Applications of Baker Theory to the Conjecture of Leopoldt
Preda Mih\u{a}ilescu
In this paper we give a short, elementary proof of the following too extreme cases of the Leopoldt conjecture: the case when $\K/\Q$ is a solvable extension and the case when it is a totally real extension in which $p$ splits completely. The first proof uses Baker theory, the second class field theory. The methods used here are a sharpening of the ones presented at the SANT meeting in G\"ottingen, 2008 and exposed in \cite{Mi2}, \cite{Mi1}.
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\begin{document}
{\obeylines \em 
\vspace*{0.5cm} 
\noindent\hspace*{4.5cm}He will always carry on \newline
\vspace*{-0.4cm}
\noindent\hspace*{4.5cm}Some things are lost, some things are found,\newline
\vspace*{-0.4cm}
\noindent\hspace*{4.5cm}They will keep on speaking his name\newline
\vspace*{-0.4cm}
\noindent\hspace*{4.5cm}Some things are changed, some still the same.\footnote{From a Hymn of \textit{Pretenders}} \newline

\hspace*{6.8cm}To Alan Baker \vspace*{0.0cm} } \title[Baker Theory]
{Applications of Baker Theory to the Conjecture of Leopoldt}
\author{Preda Mih\u{a}ilescu}
\address[P. Mih\u{a}ilescu]{Mathematisches Institut der Universit\"at
  G\"ottingen} \email[P. Mih\u{a}ilescu]{preda@uni-math.gwdg.de}
\keywords{11R23 Iwasawa Theory, 11R27 Units}
\date{Version 1.0 \today}
\vspace{1.0cm}
%\huge
\begin{abstract}
  In this paper we give a short, elementary proof of the following too extreme
cases of the Leopoldt conjecture: the case when $\K/\Q$ is a solvable extension and the case when it is a totally real extension in which $p$ splits completely. The first proof uses Baker theory, the second class field theory. The
  methods used here are a sharpening of the ones presented at the SANT
  meeting in G\"ottingen, 2008 and exposed in \cite{Mi2}, \cite{Mi1}.
\end{abstract}
\maketitle
%\tableofcontents

\section{Introduction}
Let $\K/\Q$ be a finite galois extension and $p$ be a rational prime.
It was conjected by Leopoldt in \cite{Le} that the $p$ - adic
regulator of $\K$ does not vanish. Some equivalent statements are
explained below.  The conjecture was proved for abelian extensions in
1967 by Brumer \cite{Br}, using a local version of Baker's linear
forms in logarithms: the result is known as the Baker-Brumer
theorem. A theorem proved by Ax in \cite{Ax} allows to relate the
Leopoldt conjecture for abelian extensions to transcendency theory. In
his paper, Ax mentions that he could expect his method to work also
for non - abelian extensions. This was attempted by Emsalem and
Kisilevski, who obtained in \cite{EK} results for some particular,
non abelian extensions.

The main result of this paper is
\begin{theorem}
\label{main}
Let $\K/\Q$ be a galois extension and $p$ an odd prime. The Leopoldt
conjecture holds for $\K$ and $p$ in the following cases:
\begin{itemize}
 \item [ A ] The group $\Gal(\K/\Q)$ is solvable.
\item[ B ] The extension $\K/\Q$ is totally real and splits $p$ completely.
\end{itemize}
\end{theorem}
We state from \cite{Br} the central theorem on $p$ - adic
forms in logarithms, which we shall use here:
\begin{theorem}[ Baker and Brumer ]
\label{BB}
Let $\overline{\Q}_p$ be an algebraic closure of $\Q_p$ and $\U
\subset \overline{\Q}_p$ be the units. Let $\alpha_1, \alpha_2,
\ldots, \alpha_n$ be elements of $\U$ which are algebraic over $\Q$
and whose $p$ - adic logarithms exist and are independent over
$\Q$. These logarithms are then independent over $\Q'$, the algebraic
closure of $\Q$ in $\overline{\Q}_p$.
\end{theorem}

\section{Baker theory and Leopoldt's conjecture}
Let $\K/\Q$ be an arbitrary galois field with group $G$, let $p$ be a
rational prime and $P =\{ \wp \subset \id{O}(\K) \ : \ ( p ) \subset
\wp \}$ be the set of conjugate prime ideals above $p$ in $\K$.

We shall prove in this section two important consequences of the
Theorem \ref{BB}, one for absolute and one for relative galois
extensions.

