2:3Þ
Hence WF ;w =TðnÞCAF ;w =BF ðnÞw :
ð3:2:4Þ
Rubin [9] showed the following inequality for F ¼ Qðmp Þ by using arguments of the Euler system of Gauss sums. As he mentioned the remark at the end of [9], the result is equally shown for our F ; because we assume p[½F : Q :
Proposition 1 (Rubin [9, Corollary 4.2]). For each nAS; we have jAF ;w =BF ðnÞw jpdðnÞ f ; where f ¼ ½Ow : Zp : Hence from (3.2.4), we have jWF ;w =TðnÞjpdðnÞ f ;
ð3:2:5Þ
for each nAS:
4. Proofs 4.1. In Section 4, we will give the proofs of our main results in Section 1. Recall that our aim is to determine the structure of Ow -module Xow1 =ðg kðgÞÞXow1 in Section 2.2. We will begin with the following lemma. Lemma 7. Let V G be as in Section 2.2, and WF be as in Section 2.3. Then VwG is an Ow submodule of WF ;w : Proof. By the definition, we have VwG CðFN #Z Qp =Zp ÞGw : By Lemma 3, we have an #Z Qp =Zp ÞGw CðF #Z Qp =Zp Þw : Hence we can isomorphism of Ow -modules ðFN regard VwG as an Ow -submodule of ðF #Z Qp =Zp Þw :
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Let a#1=pm ðaAF Þ be an element of VwG : Since a#1=pm gives an unramified extension, it follows that m
ðaÞ ¼ bp
in IFn ;
for some nXm; and bAIFn : Since Fn =F is unramified outside p; we have m
ðaÞ ¼ ap a0
in IF ;
where a; a0 AIF and a0 is a product of primes above p: It is sufficient to show ðaÞ ¼ ðaew Þ ¼ ew a0 ¼ 0
in IF =pm IF :
Let IF ;p be subgroup of IF which is generated by primes above p: Since our assumption wðpÞa1; we get ðIF ;p #Z Zp Þw CZp ½D=D w ¼ 0; where D is the decomposition group of p: The above assertion follows from this. & By Lemma 7, VwG is finite because WF ;w CAF ;w is a finite group. Hence from pairing (2.2.4), we have a noncanonical isomorphism of Ow -modules: Xow1 =ðg kðgÞÞXow1 CVwG
ð4:1:1Þ
(cf. [10, Lemmma 3.1]). We will study the structure of VwG instead of Xow1 =ðg kðgÞÞXow1 : By Lemma 3, we can regard WF ;w ; VwG ; TðnÞ ðnASÞ as Ow -submodules of ðFN #Z Qp =Zp ÞGw : Let m
MN ¼ FN ðfa1=p j a#1=pm AWF ;w gÞ; m
LN;0 ¼ FN ðfa1=p j a#1=pm AVwG gÞ; and for any nAS; let m
MN;n ¼ FN ðfa1=p j a#1=pm ATðnÞgÞ: MN =FN is a finite abelian p-extension unramified outside p: Every intermediate field K of FN and MN which corresponds to an Ow -submodule Y of WF ;w is a Galois extension over Q; because D acts on Y via w and G acts on Y trivially. Recall that LN is the maximal unramified abelian p-extension over FN : Since V -WF ;w ¼ VwG ; we have LN -MN ¼ LN;0 : Hence LN;0 (resp. MN;n LN;0 ) is the
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maximal unramified extension of FN (resp. MN;n ) contained in MN : MN;n
u:r:
j u:r:
FN
MN;n -LN;0
MN;n LN;0
MN
j u:r:
ð4:1:2Þ
LN;0
