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

TL;DR

This paper provides an elementary p-adic proof and improvements on the computation of the p-adic Kummer-Leopoldt constant, offering a class field theory interpretation and applications without relying on complex analytical methods.

Contribution

It introduces a new elementary approach to the p-adic Kummer-Leopoldt constant, improves existing results, and defines a normalized p-adic regulator using class field theory techniques.

Findings

Elementary proof of the p-adic Kummer-Leopoldt constant

Improved bounds and calculations for the constant

Application to generalizations of Kummer's lemma

Abstract

The p-adic Kummer--Leopoldt constant kappa\_K of a number field K is (assuming the Leopoldt conjecture) the least integer c such that for all n \textgreater{}\textgreater{} 0, any global unit of K, which is locally a p^(n+c)th power at the p-places, is necessarily the p^nth power of a global unit of K. This constant has been computed by Assim \& Nguyen Quang Do using Iwasawa's techniques,after intricate studies and calculations by many authors. We give an elementary p-adic proof and an improvement of these results, then a class field theory interpretation of kappa\_K. We give some applications (including generalizations of Kummer's lemma on regular pth cyclotomic fields) and a natural definition of the normalized p-adic regulator for any K and any p$\ge$2.This is done without analytical computations, using only class field theoryand especially the properties of the so-called p-torsion group T\_K of Abelian p-ramification theory over K.

Equations

$$\epsilon =u^{p^{n+\kappa }}$$

$$\overline{\eta }=\eta (N)\cdot u_N$$

$$\epsilon =\eta (N{)}^{p^n}\cdot u_N^{p^n}\cdot {\xi }_N$$

$$\epsilon =\eta (N{)}^{p^n}\cdot {\phi }_N^{p^n}\in E_K^{p^n}.$$

$$\log (u_0)$$

$$u_0^{p^{\kappa }}={\overline{\epsilon }}_0\cdot {\xi }_0=\epsilon (N)\cdot u_N\cdot {\xi }_0,\epsilon (N)\in E_K,u_N\equiv 1(\mathrm{mod}{p}^N),{\xi }_0\in W_K$$

$$u_0^{\prime }:=u_0\cdot v^{-p^{\kappa }},$$

$$\epsilon (N{)}^{p^n}=u_0^{p^{n+\kappa }}\cdot u_N^{-p^n}=u_0^{p^{n+\kappa }}\cdot (v^{-p^{\kappa }}{)}^{p^{n+\kappa }}=u_0^{\prime p^{n+\kappa }}\in U_K^{p^{n+(\kappa -c)+c}},$$

$$\epsilon (N{)}^{p^n}={\eta }_0^{p^{n+(\kappa -c)}},{\eta }_0\in E_K;$$

$$p^c\cdot \log (u_0^{\prime })=\log ({\eta }_0)\in \log (E_K),$$

$$1\to W_{K}\big{/}{\rm tor}_{\mathbb{Z}_{p}}(\overline{E}_{K})\relbar\mathrel{\mkern-4.0mu}\relbar\mathrel{\mkern-4.0mu}\relbar\mathrel{\mkern-4.0mu}\rightarrow{\rm tor}_{\mathbb{Z}_{p}}\big{(}U_{K}\big{/}\overline{E}_{K}\big{)}\mathop{\relbar\mathrel{\mkern-4.0mu}\relbar\mathrel{\mkern-4.0mu}\relbar\mathrel{\mkern-4.0mu}\rightarrow}^{{\rm log}}{\rm tor}_{\mathbb{Z}_{p}}\big{(}{\rm log}\big{(}U_{K}\big{)}\big{/}{\rm log}(\overline{E}_{K})\big{)}\to 0.$$

$${\mathcal{W}}_{K}:=W_{K}/\mu_{K}\ \$$

$$1\relbar\mathrel{\mkern-4.0mu}\rightarrow{\mathcal{W}}_{K}\relbar\mathrel{\mkern-4.0mu}\relbar\mathrel{\mkern-4.0mu}\relbar\mathrel{\mkern-4.0mu}\rightarrow\ {\rm tor}_{\mathbb{Z}_{p}}\big{(}U_{K}\big{/}\overline{E}_{K}\big{)}\mathop{\relbar\mathrel{\mkern-4.0mu}\relbar\mathrel{\mkern-4.0mu}\relbar\mathrel{\mkern-4.0mu}\rightarrow}^{{\rm log}}{\mathcal{R}}_{K}\relbar\mathrel{\mkern-4.0mu}\rightarrow 0.$$

