# On Leopoldt’s and Gross’s defects for Artin representations

Alexandre Maksoud  Université du Luxembourg, Unité de Recherche en
Mathématiques, Maison du nombre, 6 avenue de la Fonte, 4364 Esch-sur-Alzette,
Luxembourg<EMAIL_ADDRESS>
###### Abstract.

We generalize Waldschmidt’s bound for Leopoldt’s defect and prove a similar
bound for Gross’s defect for an arbitrary extension of number fields. As an
application, we prove new cases of the generalized Gross conjecture (also
known as the Gross-Kuz’min conjecture) beyond the classical abelian case, and
we show that Gross’s $p$-adic regulator has at least half of the conjectured
rank. We also describe and compute non-cyclotomic analogues of Gross’s defect.

###### Key words and phrases:

Leopoldt conjecture, Gross-Kuz’min conjecture, $p$-adic transcendence theory,
Iwasawa theory, Artin formalism

###### 1991 Mathematics Subject Classification:

11R27 (primary), 11R23 (secondary).

##  1\. Introduction

Dirichlet’s class number formula classically expresses the residue at $s=1$ of
the Dedekind zeta function of a number field $K$ in terms of its Dirichlet
regulator, which is a nonzero determinant involving logarithms of real units
in $K$. Perrin-Riou’s influential program [PR95] suggests that this formula
has a $p$-adic analogue which involves a $p$-adic Dirichlet regulator and an
$\mathcal{L}$-invariant, constructed in terms of Leopoldt’s and Gross’s
$p$-adic regulator maps respectively.

In fact, $p$-adic class number formulae and their appropriate generalizations
to Artin $L$-functions have already been proposed by Tate (and attributed to
Serre) in [Tat84] and Gross in [Gro81] when $K$ is a totally real field or a
CM field. In an earlier work [Mak21], we formulated a precise conjecture for
more general $p$-adic Artin $L$-functions.

Analogously to the complex setting, Leopoldt’s and Gross-Kuz’min’s conjectures
respectively predict that Leopoldt’s and Gross’s $p$-adic regulator maps have
full rank [Leo62, Gro81, Kuz72]. The purpose of this article is to deduce
lower bounds for the ranks of these regulators from classical theorems of
Brumer, Waldschmidt and Roy in $p$-adic transcendence theory [Bru67, Wal81,
Roy92]. In order to do this, we make use of Artin formalism.

We recall that the Gross-Kuz’min conjecture is known when $K/\mathbb{Q}$ is
abelian thanks to Greenberg [Gre73]. More recently, Kleine [Kle19] showed that
this conjecture is true when $K$ has at most two $p$-adic primes. (We note
that Kleine’s approach does not use $p$-adic transcendence theory). In
addition to those two results, our bounds imply many new cases of the Gross-
Kuz’min conjecture. Besides their potential applications to the Tamagawa
number conjecture (as in [BKS17]) in connection with $p$-adic Artin
$L$-functions, the Gross-Kuz’min conjecture and its analogue for non-
cyclotomic Iwasawa theory also yield information on the fine structure of
class groups attached to extensions of $K$ (see e.g. [Kol91, FMD05, Jau17]).
These conjectures are now mainstream in Iwasawa theory of number fields.

We now introduce the main notations and results of this paper.

Let $K$ be a number field and fix once and for all a prime number $p$. We
denote by $S_{p}(K)$ and $S_{\infty}(K)$ the sets of $p$-adic places and
archimedean places of $K$ respectively. Fix an algebraic closure $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}$ of
${\mathbb{Q}_{p}}$ and let $A^{\wedge}=\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}\otimes_{{\mathbb{Z}_{p}}}\varprojlim_{n}A/p^{n}A$ be the $p$-adic
completion of an abelian group $A$. Let