Here, u.r. means an unramified extension. The rest of Section 4.1 is devoted to the study of the Kummer extension MN =MN;n with nAS: Lemma 8. For any nAS; MN;n is a CM-field. Proof. Since MN;n =FN is a Galois extension and FN is a totally imaginary field, MN;n is also totally imaginary field. Let J denote the complex conjugation and /JS the Galois group which is generated by J: We have
þ /JS where FN is the maximal real subfield of FN ; and MN;n is the fixed field of J contained in MN;n : Since J acts on GalðMN;n =FN Þ via ow1 and w is an odd character, we have sJ ¼ JsJ 1 ¼ s; for every sAGalðMN;n =FN Þ: Hence J and every element of /JS þ =FN is a Galois GalðMN;n =FN Þ are commutative. It follows from this that MN;n þ /JS þ /JS is extension. Since FN is a totally real field and 2 does not divide ½MN;n : FN ; MN;n a totally real field. &
For each nAS; let cn : FN #Z Qp =Zp -MN;n #Z Qp =Zp
be the natural map. Define W ðnÞ ¼ cn ðWF ;w ÞCWM N;n
(cf. Section 2.3 for the definition of WM ). We have the following diagram of two N;n
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Kummer pairings: GalðMN =FN Þ m GalðMN =MN;n Þ
WF ;w
-mpN #Zp Ow
k jj W ðnÞ -mpN #Zp Ow
and by the definition of MN;n and (4.1.2), we have an isomorphism of Ow modules: W ðnÞCWF ;w =TðnÞ: We know from (3.2.5), ½MN : MN;n ¼ jW ðnÞjpdðnÞ f ; with f ¼ ½Ow : Zp : Next, we will define an element br;n of MN;n for each nAS and each prime ideal of r of F lying above a rational prime r 1 ðmod nNF Þ: Lemma 9. Let nAS and r be a prime ideal of F lying above a rational prime r e% 1 ðmod nNF Þ: Then ðtr w ÞdðnÞ=d is an element of MN;n : Proof. We may assume dðnÞod: By the argument of Euler system of Gauss sums (cf. [1, Proposition 3.1; 9, Proposition 2.3]), we have dðnÞrw ¼ a þ ðaÞ in ðIF #Z Zp Þw ; where a is a product of prime ideals dividing n and aAðF #Z Zp Þw : For the element uAO w defined in Section 3.1, we have dðnÞruw ¼ ða þ ðaÞÞu : By the definition of MN;n ; a is principal in MN;n : Hence we see dðnÞruw ¼ ðbÞ in IMN;n #Z Zp ; #Z Zp : For the complex conjugation J; we have for some bAMN;n
dðnÞruw ¼ ðbð1JÞ=2 Þ in IMN;n #Z Zp : ; here b% is an element of MN;n such Hence b%1J #1=ð2dðnÞÞ is an element of WM N;n e % dðnÞ w that b b% ðmod ðMN;n Þ Þ: On the other hand, by Lemma 6, tr #1=d is an element
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of WF ;w : Considering the commutative diagram
e% From (3.1.2), the image of b%1J #1=ð2dðnÞÞ and that of tr w #1=d in A MN;n are both u ½rw : Hence it follows from the isomorphism WMN;n CAMN;n that e%
tr w #1=d ¼ b%ð1JÞud=dðnÞ #1=d
in MN;n #Z Qp =Zp ; e%
Þd=dðnÞ : This completes the for some integer u40: By Lemma 4(2), we get tr w AðMN;n proof. &
For each nAS and each prime ideal r of F lying above a rational prime r 1 ðmod nNF Þ; define by the previous lemma, e%
br;n ¼ ðtr w ÞdðnÞ=d AMN;n :