$$\simeq {\mathcal{W}}_K$$

$${\mathrm{rk}}_p({\mathcal{T}}_K)={\mathrm{rk}}_p(\mathcal{C}{\ell }_K^{S_K\mathrm{res}})+\text{raise}1.5pt\#S_K-1,$$

$$\epsilon =\eta (N{)}^{p^n}\cdot u_N^{p^n}\cdot {\xi }_N$$

$${\rm rk}_{p}({\mathcal{T}}_{K})={\rm rk}_{\omega}\big{(}{\mathcal{C}}\hskip-2.0pt{\ell}_{K^{\prime}}^{S_{K^{\prime}}{\rm res}}\big{)}+\displaystyle\mathop{\raise 2.0pt\hbox{$\sum$}}\limits_{{\mathfrak{p}}\mid p}\delta_{\mathfrak{p}}-\delta,$$

$$\raise 1.5pt\hbox{${\scriptscriptstyle\#}$}{\mathcal{R}}_{K}\sim\frac{1}{2}\cdot\frac{\big{(}\mathbb{Z}_{p}:{\rm log}({\rm N}_{K/\mathbb{Q}}(U_{K}))\big{)}}{\raise 1.5pt\hbox{${\scriptscriptstyle\#}$}{\mathcal{W}}_{K}\cdot\prod_{{\mathfrak{p}}\mid p}{\rm N}{\mathfrak{p}}}\cdot\frac{R_{K}}{\sqrt{D_{K}}},$$

Full text

{altabstract}

La constante $p$-adique de Kummer–Leopoldt $\kappa_{\scriptscriptstyle K}$ d’un corps de nombres $K$ est (sous la conjecture de Leopoldt) le plus petit entier $c$ tel que pour tout $n\gg 0$, toute unité globale de $K$, qui est localement une puissance $p^{n+c}$-ième en les $p$-places, est nécessairement puissance $p^{n}$-ième d’une unité globale de $K$. Cette constante a été calculée par Assim & Nguyen Quang Do en utilisant les techniques d’Iwasawa, après des études et calculs complexes par divers auteurs. Nous donnons une preuve $p$-adique élémentaire et une généralisation de ces résultats, puis une interprétation corps de classes de $\kappa_{\scriptscriptstyle K}$. Nous donnons certaines applications (dont des généralisations du lemme de Kummer sur les $p$-corps cyclotomiques réguliers) et une définition naturelle du régulateur $p$-adique normalisé pour tous $K\ \&\ p\geq 2$. Ceci est fait sans calculs analytiques, en utilisant uniquement le corps de classes et tout spécialement les propriétés du fameux $p$-groupe de torsion ${\mathcal{T}}_{K}$ de la théorie de la $p$-ramification Abélienne sur $K$.

1. Notations

Let $K$ be a number field and let $p\geq 2$ be a prime number; we denote by ${\mathfrak{p}}\mid p$ the prime ideals of $K$ dividing $p$. Consider the group $E_{K}$ of $p$-principal global units of $K$ (i.e., units $\varepsilon\equiv 1\!\pmod{\prod_{{\mathfrak{p}}\mid p}{\mathfrak{p}}}$), so that the index of $E_{K}$ in the group of units is prime to $p$. For each ${\mathfrak{p}}\mid p$, let $K_{\mathfrak{p}}$ be the ${\mathfrak{p}}$-completion of $K$ and $\overline{\mathfrak{p}}$ the corresponding prime ideal of the ring of integers of $K_{\mathfrak{p}}$; then let

U_{K}:=\Big{\{}u\in\displaystyle\mathop{\raise 1.0pt\hbox{\bigoplus}}\limits_{{\mathfrak{p}}\mid p}K_{\mathfrak{p}}^{\times},\ \,u=1+x,\ x\in\displaystyle\mathop{\raise 1.0pt\hbox{\bigoplus}}\limits_{{\mathfrak{p}}\mid p}\overline{\mathfrak{p}}\Big{\}}\,\ \ \&\ \ \,W_{K}:={\rm tor}_{\mathbb{Z}_{p}}(U_{K}),

the $\mathbb{Z}_{p}$-module of principal local units at $p$ and its torsion subgroup.