The algebra $\eu{K}(\K) = \K \otimes_{\Q} \Q_p$ is the product of all
completions of $\K$ at the places in $P$:
\[ \K_p = \prod_{\wp \in P} \K_{\wp} .\] The global field $\K$ is
dense in $\K_{\wp}$ in the product topology and $G$ acts on this
completion faithfully, so for any $x \in \K_p, x = \lim_n x_n, x_n \in
\K$ and for all $g \in G$ we have $g ( x ) = \lim_n g ( x_n )$.  The
field $\K$ is dense in $\eu{K}$ (under the product of the topologies
of $\K_{\wp}$), the units $U \subset \K_p$ are products of the units
in $U_{\wp} \subset \K_{\wp}$ and $E$ embeds diagonally to
$\overline{E} \subset U$. We denote by $\iota_{\wp}$ the natural map
$\eu{K} \ra \K_{\wp}$ and identify $a \in \K$ with its image in
$\eu{K}$. The one - units $U^{(1)}$ are defined naturally as the
product $U^{(1)} = \prod_{\wp \in P} U^{(1)}_{\wp}$, with
$U^{(1)}_{\wp} = \{ x \in U_{\wp} : x - 1 \in \id{M}_{\wp}\}$ and
$\id{M}_{\wp}$ the maximal ideal of $U_{\wp}$. Let $\M/\K$ be the
product of all $\Z_p$ - extensions and $\Delta = \Gal(\M/\K)$. The
global Artin symbol is an homomorphism $\varphi: U^{(1)} \ra \Delta$
of $\Z_p[ G ]$ - modules and there is an exact sequence
\begin{eqnarray}
\label{glart}
1 \ra \overline{E} \ra U^{(1)} \ra \Delta \ra 1.
\end{eqnarray}

We let $U' = \{x \in U^{(1)} : \Norm_{\K/\Q}(x) = 1\}$; then
$\overline{E} \subset U'$ by definition. Therefore $U^{(1)}/U' \cong
U^{(1)} \cap \Q_p = U^{(1)}(\Q_p)$ is a quotient which is fixed by
$G$. It is known that $\td{U^{(1)}} \cong \Q_p[ G ]$, so $\Delta$ is a
quasi-cyclic $\Z_p[ G ]$ (see also \cite{Mi}). Since $\K_{\infty}/\K)$
is a $\Z_p$ - extension whose galois group is a quotient of $\Delta$,
invariant under $G$; but $\Delta$ being $\Z_p[ G ]$ - cyclic, it
follows that this quotient is unique up to quasi-isomorphism. We let
$\Delta' = \Gal(\M/\K_{\infty})$; we then have the additional exact
sequence
\begin{eqnarray}
\label{glartinf}
1 \ra \overline{E} \ra U' \ra \Delta' \ra 1.
\end{eqnarray}

We refer to \cite{Mi}, \S \S 2.1, 2.2 and 3.1 for more details on Minkowski
units, idempotents of non commutative group rings and the associated
annihilators, supports and components of $\Z_p[ G ]$ - modules.  We also refer
to \S 2.3 for the description of a choice of the base field $\K$, which
contains the \nth{p^{\kappa}} roots of unity and has some pleasant properties,
such as the fact that the $p$ - ranks of all $\Lambda$ - modules of finite
rank are stationary, all ideals that capitulate have order bounded by
$p^{\kappa}$ and $v_p(| G |) \leq \kappa$. In the same section we describe
Weierstrass modules -- which are $\Z_p$ - torsion free, infinite $\Lambda$ -
modules of finite $p$ - rank -- and prove the fundamental formula
\[ \ord(a_n) = p^{n+1+z(a)} \quad \forall n > 0, \] which
characterizes the orders of $a = (a_n)_{n \in \N} \in W \subset A$,
when $W$ is Weierstrass. Here $\Z \owns z(a) \leq \kappa$ is a
constant depending on $a$ but not on $a_n$. We use the notation
$\varsigma(x) = x^{p^{\kappa}}$ for $x$ in an abelian group; the
choice of $\varsigma$ is such that $\varsigma(A)$ is a Weierstrass
module and for $a \in \underline{A}$, the finite $p$ - torsion part of
$A$, we have $\varsigma(a) = 1$. We write $\KH, \Omega$ for the
maximal $p$ - abelian, unramified, respectively $p$ - ramified
extensions of $\K_{\infty}$. If $\F/\K_{\infty}$ is any extensions and
$F_0 = \Gal(\F/\K_{\infty})^{\circ}$ is the $\Z_p$ - torsion of its
galois group, we write $\overline{\F} = \F^{F_0}$: an extension which
is either trivial or has a Weierstrass - module as galois group; this
group may still be a free $\Lambda$ - module.
 
