The p-adic Kummer-Leopoldt constant - Normalized p-adic regulator

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by  Georges Gras

THE p-ADIC KUMMER–LEOPOLDT CONSTANT
NORMALIZED p-ADIC REGULATOR
by
7
1 Georges Gras
0
2

r
p
A

3

]
T Abstract. — The p-adic Kummer–Leopoldt constant κK of a number field K is
N (assuming the Leopoldt conjecture) the least integer c such that for all n 0, any
global unit of K, which is locally a pn+cth power at the p-places, is necess≫arily the
.
h pnth power of a global unit of K. This constant has been computed by Assim &
t Nguyen Quang Do using Iwasawa’s techniques, after intricate studies and calcula-
a
m tions by many authors. We give an elementary p-adic proof and an improvement
of these results, then a class field theory interpretation of κK. We give some ap-
[
plications (includinggeneralizations ofKummer’slemma onregular pthcyclotomic

2 fields) and a natural definition of the normalized p-adic regulator for any K and
v any p 2. This is done without analytical computations, using only class field
≥
7 theory and especially the properties of the so-called p-torsion group K of Abelian
5 p-ramification theory over K. T
8
6
0 Résumé. — La constantep-adiquedeKummer–Leopoldt κK d’un corpsdenom-
. bres K est (sous la conjecture de Leopoldt) le plus petit entier c tel que pour tout
1 n 0, toute unité globale de K, qui est localement une puissance pn+c-ième en
0 les≫p-places, est nécessairement puissance pn-ième d’une unité globale de K. Cette
7
constante a été calculée par Assim & Nguyen QuangDo en utilisant les techniques
1
d’Iwasawa, aprèsdesétudesetcalculs complexespardiversauteurs. Nousdonnons
:
v unepreuvep-adiqueélémentaire et unegénéralisation deces résultats, puisunein-
i
X terprétation corps declasses deκK. Nousdonnonscertaines applications (dont des
généralisationsdulemmedeKummersurlesp-corpscyclotomiquesréguliers)etune
r
a définition naturelle du régulateur p-adique normalisé pour tous K & p 2. Ceci
≥
est fait sans calculs analytiques, en utilisant uniquementle corps de classes et tout
spécialement les propriétés du fameux p-groupe de torsion K de la théorie de la
T
p-ramification Abéliennesur K.
1. Notations
Let K be a number field and let p 2 be a prime number; we denote by p p the
≥ |
prime ideals of K dividing p. Consider the group E of p-principal global units of
K
K (i.e., units ε 1(mod p)), so that the index of E in the group of units is
≡ p|p K
prime to p. For each p p,Qlet K bethe p-completion of K and pthe corresponding
p
|
prime ideal of the ring of integers of K ; then let
p
U := u K×, u= 1+x, x p & W := tor (U ),
K ∈ p ∈ K Zp K
n Lp|p Lp|p o
the Z -module of principal local units at p and its torsion subgroup.
p
2 NOTATIONS
The p-adic logarithm log is defined on 1 + x, x p, by means of the series
∈
Lp|p
log(1+x)= ( 1)i+1 xi K . Its kernel in U is W [15, Proposition 5.6].
p K K
iP≥1 − i ∈ Lp|p
We consider the diagonal embedding E U and its natural extension E
K K K
−−−→ ⊗
Z U whose image is E , the topological closure of E in U .
p K K K K
−−−→
In the sequel, these embeddings shall be understood; moreover, we assume in this
paper that K satisfies the Leopoldt conjecture at p, which is equivalent to the
condition rk (E ) = rk (E ) (see, e.g., [15, §5.5, p.75]).
Zp K Z K
2. The Kummer–Leopoldt constant
This notion comes from the Kummer lemma (see, e.g., [15, Theorem 5.36]), that is
to say, if the odd prime number p is “regular”, the cyclotomic field K = Q(µ ) of
p
pth roots of unity satisfies the following property stated for the whole group E′ of
K
global units of K:
any ε E′ , congruent to a rational integer modulo p, is a pth power in E′ .