The $p$-adic logarithm ${\rm log}$ is defined on $1+x$, x\in\displaystyle\mathop{\raise 1.0pt\hbox{\bigoplus}}\limits_{{\mathfrak{p}}\mid p}\overline{\mathfrak{p}}, by means of the series {\rm log}(1+x)=\displaystyle\mathop{\raise 2.0pt\hbox{\sum}}\limits_{i\geq 1}\,(-1)^{i+1}\,\hbox{\footnotesize\displaystyle\frac{x^{i}}{i}}\in\displaystyle\mathop{\raise 1.0pt\hbox{\bigoplus}}\limits_{{\mathfrak{p}}\mid p}K_{\mathfrak{p}}. Its kernel in $U_{K}$ is $W_{K}$ [15, Proposition 5.6].

We consider the diagonal embedding $E_{K}\relbar\mathrel{\mkern-4.0mu}\relbar\mathrel{\mkern-4.0mu}\relbar\mathrel{\mkern-4.0mu}\rightarrow U_{K}$ and its natural extension $E_{K}\otimes\mathbb{Z}_{p}\relbar\mathrel{\mkern-4.0mu}\relbar\mathrel{\mkern-4.0mu}\relbar\mathrel{\mkern-4.0mu}\rightarrow U_{K}$ whose image is $\overline{E}_{K}$, the topological closure of $E_{K}$ in $U_{K}$.

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 ${\rm rk}_{\mathbb{Z}_{p}}(\overline{E}_{K})={\rm rk}_{\mathbb{Z}}(E_{K})$ (see, e.g., [15, § 5.5, p. 75]).

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 $K=\mathbb{Q}(\mu_{p})$ of $p$th roots of unity satisfies the following property stated for the whole group $E^{\prime}_{K}$ of global units of $K$:

any $\varepsilon\in E^{\prime}_{K}$, congruent to a rational integer modulo $p$, is a $p$th power in $E^{\prime}_{K}$.

In fact, $\varepsilon\equiv a\pmod{p}$ with $a\in\mathbb{Z}$, implies $\varepsilon^{p-1}\equiv a^{p-1}\equiv 1\pmod{p}$. So we shall write the Kummer property with $p$-principal units in the more suitable equivalent statement:

any $\varepsilon\in E_{K}$, congruent to $1$ modulo $p$, is a $p$th power in $E_{K}$.

From [1], [11], [13], [14], [16], [17] one can study this property and its generalizations 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 $p\geq 2$. Let $E_{K}$ be the group of $p$-principal global units of $K$ and let $U_{K}$ be the group of principal local units at the $p$-places.

We call Kummer–Leopoldt constant (denoted $\kappa_{\scriptscriptstyle K}=:\kappa$), the smallest integer $c$ such that the following condition is fulfilled:

for all $n\gg 0$, any unit $\varepsilon\in E_{K}$, such that $\varepsilon\in U_{K}^{p^{n+c}}$, is necessarily in $E_{K}^{p^{n}}$.

Remark 2.1**.**

The existence of $\kappa$ 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, we can find a sequence $\varepsilon_{n}\in E_{K}\setminus E_{K}^{p}$ such that ${\rm log}(\varepsilon_{n})\to 0$ (i.e., $\varepsilon_{n}\in U_{K}^{p^{m}}\!\cdot W_{K}$, with $m\to\infty$ as $n\to\infty$); since $W_{K}$ is finite, taking a suitable $p$-power of $\varepsilon_{n}$, we see that $\kappa$ does not exist in that case.

We shall prove (Theorem 3.1) that in the above definition, the condition “for all $n\gg 0$” can be replaced by “for all $n\geq 0$”, 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 $p^{\kappa}$ under the Leopoldt conjecture:

Theorem 2.1**.**

Denote by $E_{K}$ the group of $p$-principal global units of $K$, by $U_{K}$ the $\mathbb{Z}_{p}$-module of principal local units at the $p$-places, and by $W_{K}$ its torsion subgroup. Let $\kappa_{\scriptscriptstyle K}$ be the Kummer–Leopoldt constant (Definition 2.1).

Then $p^{\kappa_{\scriptscriptstyle K}}$ is the exponent of the finite group {\rm tor}_{\mathbb{Z}_{p}}\big{(}{\rm log}(U_{K})/{\rm log}(\overline{E}_{K})\big{)}, where ${\rm log}$ is the $p$-adic logarithm and $\overline{E}_{K}$ the topological closure of $E_{K}$ in $U_{K}$ (whence the relation ${\rm log}(\overline{E}_{K})=\mathbb{Z}_{p}\,{\rm log}(E_{K})$).

Proof.

Let $p^{\kappa}$ be the exponent of {\rm tor}_{\mathbb{Z}_{p}}\big{(}{\rm log}(U_{K})/{\rm log}(\overline{E}_{K})\big{)}.