The conjecture of Leopoldt says that
\[ \zprk( \overline{E} ) = \zrk( E ) . \] Let $\delta \in E$ be a
Minkowski unit with $\delta \equiv 1 \bmod p^2$. Then the $p$ - adic
logarithms of $\delta^g$ exist in all completions $\K_{\wp}$ and for
all $g \in G$. If $A \subset \K_p$ is a multiplicative group, we write
the action of $G$ exponentially, so $a^g = g(a)$.  If $G$ is not
commutative and $g, h \in G$ we have
\begin{eqnarray}
\label{CV}
 a^{g h} = \left(a^g\right)^h = h \circ g ( a ), 
\end{eqnarray}
and the definition of a contravariant multiplication $G \times G \ra
G$ with $g \cdot h = h \circ g$ makes $A$ into a right $\Z_p[ G ]$ -
module, and likewise for $\Z[ G ]$ - modules. In particular, $U,
\overline{E}$ and are $\Z_p[ G ]$ - modules and Minkowski units
generate submodules of maximal $\Z_p$ - rank: since $\K$ is dense in
$\K_p$, it follows that $ \zprk( \overline{ E } ) = \zprk(
\delta^{\Z_p[ G ]})$.  With this structure we also define
\begin{eqnarray*}
  \delta^{\top} =  \{ x \in \Z[ G ] \ : \ \delta^x = 1 \}, \quad \delta_p^{\top} =  
  \{ x \in \Z_p[ G ] \ : \ \delta^x = 1 \},  
\end{eqnarray*}
the $\Z$ - and $\Z_p$ annihilators of $\delta$. Then Leopoldt's
conjecture is also equivalent to 
\begin{eqnarray}
\label{bul}
\delta_p^{\top} = \delta^{\top} \otimes_{\Z} \Z_p.
\end{eqnarray}

In the context of this conjecture we are interested in ranks and not
in torsion of modules over rings. It it thus a useful simplification
to tensor these modules with fields, so we introduce the following
\begin{definition}
\label{tens}
Let $G$ be a finite group and $A, B$ a $\Z$, respectively a $\Z_p$ -
module, which are torsion free.  Let $a \in A, b \in B$. We denote
\begin{eqnarray*}
\hat{A} & = & A \otimes_{\Z} \Q, \quad \hat{a} = a \otimes 1, \\ 
\td{B} & = & B \otimes_{\Z_p} \Q_p, \quad \td{b} = a \otimes 1,  
\end{eqnarray*}
Note that $\zrk(A) = \qrk(\hat{A})$ and $\zprk(B) =
\qprk(\td{B})$. We shall simply write $\rk(X)$ for
the rank of a module when the ring of definition is clear (being one
of $\Z, \Z_p$ or $\Q, \Q_p$.)
\end{definition}

For instance, $\td{E} = \overline{E} \otimes_{\Z_p} \Q_p$. The
definition of $\hat{E}$ is not important for absolute extensions, but
relevant in relative extensions $\KL/\K$, when $\Norm_{\KL/\K}(E(\KL))
\subsetneq E(\K)$.

We start with the case of an absolute extension $\K/\Q$, as introduced
above.  Let $r = r_1 + r_2 - 1 = \zrk(E)$ and $H = \{ g_1, g_2,
\ldots, g_r \} \subset G \setminus \{ 1 \}$ be a maximal set of
automorphisms, such that $\delta^{g_i}$ are $\Z$ - independent. In
particular, there is a $\Z$ - linear map $e : \Z[ G ] \ra \Z[ H ]$
such that
\begin{eqnarray}
\label{rat}
\delta^{\sigma} = \delta^{e(\sigma)}
\end{eqnarray}
for each $\sigma \in G$. The map is the identity on $H$ and
extends to $G$ due to the Minkowski property, which implies that
$\delta^{\Z[ H ]} = \delta^{\Z[ G ]}$.

We have the following consequence of Theorem \ref{BB} 
\begin{lemma}
\label{baker-abs}
Let the notations be like above and $\Z' = \Q' \cap \Z_p$ be the
integers in the algebraic closure $\Q' \subset \Q_p$ of $\Q$. Then
\[ \delta_p^{\top} \cap \Z'[ G ] = \delta^{\top}.\] In particular, if
$\delta_p^{\top} = \alpha \Z_p[ G ]$ with $\alpha \in \Z'[ G ]$, then
Leopoldt's conjecture holds for $\K$.
\end{lemma}
\begin{proof}
  Let $\wp \in P$ be fixed and $\delta_{\tau} =
  \iota_{\wp}(\delta^{\tau})$; then $\delta_{\tau} \in \Z'$. Since $\{
  \delta^{\tau} : \tau \in H\}$ are $\Z$ - independent, $\{
  \delta_{\tau} : \tau \in H\}$ are a fortiori $\Z$ -
  independent. Indeed, if $t \in \Z[ H ]$ was a linear dependence for
  $\delta_{\tau}$, such that $\iota_{\wp}(\delta^t) = 1$, then $d =
  \delta^t \in E$ verifies $\iota_{\wp}(d) = 1$. But in the diagonal
  embedding of $E$, a projection is $1$ if and only if the unit itself
  is $1$, thus $d = 1$: a contradiction of the independence of
  $\delta^{\tau}, \tau \in H$.