∈ K K
In fact, ε a (mod p) with a Z, implies εp−1 ap−1 1 (mod p). So we shall
≡ ∈ ≡ ≡
write the Kummer property with p-principal units in the more suitable equivalent
statement:
any ε E , congruent to 1 modulo p, is a pth power in E .
K K
∈
From [1], [11], [13], [14], [16], [17] one can study this property and its general-
izations with various techniques (see the rather intricate history in [2]). Give the
following definition from [2]:
Definition 2.1. — Let K be a number field satisfying the Leopoldt conjecture at
the prime p 2. Let E be the group of p-principal global units of K and let U
K K
≥
be the group of principal local units at the p-places.
We call Kummer–Leopoldt constant (denoted κ =: κ), the smallest integer c such
K
that the following condition is fulfilled:
pn+c pn
for all n 0, any unit ε E , such that ε U , is necessarily in E .
≫ ∈ K ∈ K K
Remark 2.1. — The existence of κ comes from various classical characterizations
of Leopoldt’s conjecture proved for instance in [5, Theorem III.3.6.2], after [14],
[11] and oldest Iwasawa papers. Indeed, if the Leopoldt conjecture is not satisfied,
p pm
we can find a sequence ε E E such that log(ε ) 0 (i.e., ε U W ,
n ∈ K \ K n → n ∈ K · K
with m as n ); since W is finite, taking a suitable p-power of ε , we
K n
→ ∞ → ∞
see that κ does not exist in that case.
We shall prove (Theorem 3.1) that in the above definition, the condition “for all
n 0” can be replaced by “for all n 0”, subject to introduce the group of global
≫ ≥
roots of unity of K and a suitable statement.
We have the following first p-adic result giving pκ under the Leopoldt conjecture:
Theorem 2.1. — Denote by E the group of p-principal global units of K, by
K
U the Z -module of principal local units at the p-places, and by W its torsion
K p K
subgroup. Let κ be the Kummer–Leopoldt constant (Definition 2.1).
K
Then pκK is the exponent of the finite group tor log(U )/log(E ) , where log
Zp K K
is the p-adic logarithm and E the topological clos(cid:0)ure of E in U (cid:1)(whence the
K K K
relation log(E )= Z log(E )).
K p K
2
3 THE KUMMER–LEOPOLDT CONSTANT
Proof. — Let pκ be the exponent of tor log(U )/log(E ) .
Zp K K
(cid:0) (cid:1)
(i) (κ is suitable). Let n 0 and let ε E be such that
K
≫ ∈
ε = upn+κ,u U .
K
∈
Solog(u)isoffiniteordermodulolog(E )andlog(ε) = pn (pκ log(u)) = pn log(η)
K
· · ·
with η E . By definition of E , we can write in U , for all N 0,
K K K
∈ ≫
η = η(N) u , η(N) E , u 1 (mod pN);
N K N
· ∈ ≡
we get log(ε) =pn log(η(N))+pn log(u ) giving in U
N K
· ·
ε = η(N)pn upn ξ , ξ W .
· N · N N ∈ K
But ξ is near 1 (depending on the choice of n 0), whence ξ = 1 for all N,
N N
and ε = η(N)pn u′ , u′ 1 as N ; so u′≫= ε η(N)−pn is a global unit,
· N N → → ∞ N ·
arbitrary close to 1, hence, because of Leopoldt’s conjecture [5, Theorem III.3.6.2
pn
(iii, iv)], of the form ϕ with ϕ E (recall that n is large enough, arbitrary,
N N ∈ K
but fixed), giving
ε = η(N)pn ϕpn Epn.