Remark. If dðnÞod; then br;n is not uniquely determined. Namely, there is a difference of d=dðnÞth root of unity by the definition. But ðEMN;n #Z Qp =Zp Þ ¼ 0 implies that br;n #1=dðnÞAWM is uniquely determined. N;n Recall the Kummer pairing GalðMN =MN;n Þ W ðnÞ-mpN #Zp Ow and ; W ðnÞ ¼ cn ðWF ;w ÞCWM N;n
for nAS: The next lemma follows from Lemma 6 and the definition of br;n : Lemma 10. For any nAS; we have W ðnÞ ¼ fbr;n #1=dðnÞ j r : prime ideal of F lying above a rational prime r 1 ðmod nNF Þg: 4.2. In this subsection, we will prove the key proposition (Proposition 2) for the proof of Theorem 2. We will study the Ow -module VwG CXow1 =ðg kðgÞÞXow1 : Write t
VwG ¼ " Ow =pei ; i¼1
Ow =pei C/ai #1=pmi SOw
e1 X?Xet ;
ðai #1=pmi AðF #Z Qp =Zp Þw Þ:
ð4:2:1Þ
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By Lemmas 6, 7 and the definition of TðnÞ in Section 3.2, there exists nASi such that TðnÞ ¼ "ij¼1 /aj #1=pmj SOw ; (note that Tð1Þ ¼ 0). Our aim in this subsection is the following proposition. Proposition 2. For any i; 0pipt; there exists ni ASi such that i
Tðni Þ ¼ " /aj #1=pmj SOw j¼1
and jW ðni Þj ¼ jWF ;w =Tðni Þj ¼ dðni Þ f : Proof. We will choose ni ASi inductively. There is nothing to do for i ¼ 0: Suppose that there exists ni ASi which satisfies the above conditions. By Lemma 6, we can write aiþ1 #1=pmiþ1 ¼ ðtr #1=dÞew ; for some prime ideal r of F lying above a rational prime c 1 ðmod ni NF MÞ: By the same arguments in [9, the proof of Theorem 4.1], there exists a prime ideal r of F lying above a rational prime r 1 ðmod ni NF MÞ with dðni ÞXdðni rÞ such that /½rw SOw ¼ /½rw SOw CAF ;w
ð4:2:2Þ
and in ðIF #Z Zp Þw ;
dðni Þ=dðni rÞrw ¼ a þ ðaÞ
where a is a product of prime ideals dividing ni and a is an element of ðF #Z Zp Þw : Let BF ðni Þ be the Zp ½D -submodule of AF which is defined in Section 3.2. From (3.2.3), we have WF ;w
C
, Tðni Þ"/ðtr #1=dÞew SOw
AF ;w ,
C
BF ðni Þw "/½rw SOw :
From this and (4.2.2), we have dðni Þ=dðni rÞrw ¼ ðbÞ
in ðIF #Z Zp Þw ;
for some bAðF #Z Zp Þw : For the element uAO w defined in Section 3.1, we have dðni Þ=dðni rÞruw ¼ ðbu Þ
in ðIF #Z Zp Þw :
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Hence we get bu #dðni rÞ=dðni ÞAWF ;w : The image of bu #dðni rÞ=dðni Þ and that of ðtr #1=dÞew in AF ;w are both ½ruw ; so it follows that ter%w #1=d ¼ ðtr #1=dÞew ¼ bu #dðni rÞ=dðni Þ ¼ buðddðni rÞÞ=dðni Þ #1=d: e%
By Lemma 4(1), we have trw AðF Þðddðni rÞÞ=dðni Þ : e%
k
Let kX0 be the largest integer such that trw AðF Þp : Then we have k p Xðddðni rÞÞ=dðni Þ: It follows from Lemma 5(1) that j/ter%w #1=dSOw j ¼ ðd=pk Þ f pðdðni Þ=dðni rÞÞ f ;
ð4:2:3Þ