  Let $\theta_0 \in \delta_p^{\top} \cap \Z'[ G ]$; in view of
  \rf{rat}, $\theta = e(\theta_0) \in \delta_p^{\top} \cap \Z'[ H ]$
  is also an annihilator. Let $\theta = \sum_{\tau \in H} c_{\tau}
  \tau, \ c_{\tau} \in \Z'$. We show that Theorem \ref{BB} implies
  $\theta = 0$, so $\theta_0 \in e^{-1}(0) \subset \Z[ G ]$ for all
  $\theta_0 \in \delta_p^{\top} \cap \Z'[ G ]$, which is the claim.

 We have $\iota_{\wp}(\delta^{\theta}) = \prod_{\tau \in H}
  \delta_{\tau}^{c_{\tau}} = 1 \in \K_{\wp}, $ and taking the $p$ - adic
  logarithm we find the vanishing linear form in logarithms
  \[ \sum_{\tau \in H} c_{\tau} \log_p(\delta_{\tau}) = 0.\] Since
  $c_{\tau}, \delta_{\tau} \in \Z'$ and $\{ \delta_{\tau} : \tau \in
  H\}$ are $\Z$ - independent, the Theorem of Baker and Brumer implies
  that $\theta = 0$.

  Consequently, if $\delta_p^{\top} = \theta_0 \Z_p[ G ]$ and
  $\theta_0 \in \Z'[ G ]$, then the proof above shows that $\theta_0
  \in \Z[ G ]$, which implies \rf{bul} and confirms Leopoldt's
  conjecture.
\end{proof}

The following definition brings relative annihilators into focus.
\begin{definition}
\label{rel}
Let $\KL \supset \K$ be an extension of number fields with the following
properties:
\begin{itemize}
\item[1.] $\KL/\Q$ is a galois extension with group $G$ and $H =
  \Gal(\KL/\K)$.
\item[2.] Let the \textit{relative annihilator} of $e \in E(\KL)$ be
  defined by
\begin{eqnarray*}
  \td{e}_{\KL/\K}^{\top} & = & \{ x \in \Q_p[ H ] : \td{e}^x \in \td{E(\K)}\}, \\
  e_{\KL/\K}^{\top} & = & \td{e}_{\KL/\K}^{\top} \cap \Z_p[ H ].
\end{eqnarray*}
Then for any global Minkowski unit $\delta \in E(\KL)$ we have
\[ \td{\delta}_{\KL/\K}^{\top} = \Norm_{\KL/\K} \cdot \Q_p[ H ]. \]
\end{itemize}
If points 1. and 2. hold for $\KL/\K$, we say that $\KL/\K$ is
\textit{relative Leopoldt extension, or rL - extension}. If in
addition $\KL$ is real, then the extension is a real relative
Leopoldt, or RL.
\end{definition}

\begin{remark}
  Bruno Angl\`es observed in connection with earlier attempts to use
  Baker theory, that these attempts suggest an approach using relative
  extensions, maybe even a \textit{relative Leopoldt
  conjecture}, stating that if $\KL/\K$ is RL and Leopoldt's
  conjecture holds for $\K$, then it holds for $\KL$. The use of relative
annihilators leads to a proof of statement A. in Theorem \ref{main}. However the relative conjecture encouters a severe obstruction due to the fact that Baker theory only allows statements on the relative annihilator in $\Q_p[ H ]$, but 
the relative conjecture requires annihilators in $\Q_p[ G ]$. 
\end{remark}

We consider next the case of relative abelian extensions:
\begin{lemma}
\label{baker-rel}
Abelian extensions $\KL/\K$ with $\KL/\Q$ galois are relative Leopoldt
extensions. Furthermore, if $\KL/\K$ is galois such that
$\KL_{\wp}/\K_{\wp}$ is abelian for all prime ideals $\wp \in P$, then
$\KL_{\wp}/\K_{\wp}$ is a local relative Leopoldt extension with
respect to $\iota_{\wp}(e)$ for all Minkowski units $e \in E(\KL)$.
\end{lemma}
\begin{proof}
  Let $H = \Gal(\KL/\K)$ be abelian; the extension $\KL/\K$ arises
  from a succession of cyclic extensions of prime degree, so it
  suffices to assume this case. Let $H = \lan \sigma \ran$ with $| H |
  = [ \KL : \K ] = q$, for a prime $q$ which is not necessarily
  different from $p$. The group $\Q_p[ H ]$ decomposes as $\Q_p[ H ] =
  e_1 \Q_p[ H ] \oplus (1-e_1) \Q_p[ H ]$, where $e_1$ is the
  idempotent $\frac{N}{q}$, with $N =\Norm_{\KL/\K}$ . Suppose thus
  that $\td{\delta}_{\KL/\K}^{\top} = (a e_1 + b e_{\chi}) \Q_p[ H ]$,
  where $e_{\chi}$ is a (non trivial) sum of central idempotents for
  the augmentation part $\Q_p[ I_{\KL/\K} ]$ and $a, b \in \{0,
  1\}$. We shall show that $a = 1$ and $b = 0$.