· N ∈ K
(ii) (κ is the least solution). Suppose there exists an integer c < κ having the
property given in Definition 2.1. Let u U be such that
0 K
∈
log(u ) is of order pκ in tor log(U )/log(E ) ;
0 Zp K K
(cid:0) (cid:1)
pκ
then log(u )= log(ε ), ε E . This is equivalent to
0 0 0 ∈ K
upκ = ε ξ = ε(N) u ξ , ε(N) E , u 1 (mod pN), ξ W ,
0 0· 0 · N · 0 ∈ K N ≡ 0 ∈ K
hence, for any n 0, upn+κ = ε(N)pn upn. Taking N large enough, but fixed, we
≫ 0 · N
can suppose that u = vp2κ, v U near 1; because of the above relations, log(v)
N K
is of finite order modulo log(E ∈), thus log(vpκ) log(E ). This is sufficient, for
K K
∈
u′ := u v−pκ,
0 0·
to get log(u′) of order pκ modulo log(E ). So we can write:
0 K
ε(N)pn = upn+κ u−pn = upn+κ (v−pκ)pn+κ = u′pn+κ Upn+(κ−c)+c,
0 · N 0 · 0 ∈ K
but, by assumption on c applied to the global unit ε(N)pn, we obtain
ε(N)pn = ηpn+(κ−c), η E ;
0 0 ∈ K
thus, the above relation u′pn+κ = ε(N)pn = ηpn+(κ−c) yields:
0 0
pc log(u′) = log(η ) log(E ),
· 0 0 ∈ K
which is absurd since log(u′) is of order pκ modulo log(E ).
0 K
3. Interpretation of κ – Fundamental exact sequence
K
The following p-adic result is valid without any assumption on K and p:
Lemma 3.1. — We have the exact sequence (from [5, Lemma 4.2.4]):
log
1 W tor (E ) tor U E tor log U log(E ) 0.
→ K Zp K −−−→ Zp K K −−−→ Zp K K →
(cid:14) (cid:0) (cid:14) (cid:1) (cid:0) (cid:0) (cid:1)(cid:14) (cid:1)
3
3 INTERPRETATION OF κ – FUNDAMENTAL EXACT SEQUENCE
K
Proof. — The surjectivity comes from the fact that if u U is such that
K
pnlog(u) = log(ε), ε E , then upn = ε ξ for ξ W , hence∈there exists m n
K K
such that upm E ,∈whence u gives a pr·eimage in∈tor U E . ≥
∈ K Zp K K
(cid:0) (cid:14) (cid:1)
If u U is such that log(u) log(E ), then u= ε ξ as above, giving the kernel
K K
∈ ∈ ·
equal to E W /E = W /tor (E ).
K · K K K Zp K
Corollary 3.1. — Let µ be the group of global roots of unity of p-power order of
K
K.
Then, under the Leopoldt conjecture for p in K, we have tor (E ) = µ ; thus in
Zp K K
that case W tor (E ) = W /µ .
K Zp K K K
(cid:14)
Proof. — From [5, Corollary III.3.6.3], [9, Définition 2.11, Proposition 2.12].
Put
:= W /µ & := tor log(U )/log(E ) .
WK K K RK Zp K K
(cid:0) (cid:1)
Then the exact sequence of Lemma 3.1 becomes:
log
1 tor U E 0.
−→ WK −−−→ Zp K K −−−→RK −→
(cid:0) (cid:14) (cid:1)
Consider the following diagram (see [5], §III.2, (c), Fig.2.2), under the Leopoldt
conjecture for p in K:
K
T
′
TK
reg pr
K KH H H
K K K
K K
≃R ≃W
e e
K H H ≃UK/EK
K K
∩
e
ℓK
≃C
K
where K is the compositum of the Z -extensions, ℓ the p-class group, H the p-
p K K
pr C
Hilbert class field, H the maximal Abelian p-ramified (i.e., unramified outside p)
e K
pro-p-extension, of K. These definitions are given in the ordinary sense when p = 2
(so that the real infinite places of K are not complexified (= are unramified) in the
class fields under consideration which are “real”).
pr
By class field theory, Gal(H /H ) U /E in which the image of fixes
K K ≃ K K WK
reg reg
H , the Bertrandias–Payan field, Gal(H /K) beingthentheBertrandias–Payan
K K
module, except possibly if p = 2 in the “special case” (cf. [2] about the calculation
e
of κ and the Références in [6] for some history about this module).