where f ¼ ½Ow : Zp : Let niþ1 ¼ ni rASiþ1 : From (4.2.2) and (4.2.3) we have Tðni Þ"/aiþ1 #1=pmiþ1 SOw ¼ Tðni Þ"/ðtr #1=dÞew SOw ¼ Tðniþ1 Þ and jW ðniþ1 Þj ¼ jWF ;w =Tðniþ1 Þj ¼ jW ðni Þj=j/ðtr #1=dÞew SOw j X dðni Þ f ðdðniþ1 Þ=dðni ÞÞ f ¼ dðniþ1 Þ f : Using (3.2.5), we get the conclusion. & 4.3. Let MN and LN;0 be as in Section 4.1. Recall that LN;0 is the maximal unramified extension of FN contained in MN (see (4.1.2)). In addition, from Section 2.1 we know that LN;0 is the maximal extension of FN contained in MN in which every prime divisor of FN lying above p is completely decomposed. Since MN =FN is an extension unramified outside p; the group GalðMN =LN;0 Þ is generated by decomposition groups in GalðMN =FN Þ of prime divisors above p: Fix a prime ideal p of F lying above p; and let MN;p =FN;p denote the completion of the extension at a prime divisor lying above p: Let D denote the decomposition group of p in D ¼ GalðF =QÞ; and for any character c of D; let cD denote the restriction of c to D: GalðMN;p =FN;p Þ is an OwD -module because D acts on it via ow1 From the above arguments, we have GalðMN =LN;0 Þ ¼ D : /GalðMN;p =FN;p ÞSOw :
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Lemma 11. GalðMN;p =FN;p Þ is a finite cyclic OwD -module. t:r: Proof. The finiteness follows from GalðMN =FN ÞoN: Let FN;p denote the maximal totally ramified abelian p-extension of FN;p : By local class field theory, we get an ð1Þ ð1Þ t:r: isomorphism GalðFN;p =FN;p ÞCUð1Þ p;N ¼ lim Up;m ; where Up;m denotes the group of ’
ow1
principal units of the completion of Fm for the prime ideal above p: Let FN D be as in ow1
Section 2.1. Since D acts on GalðMN =FN Þ via ow1 ; we see FN CMN CFN D : By the arguments in Section 2.1, the decomposition group of prime divisor above p in GalðMN =FN Þ coincides with the inertia group. Hence, MN;p =FN;p is a totally ramified extension and D acts on GalðMN;p =FN;p Þ via ow1 D : We have a surjection of Lw ¼ Ow ½½G
-modules ðUð1Þ p;N Þow1 #OwD Ow -GalðMN;p =FN;p Þ#OwD Ow : D
ð4:3:1Þ
#OwD Ow is isomorphic to Lw From our assumption wðpÞa1; we know that ðUð1Þ p;N Þow1 D [3, Proposition 1]. On the other hand, G acts on GalðMN =FN Þ via the cyclotomic character k; because of the Kummer pairing GalðMN =FN Þ WF ;w -mpN #Zp Ow ; and WF ;w CðF #Z Qp =Zp Þw : Hence from (4.3.1), we have the surjection of Lw modules: Lw =ðg kðgÞÞLw -GalðMN;p =FN;p Þ#OwD Ow : By the isomorphism Lw =ðg kðgÞÞLw COw ; GalðMN;p =FN;p Þ#OwD Ow is a cyclic Ow module, and hence GalðMN;p =FN;p Þ is a cyclic OwD -module. & For each nAS and each prime ideal r of F lying above a rational prime r 1 ðmod nNF Þ; let br;n be the element of MN;n which is defined in Section 4.1. Fix a prime ideal p of F lying above p; and let MN;n;p denote the completion of MN;n for a prime divisor lying above p: Let gr ðnÞAZ be the largest power of p which satisfies br;n AðMN;n;p Þgr ðnÞ ; and define gðnÞ ¼ minfgr ðnÞ j r: prime ideals of F lying above a rational prime r 1 ðmod nNF Þg: Remark. We can easily see that the integer gðnÞ does not depend on a prime divisor of MN;n lying above p: Proposition 3. Let nAS: dðnÞ f =½MN : MN;n LN;0 ¼ minfgðnÞ; dðnÞg f :