  From the definition of $\td{\delta}_{\KL/\K}^{\top}$ we have
\[ \td{\delta}^{a e_1 + b e_2} = N(\td{\delta})^a \cdot
\td{\delta}^{b e_2} \in \td{E}(\K). \] Since $N(\td{\delta}) \in \td{E}(\K)$,
we also have $d := \td{\delta}^{b e_2} \in \td{E}(\K)$. The group $H$ is
cyclic and $e_2$ is in the augmentation, so $e_2 N = 0$. Taking the norm in
the definition of $d$ and using the fact that $d^{\sigma} = d$ and thus $N(d)
= d^q$, we find that
\[ \td{\delta}^{b e_2 N } = d^q = \td{\delta}^{b e_2 q } = 1. \] But
$e_2 q \in \Z_p[ H ]$ and thus $\delta^{b e_2 q} = 1$: starting from a
relative relation we deduced an absolute annihilator of $\delta$ which
is algebraic. We may apply the Lemma \ref{baker-abs}, concluding that
$e_2 = 0$, since by hypothesis there is no rational dependence for
$\delta$ in the augmentation. This completes the proof.
\end{proof}

As a consequence, we have
\begin{lemma}
\label{relsolv}
Solvable extensions $\KL/\K$ with $\KL/\Q$ real and galois are RL -
extensions.
\end{lemma}
\begin{proof}
  Since $H$ is solvable, there is a chain of intermediate extensions
  $\K_0 = \K \subset \K_1 \subset \K_2 \subset \ldots \subset \K_r =
  \KL$ such that $\K_{i+1}/\K_i$ is abelian for $i = 0, 1, \ldots,
  r-1$ and $\KL/\K_i$ is solvable for all $i$.  The Lemma
  \ref{baker-rel} holds for all $\K_{i+1}/\K_i$. Let $N_i =
  \sum_{\sigma \in \Gals(\K_{i+1}/\K_i)} \sigma$; then $N = N_0 \circ
  N_1 \circ \ldots \circ N_{r-1}$. The claim follows by induction and
  we illustrate this for the case $r = 2$, so $\Gal(\KL/\K_1) = H_1,
  \Gal(\K_1/\K) = H_0$ and $H = H_0 \ltimes H_1$. Furthermore, $\Q_p[
  H ] = \Q_p[ H_0 ] \ltimes\Q_p[ H_1 ]$ where the semidirect product
  $a_0 \ltimes a_1$, with $a_i \in \Q_p[ H_i ], i = 0, 1$ is defined
  term-wise; $N = N_0 \ltimes N_1$ follows from this definition.

  We know from the lemma that $\td{\delta}_{\KL/\K_1}^{\top} = N_1
  \Q_p[ H_1 ]$ and letting $\delta_1 = N_1(\delta) \in \K_1$, the same
  lemma yields $\td{\delta_1}_{\K_1/\K}^{\top} = N_0 \Q_p[ H_0 ]$. It
  follows that $\td{\delta}_{\KL/\K}^{\top} \subset
  \td{\delta_1}_{\K_1/\K}^{\top} \ltimes N_1 \Q_p[ H_1 ] = N_0 \Q_p[
  H_0 ] \ltimes N_1 \Q_p[ H_1 ] = N \Q_p[ H ]$. This way we may prove
  inductively that $\KL/\K_i$ is RL for $i = r-2, r-3, \ldots, 0$.
\end{proof}

The last lemma leads to:
\begin{corollary}
\label{A}
Point A. in Theorem \ref{main} is true.
\end{corollary} 
\begin{proof}
Suppose that $\KL/\Q$ is a real extension with solvable group $H$. If $\delta \in E(\KL)$ is a Minkowski unit, then Lemma \ref{relsolv} implies that its relative annihilator $\td{\delta}_{\KL/\Q}^{\top} = \Norm_{\KL/\Q} \Q_p[ H ]$. Since the base field is $\Q$, the relative annihilator is equal to the absolute one and it follows that $\zprk(\overline{E}) = \zrk(E)$ and Leopoldt's conjecture is true.