But givingκ has,apriori,nothingtodowiththedefinitionoftheBertrandias–
RK K
Payan module associated with pr-cyclic extensions of K, r 1, which are embed-
≥
dable in cyclic p-extensions of K of arbitrary large degree.
Then we put ′ := tor (Gal(Hpr/H )) := tor (Gal(Hpr/K)). The group
TK Zp K K ⊆ TK Zp K
reg
is then isomorphic to Gal(H /KH ). Of course, for p p (explicit), =
RK K K ≥ 0 WK
[H : K H ] = 1, whence = . We shall see in the Section 5 that
K ∩ K RK e TK
′ / is closely related to the classical p-adic regulator of K.
RK ≃ TeK WK
4
3 INTERPRETATION OF κ – FUNDAMENTAL EXACT SEQUENCE
K
Corollary 3.2. — Under the Leopoldt conjecture for p in K, the Kummer–
reg
Leopoldt constant κ of K is 0 if and only if = 1 (i.e., H = KH ).
K RK K K
Proof. — From Theorem 2.1 usingthenew terminology of the“algeberaic regulator”
:= tor log(U )/log(E ) whose exponent is pκ.
RK Zp K K
(cid:0) (cid:1)
Corollary 3.3. — If the prime number p is regular, then κ = 0 for the field
K
K = Q(µ ) of pth roots of unity, and any unit ε E such that ε 1 (mod p) is
p K
p ∈ ≡
in E (Kummer’s lemma).
K
Proof. — (i) We first prove that if the real unit ε is congruent to 1 modulo p then
it is a pth power in U . Put ε = 1 + α p for a p-integer α K×. Let K
K 0
· ∈
be the maximal real subfield of K and let π be an uniformizing parameter of its
0
p-completion. Put α = a + β π with a [0,p 1] and a p-integer β. Since
0 0 0
· ∈ −
N (ε) = 1, this yields a = 0, whence ε= 1+β p π . The valuation of p π ,
K0/Q 0 · · 0 · 0
calculated in K, is p+1, which is sufficient to get ε Up (use [15, Proposition
∈ K
5.7]).
(ii) Then we prove that κ = 0. The cyclotomic field K = Q(µ ) is p-regular and
p
p-rationalinthemeaningof[4,Théorème&Définition2.1],so = 1givingκ= 0.
K
T
In other words, κ = 0 is given by a stronger condition (p-rationality of K) than
= 1.
K
R
One may preferably use the general well-known p-rank formula (the p-rank rk (A)
p
ofafiniteAbeliangroupAistheF -dimensionofA/Ap),validforanyfieldK under
p
the Leopoldt conjecture, when the group µ of pth roots of unity is nontrivial [5,
K
Proposition III.4.2.2]:
rkp(TK) = rkp(CℓSKKres)+#SK −1,
where S is the set of prime ideals of K above p and ℓSKres the S -class group in
K C K K
the restricted sense (when p = 2) equal to the quotient of the p-class group of K in
the restricted sense by the subgroup generated by the classes of ideals of S ; so for
K
K = Q(µ ), we immediately get rk ( ) = rk ( ℓ ), which is by definition trivial
p p TK p C K
for regular primes.
Theorem 3.1. — Let κ be the Kummer-Leopoldt constant of K (Definition 2.1)
K
and let pν be the exponent of = W /µ , where W = tor (U ) and µ is
WK K K K Zp K K
the group of global roots of unity of K of p-power order.(1)
The property defining κ can be improved as follows:
K
(i) If ν 1, for all n 0, any ε E such that ε Upn+κK is necessarily of the
form ε=≥ζ ηpn, with ζ≥ µ W∈pn,Kη E . ∈ K
· ∈ K ∩ K ∈ K
(ii) If ν = 0, for all n 0, any ε E being in Upn+κK is necessarily in Epn.