(i) ($\kappa$ is suitable). Let $n\gg 0$ and let $\varepsilon\in E_{K}$ be such that

[TABLE]

So ${\rm log}(u)$ is of finite order modulo ${\rm log}(\overline{E}_{K})$ and ${\rm log}(\varepsilon)=p^{n}\cdot(p^{\kappa}\cdot{\rm log}(u))=p^{n}\cdot{\rm log}(\overline{\eta})$ with $\overline{\eta}\in\overline{E}_{K}$. By definition of $\overline{E}_{K}$, we can write in $U_{K}$, for all $N\gg 0$,

[TABLE]

we get ${\rm log}(\varepsilon)=p^{n}\cdot{\rm log}(\eta(N))+p^{n}\cdot{\rm log}(u_{N})$ giving in $U_{K}$

[TABLE]

But $\xi_{N}$ is near $1$ (depending on the choice of $n\gg 0$), whence $\xi_{N}=1$ for all $N$, and $\varepsilon=\eta(N)^{p^{n}}\!\!\cdot u^{\prime}_{N}$, $\ u^{\prime}_{N}\to 1$ as $N\to\infty$; so $u^{\prime}_{N}=\varepsilon\cdot\eta(N)^{-p^{n}}$ is a global unit, arbitrary close to $1$, hence, because of Leopoldt’s conjecture [5, Theorem III.3.6.2 (iii, iv)], of the form $\varphi_{N}^{p^{n}}$ with $\varphi_{N}\in E_{K}$ (recall that $n$ is large enough, arbitrary, but fixed), giving

[TABLE]

(ii) ($\kappa$ is the least solution). Suppose there exists an integer $c<\kappa$ having the property given in Definition 2.1. Let $u_{0}\in U_{K}$ be such that

[TABLE]

then ${\rm log}(u_{0}^{p^{\kappa}})={\rm log}(\overline{\varepsilon}_{0})$, $\overline{\varepsilon}_{0}\in\overline{E}_{K}$. This is equivalent to

[TABLE]

hence, for any $n\gg 0$, $u_{0}^{p^{n+\kappa}}=\varepsilon(N)^{p^{n}}\!\!\cdot u_{N}^{p^{n}}$. Taking $N$ large enough, but fixed, we can suppose that $u_{N}=v^{p^{2\kappa}}$, $v\in U_{K}$ near $1$; because of the above relations, ${\rm log}(v)$ is of finite order modulo ${\rm log}(\overline{E}_{K})$, thus ${\rm log}(v^{p^{\kappa}})\in{\rm log}(\overline{E}_{K})$. This is sufficient, for

[TABLE]

to get ${\rm log}(u^{\prime}_{0})$ of order $p^{\kappa}$ modulo ${\rm log}(\overline{E}_{K})$. So we can write:

[TABLE]

but, by assumption on $c$ applied to the global unit $\varepsilon(N)^{p^{n}}$, we obtain

[TABLE]

thus, the above relation $u^{\prime p^{n+\kappa}}_{0}=\varepsilon(N)^{p^{n}}=\eta_{0}^{p^{n+(\kappa-c)}}$ yields:

[TABLE]

which is absurd since ${\rm log}(u^{\prime}_{0})$ is of order $p^{\kappa}$ modulo ${\rm log}(\overline{E}_{K})$. ∎

3. Interpretation of $\kappa_{\scriptscriptstyle K}$ –

Fundamental exact sequence

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]):

[TABLE]

Proof.

The surjectivity comes from the fact that if $u\in U_{K}$ is such that $p^{n}{\rm log}(u)={\rm log}(\overline{\varepsilon})$, $\overline{\varepsilon}\in\overline{E}_{K}$, then $u^{p^{n}}=\overline{\varepsilon}\cdot\xi$ for $\xi\in W_{K}$, hence there exists $m\geq n$ such that $u^{p^{m}}\in\overline{E}_{K}$, whence $u$ gives a preimage in {\rm tor}_{\mathbb{Z}_{p}}\big{(}U_{K}\big{/}\overline{E}_{K}\big{)}.

If $u\in U_{K}$ is such that ${\rm log}(u)\in{\rm log}(\overline{E}_{K})$, then $u=\overline{\varepsilon}\cdot\xi$ as above, giving the kernel equal to $\overline{E}_{K}\cdot W_{K}/\overline{E}_{K}=W_{K}/{\rm tor}_{\mathbb{Z}_{p}}(\overline{E}_{K})$. ∎

Corollary 3.1**.**

Let $\mu_{K}$ be the group of global roots of unity of $p$-power order of $K$.