The restriction that $\KL$ be real is not important. If $\KL$ is complex, with solvable group, then $< \jmath > = \Gal(\KL/\KL^+) \subset H$ is a normal subgroup, so $\K^+/\Q$ is galois, and Leopoldt's conjecture holds in this case too. 
\end{proof}

We are prepared to prove
\begin{theorem}
\label{lmain}
Let $\K/\Q$ be a totally real extension with group $G$ and $\M/\K$ be
the product of all $\Z_p$ extensions of $\K$, $\K_{\infty}$ the cyclotomic $\Z_p$ - extension of $\K$ and $\KH/\K_{\infty}$ be the maximal $p$ - abelian $p$ - unramified extension. Then $\KH \cap \M = \K_{\infty}$. 
\end{theorem}
\begin{proof}
We adopt a class field theoretic approach for our proof. Let
$\K$ be like in the hypothesis and $(\K_n)_{n \in \N}$ be the
intermediate fields of its cyclotomic $\Z_p$ - extension. We assume
that $\K$ is such that the Leopoldt defect $\id{D}(\K_n) = \id{D}(\K)$
is constant for all $n \geq 0$ and let $\KL = \K[\zeta_p]$.

Firstly, we note that we can exclude the case that $\M/\K_{\infty}$
contains subextensions which split the primes above $p$: this follows
by using the Iwasawa skew symmetric pairing and was proved in
\cite{Mi}, Theorem 3 of \S 3.3. It remains that $\M \subset \Omega_E$
and $\Gal(\M/\K_{\infty}) \hookrightarrow \rg{B}$, where $\rg{B}
\subset A$ is the submodule generated by classes containing ramified
primes above $p$, as defined in \cite{Mi}.

Let $E_n = E(\K_n)$ and $U'_n = U'_n(\K_n)$; we define $U'_{\infty} =
\cup_n U'_n$ and $E_{\infty} = \cup_n E_n$. Then it is known that
$U'_{\infty}/E_{\infty}$ is a torsion $\Lambda$ - module and thus, by
choice of $\varsigma$, we obtain a Weierstrass module
$\varsigma(U'_{\infty}/E_{\infty})$; since $\K$ is totally real, by
reflection we see that
  \[ \varsigma(U'_{\infty}/E_{\infty})^{\bullet} \hookrightarrow
  A^-(\KL).\] Let $X''_n = \{ x \in X_n : N_{\K_n,\K}(x) = 1, n >
  \kappa\}$ for $X \in \{U'_n, E_n\}$. We let $W_n =
  \varsigma(U''_{n}/\overline{E}''_{n})$ and $W =
  \varsigma(U''_{\infty}/\overline{E}''_{\infty})$. Then $W$ is a
  Weierstrass module and we denote by $F$ its characteristic
  polynomial. Since the $T$ - part of $W$ is trivial, $T \nmid F(T)$.
  The Leopoldt defect is stationarity, so $\td{E}_n'' =
  (\td{U}_n'')^+$. Applying $F$ annihilates the diverging part in
  $W_n$ and we obtain:
\begin{eqnarray}
\label{mb}
 \left[ \left((U''_n)^+\right)^{F(T)} : (E''_n)^{F(T)} \right] < M, 
\end{eqnarray}
for a fixed upper bound $M$. In particular, since $(T, F(T)) = 1$, there is a
fixed $m \geq \kappa$, depending on $F(T)$ and $| G |$, such that for all $n >
0$ we have
\begin{eqnarray} 
\label{ucyc}
((U''_n)^+)^{p^m} \subset E''_n \cdot {U''_n}^{T}. 
\end{eqnarray}

Note that $E_n'' \cap {U_n''}^T = {E_n''}^T$; indeed, let $e \in E_n''
\cap {U_n''}^T$.  By Hilbert 90 and the choice of $e$, there are a $w
\in \K_n^{\times}$ and $\xi \in U_n''$ with $e = w^T = \xi^T$, and it
follows that $w = w_0 \cdot \xi$ for some $w_0 \in U(\K)$. The ideal
$(w)$ is ambig above $\K$; since $p^{\kappa}$ annihilates the class
group $A_0$, we have $(w^{p^{\kappa}}) = (\pi \cdot \gamma)$, with
$\gamma \in \K, (\gamma, p) = 1$ and $\pi$ a product of ideals above
$p$. There is a unit $e_1 \in E_n$ such that $e_1 \pi \gamma =
w^{p^{\kappa}} = (w_0 \xi)^{p^{\kappa}}$ and since $\xi$ is a local
unit, it follows that $\pi | w_0$ or $\pi = 1$; moreover, $(\pi
\gamma)^T = \gamma^T = 1$. It remains that $e^{p^{\kappa}} = w^{T
  p^{\kappa}} = e_1^T$, so $e \in E_n^T \cap E_n''$.  Finally we show
that we may choose $e_1 \in E_n''$.  From $(\xi/e_1)^T = 1$ we have
$e_1 = c \xi, c \in \K$ and $N_{n,0}(e_1) = c^{p^{n-\kappa}}$, so $c
\in E(\K)$; the unit $e_2 := e_1/c \in E''_n$ verifies $e_2^T = e$,
which confirms this claim.