≥ ∈ K K K
Proof. — Let n 0. Supposethat ε = upn+κ, u U . So log(ε) =pn pκ log(u) =
K
≥ ∈ · ·
pn log(η), η E ; thus η = η(N) u , with η(N) E , u 1 (mod pN), for
K N K N
· ∈ · ∈ ≡
all N 0, and log(ε) = pn log(η(N))+pn log(u ) giving in U
N K
≫ · ·
ε = η(N)pn upn ξ , ξ W , for all N 0.
· N · N N ∈ K ≫
Taking N in a suitable infinite subset of N, we can suppose ξ = ξ independent of
N
N . Then ξ = ε η(N)−pn u−pn tor (E ), whence ξ = ζ µ because
→ ∞ · · N ∈ Zp K ∈ K
of Leopoldt’s conject(cid:0)ure (loc. cit(cid:1). in proof of Corollary 3.1). Then
(1)In thecase ν =0, if µK =1, then µKp =1 ∀p∈SK; if µK 6=1, then SK ={p} & µKp =µK.
5
4 INTERPRETATION OF κ – FUNDAMENTAL EXACT SEQUENCE
K
upn+κ = ε = η(N)pn upn ζ = η(N)pn u′ ζ, u′ ( E ) 1 as N ,
· N · · N · N ∈ K → → ∞
whence ε of the form η(N)pn ϕpn ζ, ϕ E , for N 0. So ε = ζ ηpn, with
η E and ζ = ε η−pn µ · NW·pn, siNnc∈e ε iKs a local p≫nth power. ·
∈ K · ∈ K ∩ K
If ν = 0, W = µ and ζ µ Wpn = µpn is a pnth power.
K K ∈ K ∩ K K
4. Remarks and applications
Asabove, weassumetheLeopoldtconjectureforpinthefieldsunderconsideration.
(a) The condition ε Upn+κ = Upn+κ, where U := 1+p, may be translated,
∈ K p p
Lp|p
in the framework of Kummer’s lemma, into a less precise condition of the form
ε 1 (mod pmp(n,κ)) for suitable minimal exponents m (n,κ) giving local
≡ p|p p
pn+κth powersQ. This was usedby most ofthe cited references with p-adic analytical
calculations using the fact that # K is, roughly speaking, a product “class number”
T
“regulator” fromp-adicL-functions,givinganupperboundforκ(itistheanalytic
×
way used in [16] and [13] to generalize Kummer’s lemma when p is not regular).
(b) If = 1 (in which case κ = 0), the field K is said to be a p-rational field (see
K
T
[5, §IV.3], [4], [10], [12]). Then in any p-primitively ramified p-extension L of K
(definition and examples in [5, §IV.3, (b); §IV.3.5.1], after [8, Theorem 1, §II.2]),
we get = 1 whence κ = 0.
TL L
The following examples illustrate this principle:
(i) The pm-cyclotomic fields. The above applies for the fields Km := Q(µpm) of
pm-roots of unity when the prime p is regular, since we have seen that = 1.
TQ(µp)
(ii) Some p-rational p-extensions of Q (p = 2 and p = 3). The following fields have
a Kummer-Leopoldt constant κ = 0 ([5, Example IV.3.5.1], after [8, §III]):
– The real Abelian 2-extensions of Q, subfields of the fields Q(µ2∞)·Q(√ℓ), ℓ ≡ 3
(mod 8), and Q(µ2∞)·Q(cid:18)r√ℓ a−2√ℓ (cid:19), ℓ =a2+4b2 ≡ 5 (mod 8).
– The real Abelian 3-extensions of Q, subfields of the fields Q(µ3∞)·kℓ, ℓ ≡ 4, 7
(mod 9), where k is the cyclic cubic field of conductor ℓ.