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Proof. Recall the Kummer pairing in Section 4.1: GalðMN =MN;n Þ W ðnÞ-mpN #Zp Ow : By Lemma 10, we can write k
W ðnÞ ¼ " OwD =pmi ; i¼1
OwD =pmi C/bri ;n #1=dðnÞSOwD ; with prime ideals ri of F lying above a rational prime ri 1 ðmod nNF Þ: m For every i with 1pipk; let Ki ¼ MN;n ðfa1=p j a#1=pm A/bri ;n #1=dðnÞSOwD gÞ: Then MN is the composite field of all Ki ; 1pipk: For a fixed prime ideal p of F ; let MN;n;p ; Ki;p and MN;p be the completions of the extensions at a prime divisor lying above p: Clearly, Ki;p ¼ m
MN;n;p ðfa1=p j a#1=pm A/bri ;n #1=dðnÞSOwD gÞ and MN;p is the composite of all Ki;p ; 1pipk: For any prime ideal r of F lying above a rational prime r 1 ðmod nNF Þ; we have br;n #1=dðnÞAW ðnÞ by Lemma 10. Since W ðnÞ is generated by all bri ;n #1=dðnÞ with 1pipk; we get gr ðnÞXminfgri ðnÞ; dðnÞ j 1pipkg by Lemma 4(2). Hence we get minfgðnÞ; dðnÞg ¼ minfgri ðnÞ; dðnÞ j 1pipkg: By Lemma 11, GalðMN;p =MN;n;p Þ is a cyclic OwD -module. Hence we get ½MN;p : MN;n;p ¼ maxf½ðKi Þp :ðMN;n Þp j 1pipkg: (1) Case dðnÞpgri ðnÞ for all i such that 0pipk: In this case, we have Ki;p ¼ MN;n;p for all i: Hence we get MN;p ¼ MN;n;p : Since GalðMN =MN;n LN;0 Þ is generated by decomposition groups in GalðMN =MN;n Þ of prime divisors above p; we have ½MN : MN;n LN;0 ¼ 1; and then we complete the proof in this case. (2) Case dðnÞ4gri ðnÞ for some i: Let i0 be such that gri0 ðnÞ ¼ minfgri ðnÞ j 1pipkg: We have by Lemma 4(2) GalððMN Þp =ðMN;n Þp Þ ¼ GalððKi0 Þp =ðMN;n Þp C OwD =ðdðnÞ=gri0 ðnÞÞ and GalðMN =MN;n LN;0 ÞCOw =ðdðnÞ=gri0 ðnÞÞ: Hence it follows that ½MN : MN;n LN;0 ¼ ðdðnÞ=gri0 ðnÞÞ f ; and the proof if complete. &
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Q Q 4.4. Let JF be the ide`le group of F ; and F v[p Uv be the closure of F v[p Uv in JF where Uv is the group of units of Fv (for an infinite prime divisor, Uv is defined to Q be the multiplicative group of Fv ). Let YF ¼ JF =F v[p Uv and YF fpg denotes the p-primary component of YF : By class field theory, we have the following exact sequence. 0-Uð1Þ =Uð1Þ -EF -YF fpg-AF -0; Q ð1Þ where Uð1Þ ¼ pjp Uð1Þ p ; Up is the group of principal units of Fp ; and EF is the group of units of F : By taking the w-part of the above exact sequence, we get the f isomorphism AF ;w CYF fpgw =Uð1Þ w : Let dAZ40 be as before, that is, d ¼ jAF ;w j and e%w AZ½D such that e%w ew ðmod dÞ: Since dAF ;w ¼ 0; we have d ð1Þ ðUð1Þ w Þ CdðYF fpgw ÞCUw :
By the assumption ow1 ðpÞa1; we have Uð1Þ w COw [3, Proposition 1]. Let n be the order of the residue field of a prime ideal of F lying above p; and r a prime ideal Q of F lying above a rational prime r 1 ðmod NF Þ: By the natural injection F + pjp Fp ; Q e% we can regard ðtr w Þn1 as an element of Uð1Þ ¼ pjp Uð1Þ p : For any integer k40; we see e%