Then, under the Leopoldt conjecture for $p$ in $K$, we have ${\rm tor}_{\mathbb{Z}_{p}}(\overline{E}_{K})=\mu_{K}$; thus in that case W_{K}\big{/}{\rm tor}_{\mathbb{Z}_{p}}(\overline{E}_{K})=W_{K}/\mu_{K}.

Proof.

From [5, Corollary III.3.6.3], [9, Définition 2.11, Proposition 2.12]. ∎

Put

[TABLE]

Then the exact sequence of Lemma 3.1 becomes:

[TABLE]

Consider the following diagram (see [5], § III.2, (c), Fig. 2.2), under the Leopoldt conjecture for $p$ in $K$:

[TABLE]

where $\widetilde{K}$ is the compositum of the $\mathbb{Z}_{p}$-extensions, ${\mathcal{C}}\hskip-2.0pt{\ell}_{K}$ the $p$-class group, $H_{K}$ the $p$-Hilbert class field, $H_{K}^{\rm pr}$ the maximal Abelian $p$-ramified (i.e., unramified outside $p$) 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”).

By class field theory, ${\rm Gal}(H_{K}^{\rm pr}/H_{K})\simeq U_{K}/\overline{E}_{K}$ in which the image of ${\mathcal{W}}_{K}$ fixes $H_{K}^{\rm reg}$, the Bertrandias–Payan field, ${\rm Gal}(H_{K}^{\rm reg}/\widetilde{K})$ being then the Bertrandias–Payan module, except possibly if $p=2$ in the “special case” (cf. [2] about the calculation of $\kappa$ and the Références in [6] for some history about this module).

But ${\mathcal{R}}_{K}$ giving $\kappa_{\scriptscriptstyle K}$ has, a priori, nothing to do with the definition of the Bertrandias–Payan module associated with $p^{r}$-cyclic extensions of $K$, $r\geq 1$, which are embeddable in cyclic $p$-extensions of $K$ of arbitrary large degree.

Then we put ${\mathcal{T}}^{\prime}_{K}:={\rm tor}_{\mathbb{Z}_{p}}({\rm Gal}(H_{K}^{\rm pr}/H_{K}))\subseteq{\mathcal{T}}_{K}:={\rm tor}_{\mathbb{Z}_{p}}({\rm Gal}(H_{K}^{\rm pr}/K))$. The group ${\mathcal{R}}_{K}$ is then isomorphic to ${\rm Gal}(H_{K}^{\rm reg}/\widetilde{K}H_{K})$. Of course, for $p\geq p_{0}$ (explicit), ${\mathcal{W}}_{K}=[H_{K}:\widetilde{K}\cap H_{K}]=1$, whence ${\mathcal{R}}_{K}={\mathcal{T}}_{K}$. We shall see in the Section 5 that ${\mathcal{R}}_{K}\simeq{\mathcal{T}}^{\prime}_{K}/{\mathcal{W}}_{K}$ is closely related to the classical $p$-adic regulator of $K$.

Corollary 3.2**.**

Under the Leopoldt conjecture for $p$ in $K$, the Kummer–Leopoldt constant $\kappa_{\scriptscriptstyle K}$ of $K$ is [math] if and only if ${\mathcal{R}}_{K}=1$ (i.e., $H_{K}^{\rm reg}=\widetilde{K}H_{K}$).

Proof.

From Theorem 2.1 using the new terminology of the “algebraic regulator” {\mathcal{R}}_{K}:={\rm tor}_{\mathbb{Z}_{p}}\big{(}{\rm log}(U_{K})/{\rm log}(\overline{E}_{K})\big{)} whose exponent is $p^{\kappa}$. ∎

Corollary 3.3**.**

If the prime number $p$ is regular, then $\kappa_{\scriptscriptstyle K}=0$ for the field $K=\mathbb{Q}(\mu_{p})$ of $p$th roots of unity, and any unit $\varepsilon\in E_{K}$ such that $\varepsilon\equiv 1\pmod{p}$ is in $E_{K}^{p}$ (Kummer’s lemma).

Proof.

(i) We first prove that if the real unit $\varepsilon$ is congruent to $1$ modulo $p$ then it is a $p$th power in $U_{K}$. Put $\varepsilon=1+\alpha\cdot p$ for a $p$-integer $\alpha\in K^{\times}$. Let $K_{0}$ be the maximal real subfield of $K$ and let $\pi_{0}$ be an uniformizing parameter of its $p$-completion. Put $\alpha=a_{0}+\beta\cdot\pi_{0}$ with $a_{0}\in[0,p-1]$ and a $p$-integer $\beta$. Since ${\rm N}_{K_{0}/\mathbb{Q}}(\varepsilon)=1$, this yields $a_{0}=0$, whence $\varepsilon=1+\beta\cdot p\cdot\pi_{0}$. The valuation of $p\cdot\pi_{0}$, calculated in $K$, is $p+1$, which is sufficient to get $\varepsilon\in U_{K}^{p}$ (use [15, Proposition 5.7]).