ℓ
(c) When µ = 1, the formula giving rk ( ), used in the proof of Kummer’s
K p TK
lemma (Corollary 3.3), must be replaced by a formula deduced from the “reflection
theorem”: let K′ := K(ζ ), where ζ is a primitive pth root of unity; then
p p
rk ( ) = rk ℓSK′res + δ δ,
p TK ω(cid:0)C K′ (cid:1) Pp|p p−
which links the p-rank of K tothat of the ω-component of the p-group of SK′-ideal
T
classes of the field K′, where ω is the Teichmüller character of Gal(K′/K), δ := 1
p
or 0 according as the completion K contains ζ or not, δ := 1 or 0 according as
p p
ζ K or not (so that ω = 1 if and only if ζ K).
p p
∈ ∈
(d) Unfortunately, pκ may be less than # K (hence a fortiori less than # K) due
R T
to the unknown group structure of ; as usual, when K/Q is Galois with Galois
K
R
group G, the study of its G-structure may give more precise information:
Indeed, to simplify assume p > 2 unramified in K, so that log(U ) is isomorphic
K
to O , the direct sum of the rings of integers of the K , p p; if η =:1+p α E
K p K
| · ∈
generates a sub-G-module of E , of index prime to p (such a unit does exist since
K
E Q is a monogenic Q[G]-module; cf. [5, Corollary I.3.7.2 & Remark I.3.7.3]),
K
⊗
6
5 REMARKS AND APPLICATIONS
the structure of can be easily deduced from the knowledge of P(α) 1log(η)
RK ≡ p
modulo a suitable power of p, where P(α) is a rational polynomial expression of α
generatingasub-G-moduleofO ;thusmanynumerical examplesmaybeobtained.
K
(e) We have given in [7, §8.6] a conjecture saying that, in any fixed number field
K, we have = 1 for all p 0, giving conjecturally κ= 0 for all p 0.
K
T ≫ ≫
5. Normalized p-adic regulator of a number field
The previous Section 3 shows that the good notion of p-adic regulator comes from
the expression of the p-adic finite group associated with the class field theory
K
reg R
interpretation of Gal(H /KH ).
K K
For this, recall that E is the topological closure, in the Z -module U of principal
K e p K
local units at p, of the group of p-principal global units of K, and log the p-adic
logarithm:
Definition 5.1. — Let K be any number field and let p 2 be any prime number.
≥
Under the Leopoldt conjecture for p in K, we call := tor log(U )/log(E )
RK Zp K K
(or its order # K) the normalized p-adic regulator of K. (cid:0) (cid:1)
R
We have in the simplest case of totally real number fields (from Coates’s formula
[3, Appendix] and also [5, Remarks III.2.6.5] for p = 2):
Proposition 5.1. — For any totally real number field K = Q, we have, under the
6
Leopoldt conjecture for p in K,
1 Zp :log(NK/Q(UK)) RK
#RK ∼ 2 · (cid:0) #WK · p|pNp (cid:1) · √DK,
Q
where means equality up to a p-adic unit factor, where R is the usual p-adic
K
∼
regulator [15, §5.5] and D the discriminant of K.
K
With this expression, we find again classical results obtained by means of analytic
computations (e.g., [1, Theorem 6.5]). In the real Galois case, with p unramified
R
in K/Q, we get, as defined in [7, Définition 2.3], #RK ∼ p[K:QK]−1 for p 6= 2 and
1 R
K
#RK ∼ 2d−12[K:Q]−1 for p = 2, where d is the number of prime ideals p| 2 in K.
Of course, #RK = #torZp(log(UK)) = 1 for Q and any imaginary quadratic field.
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GeorgesGras,VillalaGardette,CheminChâteauGagnière,F–38520LeBourgd’Oisans,France
– https://www.researchgate.net/profile/Georges_Gras E-mail : [email protected]
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