In particular, for sufficiently large $n$, we have 
\begin{eqnarray}
\label{perk}
\quad \quad \prk\left(E''_n/\left({E''}_n^{p^n} \cdot ({U''_n}^T \cap E''_n)
\right)\right) & = & \prk\left(E''_n/{E''}_n^{(p^n, T)}\right) \\ 
& = & r_2 - 1.\nonumber
\end{eqnarray}
Assume that $\rg{B}$ is infinite, and let
$\alpha^{\top} \in \Q_p[ G ]$ be its canonic annihilator\footnote{see
  \cite{Mi} \S 7.1 for a definition of canonic annihilators for cyclic
  modules over non - abelian group rings}. We shall write $\zprk(\varsigma(\rg{B})) = \id{D}(\K)$: this is the case for the totally
real extensions $\K$ which split the primes above $p$, as we show in the corollary below; note however that the claim of this proposition is more general and holds independently of this assumption. It does follow from the general case of Leopoldt's conjecture too.

We shall use \rf{ucyc} and \rf{perk} for proving that $\id{D}(\K) = 0$. The
core observation is that, if $\id{D}(\K) > 0$, then there is a defect
emerging in the $\Z$ - rank of $E''_n$, which raises a contradiction
to \rf{perk}.

For $\wp \in P$, we let $\wp_n \in A_n$ be the primes above $\wp$ and
$a_n = [\wp_n] \in A_n$ be their classes, with diverging orders
$\ord(a_n) = p^{n+1+z(a)}$. If $\alpha_n$ approximates $| G | \alpha$
to the power $p^{n+\kappa+1}$, say, then there is a $\nu_n \in \K_n$
such that $(\nu_n) = \wp_n^{p^{\kappa} \cdot \alpha_n}$ and $\nu_n^T =
e_0 \in E''_n$. Since $\nu_n^{| G | (1-\alpha_n)} \in
(\K_n^{\times})^{p^n}$, it follows that the unit $e_0$ is annihilated
in $E''_n/{E''}_n^{(T, p^n)}$ by $|G|(1-\alpha_n)$. We shall show that
$\prk\left(E''_n/{E''}_n^{T, p^n} \right) = r_2 - 1 - \id{D}(\K)$, so
\rf{perk} implies $\id{D}(\K ) = 0$.

Let $B \subset \rg{B}^{\top}$ be an irreducible elementary module with
$\td{B}^{\top} = \beta \in \Q_p[ G ]$, an idempotent dividing
$\alpha$, and let $\beta_n$ be rational approximants of $| G | \cdot
\beta$. Suppose that there is a unit $e \in E''_n$ with $e^{\beta_n}
\in \nu_n^{T \Z[ G ]}$; since $N_n(e) = 1$, it follows from \cite{Mi},
Lemma 16, that $\varsigma(e) = \pi^T$ for some $p$ - unit $\pi$, so
there is a $\theta \in \Z[ G ]$, with $(\pi) = \wp_n^{\theta}$. We may
write
\[ | G | \theta \equiv c \alpha_n + b(1-\alpha_n) \ \bmod p^{n} \Z[ G ];
\ c, b \in p^{\kappa} \Z[ G ], \] and claim that $b \equiv 0$ modulo a large
power of $p$. Upon multiplication with $| G |(1-\alpha_n)$ we obtain a unit
$e_1 = e^{| G |(1-\alpha_n)} = \pi_1^T$ with $(\pi_1) = \wp_n^{b| G
|(1-\alpha_n) + O(p^{n})}$. The identity requires that the ideal $\wp_n^{b| G |(1-\alpha_n)}$
be principal and since $\alpha$ is the minimal annihilator of $a$, for large
$n$, this implies $b \equiv 0 \bmod p^{n-(m+\kappa)}$, which was our claim on
$b$.  It follows that $\beta_n \in \alpha_n \Z[ G ] + p^{n-(m+\kappa)} \Z[ G
]$.  But then, for $m' = 2(m+\kappa)$ and $n > m'$, the quotient
$(E_n'')/((E_n'')^T \cdot (E_n'')^{p^{n-m'}})$ has $p$ - rank $r_2 - 1 -
\id{D}(\K)$ and \rf{perk} implies $\id{D}(\K) = 0$, which completes this
proof.
\end{proof}
As a consequence, 
\begin{corollary}
\label{B}
Point B. in Theorem \ref{main} is true.
\end{corollary}
\begin{proof}
Assume that $\K/\Q$ is totally real and splits $p$ completely and let $\wp \in P$ be any prime above $p$. Then $\K_{\wp} = \Q_p$ and $\M_{\wp}/\K_{\wp}$ is in the product of the two $\Z_p$ - extensions of $\Q_p$: the unramified and the cyclotomic. Consequently $\M/\K_{\infty}$ must be unramified at $p$. This holds for all primes above $p$, so $\M/\K_{\infty}$ is totally unramified. However, by Theorem \ref{lmain} we know that $\M \cap \KH = \K_{\infty}$, so we must have $\M = \K_{\infty}$ and the Leopoldt conjecture holds in this case.