(ii) Then we prove that $\kappa=0$. The cyclotomic field $K=\mathbb{Q}(\mu_{p})$ is $p$-regular and $p$-rational in the meaning of [4, Théorème & Définition 2.1], so ${\mathcal{T}}_{K}=1$ giving $\kappa=0$. In other words, $\kappa=0$ is given by a stronger condition ($p$-rationality of $K$) than ${\mathcal{R}}_{K}=1$.

One may preferably use the general well-known $p$-rank formula (the $p$-rank ${\rm rk}_{p}(A)$ of a finite Abelian group $A$ is the $\mathbb{F}_{p}$-dimension of $A/A^{p}$), valid for any field $K$ under the Leopoldt conjecture, when the group $\mu_{K}$ of $p$th roots of unity is nontrivial [5, Proposition III.4.2.2]:

[TABLE]

where $S_{K}$ is the set of prime ideals of $K$ above $p$ and ${\mathcal{C}}\hskip-2.0pt{\ell}_{K}^{S_{K}\rm res}$ the $S_{K}$-class group in 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 $S_{K}$; so for $K=\mathbb{Q}(\mu_{p})$, we immediately get ${\rm rk}_{p}({\mathcal{T}}_{K})={\rm rk}_{p}({\mathcal{C}}\hskip-2.0pt{\ell}_{K})$, which is by definition trivial for regular primes. ∎

Theorem 3.1**.**

Let $\kappa_{\scriptscriptstyle K}$ be the Kummer-Leopoldt constant of $K$ (Definition 2.1) and let $p^{\nu}$ be the exponent of ${\mathcal{W}}_{K}=W_{K}/\mu_{K}$, where $W_{K}={\rm tor}_{\mathbb{Z}_{p}}(U_{K})$ and $\mu_{K}$ is the group of global roots of unity of $K$ of $p$-power order. 111In the case $\nu=0$, if $\mu_{K}=1$, then $\mu_{K_{\mathfrak{p}}}=1\ \forall{\mathfrak{p}}\in S_{K}$; if $\mu_{K}\neq 1$, then $S_{K}=\{{\mathfrak{p}}\}$ & $\mu_{K_{\mathfrak{p}}}=\mu_{K}$.

The property defining $\kappa_{\scriptscriptstyle K}$ can be improved as follows:

(i) If $\nu\geq 1$, for all $n\geq 0$, any $\varepsilon\in E_{K}$ such that $\varepsilon\in U_{K}^{p^{n+\kappa_{\scriptscriptstyle K}}}$ is necessarily of the form $\varepsilon=\zeta\cdot\eta^{p^{n}}$, with $\zeta\in\mu_{K}\cap W_{K}^{p^{n}}$, $\eta\in E_{K}$.

(ii) If $\nu=0$, for all $n\geq 0$, any $\varepsilon\in E_{K}$ being in $U_{K}^{p^{n+\kappa_{\scriptscriptstyle K}}}$ is necessarily in $E_{K}^{p^{n}}$.

Proof.

Let $n\geq 0$. Suppose that $\varepsilon=u^{p^{n+\kappa}}$, $u\in U_{K}$. So ${\rm log}(\varepsilon)=p^{n}\cdot p^{\kappa}\cdot{\rm log}(u)=p^{n}\cdot{\rm log}(\overline{\eta})$, $\overline{\eta}\in\overline{E}_{K}$; thus $\overline{\eta}=\eta(N)\cdot u_{N}$, with $\eta(N)\in E_{K}$, $u_{N}\equiv 1\!\pmod{p^{N}}$, for all $N\gg 0$, and ${\rm log}(\varepsilon)=p^{n}\cdot{\rm log}(\eta(N))+p^{n}\cdot{\rm log}(u_{N})$ giving in $U_{K}$

[TABLE]

Taking $N$ in a suitable infinite subset of $\mathbb{N}$, we can suppose $\xi_{N}=\xi$ independent of $N\to\infty$. Then \xi=\big{(}\varepsilon\cdot\eta(N)^{-p^{n}}\big{)}\cdot u_{N}^{-p^{n}}\in{\rm tor}_{\mathbb{Z}_{p}}(\overline{E}_{K}), whence $\xi=\zeta\in\mu_{K}$ because of Leopoldt’s conjecture (loc. cit. in proof of Corollary 3.1). Then