dðYF fpgw Þ ¼ /fðtr w Þn1 j r: prime ideals of F lying above a rational prime r 1 ðmod kNF ÞgSZp ;
ð4:4:1Þ
(cf. [4, Lemma 4.4]). Lemma 12. For any integer k40; there exists a prime ideal r of F lying above a rational prime r 1 ðmod kNF Þ such that gr ð1Þpd: Proof. For mX0 we put Gm ¼ GalðFm =F Þ: By the spectral sequence, we have an exact sequence m
m
p Gm =ðFm;p Þ Þ 0- H 1 ðGm ; " mpm ðFm;p ÞÞ- " Fp =ðFp Þp -ð" Fm;p pjp
pjp
pjp
- H ðGm ; " mpm ðFm;p ÞÞ-?: 2
pjp
By taking the w-part of the above exact sequence, we get ! 8 !Gm 9 < = m m p C " Fm;p =ðFm;p Þ ; " Fp =ðFp Þp : pjp ; pjp w
w
ð4:4:2Þ
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because of ð"pjp mpm ðFm;p ÞÞw ¼ 0 from ow1 ðpÞa1 [3, Proposition 1]. The assertion follows this and (4.4.1). & We write as in Section 4.2, t
i
VwG ¼ " Ow =pe ; i¼1
Ow =pei C/ai #1=pmi SOw ;
e1 X?Xet ;
ðai #1=pmi AðF #Z Qp =Zp Þw Þ:
Proof of Theorem 2. Let nASi ; 0pipt: We can easily see from (4.1.2) that jVwG =TðnÞ-VwG j ¼ ½LN;0 : MN;n -LN;0
¼ ½MN;n LN;0 : MN;n
¼ ½MN : MN;n =½MN : MN;n LN;0
¼ jW ðnÞj=½MN : MN;n LN;0 : From this and Proposition 3, we get jVwG =TðnÞ-VwG j dðnÞ f =jW ðnÞj ¼ dðnÞ f =½MN : MN;n LN;0
¼ minfgðnÞ; dðnÞg f : Note that dðnÞ f =jW ðnÞjX1 from (3.2.5). We can choose ni ASi from Proposition 2 which satisfies i
Tðni Þ ¼ " /aj #1=pmj SOw j¼1
and jW ðni Þj ¼ dðni Þ f : Hence we have minfjVwG =TðnÞ-VwG j j nASi g ¼ jVwG =Tðni Þj ¼ pðeiþ1 þ?þet Þf : Further, since jW ðni Þj ¼ dðni Þ f we have minfjVwG =TðnÞ-VwG j dðnÞ f =jW ðnÞj j nASi g ¼ pðeiþ1 þ?þet Þf :
ð4:4:3Þ
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From (4.4.3), we get minfgðnÞ; dðnÞ j nASi g ¼ peiþ1 þ?þet :
ð4:4:4Þ
On the other hand, we have FN;p ¼ MN;ni ;p for every prime ideal p of F lying above p because FN CMN;ni CLN;0 and p is totally decomposed in LN;0 =FN : By Lemma 12, there exists a prime ideal r of F lying above a rational prime r 1 ðmod ni NF Þ dðn Þ=d such that gr ð1Þpd: By the definition, br;ni ¼ br;1 i is an element of MN;n and gr ðni Þ i Þgr ðni Þ ¼ ðFN;p Þgr ðni Þ : Hence we get is the largest power of p satisfying br;ni AðMN;n;p from gr ð1Þpd that
gr ðni Þ ¼ gr ð1Þ dðni Þ=d p dðni Þ: It concludes from this that minfgðnÞ; dðnÞ j nASi g ¼ minfgðni Þ; dðni Þg ¼ gðni Þ ¼ minfgðnÞ j nASi g; and using (4.4.4), we complete the proof.
&
Proof of Corollary 2. It is sufficient to show the isomorphism Xow1 =ðg kðgÞXow1 CTw : Since we assumed ow1 ðpÞa1; this follows from the isomorphism Tw CHomOw ðXow1 =ðg kðgÞÞXow1 ; mpN #Zp Ow Þ (cf. [8]).
&
Proof of Corollary 3. Let D be the decomposition group of p in D ¼ GalðF =QÞ; and ð1Þ wD the restriction of w to D: Since Uð1Þ w CðUp ÞwD #OwD Ow and (4.4.2), we get the conclusion. &
Acknowledgments I thank the thesis advisor Professor M. Kurihara for valuable discussions and advice.
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