The same argument was used by Greenberg in \cite{Gr} for showing that $\lambda$ may take arbitrarily large values in abelian extenions $\K/\Q$. 
\end{proof}
\begin{remark}
It has been believed for a longer time that the two extreme cases treated by Theorem \ref{main} are the easire one for Lepoldt's conjecture, so this short proof only confirms this general belief. The general proof, given in \cite{Mi}, requires deeper class field theory.

The obstruction encountered, when trying to generalize the results of this paper is the following: let $\K/\Q$ have group $G$ and $\wp \in P$. The facts proven on relative annihilators imply quite easily that $\iota_{\wp}(\overline{E} \cdot \Q_p = \K_{\wp}$. Let $\Delta_{\wp} \subset \Delta = \Gal(\M/\K)$; Leopoldt's conjecture would follow from Theorem \ref{lmain}, if we can prove that 
\[ \Delta_{\wp} \cong U^{(1)}_{\wp}/\iota_{\wp}(\overline{E}), \]
which may appear as a 'reasonable' localization of \rf{glart}. It needs however not be true and all we can say is the following: if $D \subset G$ is the decomposition group of $\wp$, $\td{\delta} = \td{\xi}^{\alpha}$ and
$\alpha = \sum_{\tau \in C} c_{\tau} \tau$ with $c_{\tau} \in \Q_p[ D ]$ and $C = G/D$, then $\sum_{\tau \in C} c_{\tau} \in \Norm_{\M_{\wp}/\K_{\wp}}$. 
\end{remark}

\textbf{Acknowledgments: } Much of the material presented here was completed
after a two day visit of intensive work at the Laboratoire de Math\'ematique
Nicolas Oresme of the University of Caen. I am most grateful to Bruno Angl\'es
and David Vauclair for the helfpul and stimulating discussions which had an
important contribution for clarifying the central ideas of these two papers.

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\end{document}

baknew.bbl
\begin{thebibliography}{1}

\bibitem{Ax}
J.~Ax.
\newblock On the units of an algebraic number field.
\newblock {\em Illinois Journal of Mathematics}, 9:584--589, 1965.

\bibitem{Br}
A.~Brumer.
\newblock On the units of algebraic number fields.
\newblock {\em Mathematika}, 14:121--124, 1967.

\bibitem{EK}
M.~Emsalem, H.~Kisilevsky, and D.~Wales.
\newblock Ind\'ependance lin\'eaire sur $\overline{\Q}$ de logarithmes $p$ -
  adiques de nombres alg\'ebriques et rang $p$ - adique du groupe des unit\'es
  d'un corps de nombres.
\newblock {\em Journal of Number Theory}, 19:384--391, 1984.

\bibitem{Gr}
R.~Greenberg.
\newblock On the iwasawa invariants of totally real fields.
\newblock {\em American Journal of Mathematics}, 98:263--284, 1973.

\bibitem{Le}
H.-W. Leopoldt.
\newblock Zur {Arithmetik} in abelschen {Zahlk\"orpern}.
\newblock {\em J. Reine Angew. Mathematik}, 209:54--71, 1962.

\bibitem{Mi2}
P.~Mih\u{a}ilescu.
\newblock Leopoldt's conjecture for some galois extensions.
\newblock In {\em Proceedings SANT}. Universit\"atsverlag G\"ottingen, 2009.

\bibitem{Mi1}
P.~Mih\u{a}ilescu.
\newblock On {Leopoldt's} conjecture and a special case of {Greenberg's}
  conjecture.
\newblock In {\em Proceedings SANT}. Universit\"atsverlag G\"ottingen, 2009.

\bibitem{Mi}
P.~Mih\u{a}ilescu.
\newblock The {$T$} and {$T^*$} components of {$\Lambda$} - modules and
  {Leopoldt's} conjecture.
\newblock arxiv.org/abs/0905.1274, September 2009.

\end{thebibliography}