$u^{p^{n+\kappa}}=\varepsilon=\eta(N)^{p^{n}}\!\cdot u_{N}^{p^{n}}\!\cdot\zeta=\eta(N)^{p^{n}}\!\cdot u^{\prime}_{N}\cdot\zeta$, $\ \ u^{\prime}_{N}(\in E_{K})\to 1$ as $N\to\infty$,

whence $\varepsilon$ of the form $\eta(N)^{p^{n}}\!\cdot\varphi_{N}^{p^{n}}\!\cdot\zeta$, $\varphi_{N}\in E_{K}$, for $N\gg 0$. So $\varepsilon=\zeta\cdot\eta^{p^{n}}$, with $\eta\in E_{K}$ and $\zeta=\varepsilon\cdot\eta^{-p^{n}}\in\mu_{K}\cap W_{K}^{p^{n}}$, since $\varepsilon$ is a local $p^{n}$th power.

If $\nu=0$, $W_{K}=\mu_{K}$ and $\zeta\in\mu_{K}\cap W_{K}^{p^{n}}=\mu_{K}^{p^{n}}$ is a $p^{n}$th power. ∎

4. Remarks and applications

As above, we assume the Leopoldt conjecture for $p$ in the fields under consideration.

(a) The condition \varepsilon\in U_{K}^{p^{n+\kappa}}=\displaystyle\mathop{\raise 1.0pt\hbox{\bigoplus}}\limits_{{\mathfrak{p}}\mid p}U_{\mathfrak{p}}^{p^{n+\kappa}}, where $U_{\mathfrak{p}}:=1+\overline{\mathfrak{p}}$, may be translated, in the framework of Kummer’s lemma, into a less precise condition of the form $\varepsilon\equiv 1\pmod{\prod_{{\mathfrak{p}}\mid p}{\mathfrak{p}}^{m_{\mathfrak{p}}(n,\kappa)}}$ for suitable minimal exponents $m_{\mathfrak{p}}(n,\kappa)$ giving local $p^{n+\kappa}$th powers. This was used by most of the cited references with $p$-adic analytical calculations using the fact that \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{T}}_{K} is, roughly speaking, a product “class number” $\times$ “regulator” from $p$-adic $L$-functions, giving an upper bound for $\kappa$ (it is the analytic way used in [16] and [13] to generalize Kummer’s lemma when $p$ is not regular).

(b) If ${\mathcal{T}}_{K}=1$ (in which case $\kappa=0$), the field $K$ is said to be a $p$-rational field (see [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 ${\mathcal{T}}_{L}=1$ whence $\kappa_{\scriptscriptstyle L}=0$.

The following examples illustrate this principle:

(i) The $p^{m}$-cyclotomic fields. The above applies for the fields $K_{m}:=\mathbb{Q}(\mu_{p^{m}})$ of $p^{m}$-roots of unity when the prime $p$ is regular, since we have seen that ${\mathcal{T}}_{\mathbb{Q}(\mu_{p})}=1$.

(ii) Some $p$-rational $p$-extensions of $\mathbb{Q}$ ($p=2$ and $p=3$). The following fields have a Kummer-Leopoldt constant $\kappa=0$ ([5, Example IV.3.5.1], after [8, § III]):

– The real Abelian 2-extensions of $\mathbb{Q}$, subfields of the fields $\mathbb{Q}(\mu_{2^{\infty}})\cdot\mathbb{Q}(\sqrt{\ell}\,)$, $\ell\equiv 3\pmod{8}$, and \mathbb{Q}(\mu_{2^{\infty}})\cdot\mathbb{Q}\bigg{(}\hbox{\sqrt{\sqrt{\ell}\ \hbox{\footnotesize$\displaystyle\frac{a-\sqrt{\ell}}{2}$}}}\,\bigg{)},\ \ell=a^{2}+4\,b^{2}\equiv 5\pmod{8}.

– The real Abelian $3$-extensions of $\mathbb{Q}$, subfields of the fields $\mathbb{Q}(\mu_{3^{\infty}})\cdot k_{\ell}$, $\ell\equiv 4,\ 7\pmod{9}$, where $k_{\ell}$ is the cyclic cubic field of conductor $\ell$.

(c) When $\mu_{K}=1$, the formula giving ${\rm rk}_{p}({\mathcal{T}}_{K})$, used in the proof of Kummer’s lemma (Corollary 3.3), must be replaced by a formula deduced from the “reflection theorem”: let $K^{\prime}:=K(\zeta_{p})$, where $\zeta_{p}$ is a primitive $p$th root of unity; then

[TABLE]

which links the $p$-rank of ${\mathcal{T}}_{K}$ to that of the $\omega$-component of the $p$-group of $S_{K^{\prime}}$-ideal classes of the field $K^{\prime}$, where $\omega$ is the Teichmüller character of ${\rm Gal}(K^{\prime}/K)$, $\delta_{\mathfrak{p}}:=1$ or [math] according as the completion $K_{\mathfrak{p}}$ contains $\zeta_{p}$ or not, $\delta:=1$ or [math] according as $\zeta_{p}\in K$ or not (so that $\omega=1$ if and only if $\zeta_{p}\in K$).

(d) Unfortunately, $p^{\kappa}$ may be less than \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{R}}_{K} (hence a fortiori less than \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{T}}_{K}) due to the unknown group structure of ${\mathcal{R}}_{K}$; as usual, when $K/\mathbb{Q}$ is Galois with Galois group $G$, the study of its $G$-structure may give more precise information:

Indeed, to simplify assume $p>2$ unramified in $K$, so that ${\rm log}(U_{K})$ is isomorphic to $O_{K}$, the direct sum of the rings of integers of the $K_{\mathfrak{p}}$, ${\mathfrak{p}}\mid p$; if $\eta=:1+p\cdot\alpha\in E_{K}$ generates a sub-$G$-module of $E_{K}$, of index prime to $p$ (such a unit does exist since $E_{K}\otimes\mathbb{Q}$ is a monogenic $\mathbb{Q}[G]$-module; cf. [5, Corollary I.3.7.2 & Remark I.3.7.3]), the structure of ${\mathcal{R}}_{K}$ can be easily deduced from the knowledge of $P(\alpha)\equiv\frac{1}{p}{\rm log}(\eta)$ modulo a suitable power of $p$, where $P(\alpha)$ is a rational polynomial expression of $\alpha$ generating a sub-$G$-module of $O_{K}$; thus many numerical examples may be obtained.

(e) We have given in [7, § 8.6] a conjecture saying that, in any fixed number field $K$, we have ${\mathcal{T}}_{K}=1$ for all $p\gg 0$, giving conjecturally $\kappa=0$ for all $p\gg 0$.

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 ${\mathcal{R}}_{K}$ associated with the class field theory interpretation of ${\rm Gal}(H_{K}^{\rm reg}/\widetilde{K}H_{K})$.

For this, recall that $\overline{E}_{K}$ is the topological closure, in the $\mathbb{Z}_{p}$-module $U_{K}$ of principal local units at $p$, of the group of $p$-principal global units of $K$, and ${\rm log}$ the $p$-adic logarithm:

Definition 5.1**.**

Let $K$ be any number field and let $p\geq 2$ be any prime number. Under the Leopoldt conjecture for $p$ in $K$, we call {\mathcal{R}}_{K}:={\rm tor}_{\mathbb{Z}_{p}}\big{(}{\rm log}(U_{K})/{\rm log}(\overline{E}_{K})\big{)} (or its order \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{R}}_{K}) the normalized $p$-adic regulator of $K$.

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 $K\neq\mathbb{Q}$, we have, under the Leopoldt conjecture for $p$ in $K$,

[TABLE]

where $\sim$ means equality up to a $p$-adic unit factor, where $R_{K}$ is the usual $p$-adic regulator **[15, § 5.5]** and $D_{K}$ the discriminant of $K$.

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 in $K/\mathbb{Q}$, we get, as defined in [7, Définition 2.3], \displaystyle\raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{R}}_{K}\sim\frac{R_{K}}{p^{[K:\mathbb{Q}]-1}} for $p\neq 2$ and \displaystyle\raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{R}}_{K}\sim\frac{1}{2^{d-1}}\frac{R_{K}}{2^{[K:\mathbb{Q}]-1}} for $p=2$, where $d$ is the number of prime ideals ${\mathfrak{p}}\mid 2$ in $K$.

Of course, \raise 1.5pt\hbox{{\scriptscriptstyle#}}{\mathcal{R}}_{K}=\raise 1.5pt\hbox{{\scriptscriptstyle#}}{\rm tor}_{\mathbb{Z}_{p}}({\rm log}(U_{K}))=1 for $\mathbb{Q}$ and any imaginary quadratic field.