(1)
$\iota_{K}\colon\mathcal{O}_{K}^{\times,\wedge}\longrightarrow\prod_{\mathfrak{P}|p}\mathcal{O}_{K_{\mathfrak{P}}}^{\times,\wedge}$

be the $\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}$-linear map induced by the diagonal embedding of the units of $K$
into all its $p$-adic completions. For $K$ and $p$, the classical Leopoldt’s
conjecture asserts that the map $\iota_{K}$ is injective. The dimension of
$\ker\iota_{K}$ is called Leopoldt’s defect and is denoted by
$\delta^{\textbf{L}}_{K}$ (we will omit its dependence on $p$). Next, we
define the Gross regulator map $\mathscr{L}_{K}$, which is closely related to
$\iota_{K}$. Consider the ${\mathbb{Z}_{p}}$-hyperplane $\mathcal{H}$ of
$\bigoplus_{\mathfrak{P}|p}{\mathbb{Z}_{p}}$ given by the equation
$\sum_{\mathfrak{P}|p}s_{\mathfrak{P}}=0$, and the map

(2)
$\mathscr{L}_{K}\colon\left\\{\begin{array}[]{ccl}\mathcal{O}_{K}[\tfrac{1}{p}]^{\times,\wedge}&\longrightarrow&\mathcal{H}^{\wedge}\\\
x&\mapsto&(-\log_{p}(\mathcal{N}_{\mathfrak{P}}(x)))_{\mathfrak{P}},\end{array}\right.$

where $\mathcal{O}_{K}[\tfrac{1}{p}]^{\times}$ is the group of $p$-units of
$K$, $\mathcal{N}_{\mathfrak{P}}$ is the local norm map for the extension
$K_{\mathfrak{P}}/{\mathbb{Q}_{p}}$, and
$\log_{p}:\mathbb{Q}_{p}^{\times}\to{\mathbb{Q}_{p}}$ is the usual Iwasawa
$p$-adic logarithm. By the usual product formula, $\mathscr{L}_{K}$ is well-
defined. The dimension of $\operatorname{coker}\mathscr{L}_{K}$ is called
Gross’s defect and is denoted by $\delta^{\textbf{G}}_{K}$. Waldschmidt’s
classical bound for Leopoldt’s defect reads
$\delta^{\textbf{L}}_{K}\leq(|S_{\infty}(K)|-1)/2$ [Wal81]. The following
theorem generalizes this to a relative setting and also states a similar bound
for $\delta^{\textbf{G}}_{K}$.

######  Theorem 1.1.

Let $K/k$ be an extension of number fields. The following inequalities hold.

$\begin{array}[]{ccc}\delta^{\textbf{L}}_{K}&\leq&\delta^{\textbf{L}}_{k}+\left(|S_{\infty}(K)|-|S_{\infty}(k)|\right)/2,\\\
\delta^{\textbf{G}}_{K}&\leq&\delta^{\textbf{G}}_{k}+\left(|S_{p}(K)|-|S_{p}(k)|\right)/2.\end{array}$

Moreover, if $K$ has at least one real place and $|S_{p}(K)|\neq|S_{p}(k)|$,
then the second bound is strict.

In particular, the Gross-Kuz’min conjecture holds for all cubic number fields.

A key observation that we use in the computation of $\delta^{\textbf{L}}_{K}$
and $\delta^{\textbf{G}}_{K}$ is that they are compatible with Artin
formalism. For any number field $k\subset\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu$ of absolute
Galois group $G_{k}=\operatorname{Gal}(\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu/k)$, let
$\textrm{Art}_{\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}}(G_{k})$ be the set of finite dimensional $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}$-valued
representations of $G_{k}$ of finite image. We will define defects
$\delta^{\textbf{L}}_{k}(\rho)$ and $\delta^{\textbf{G}}_{k}(\rho)$ associated
with $\rho\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}}(G_{k})$
which satisfy the usual Artin formalism. In particular, when
$\rho=\mathrm{Ind}^{k}_{K}\mathds{1}_{K}$ is the induction from $G_{K}$ to
$G_{k}$ of the trivial representation, they coincide with
$\delta^{\textbf{L}}_{K}$ and $\delta^{\textbf{G}}_{K}$ respectively. We will
also define quantities $d(\rho)$, $d^{+}(\rho)$ and $f(\rho)$ which compute
$[K:\mathbb{Q}]$, $|S_{\infty}(K)|$ and $|S_{p}(K)|$, respectively, when
$\rho=\mathrm{Ind}^{k}_{K}\mathds{1}_{K}$ (see (5)). Our main theorem is the
following.

######  Theorem 1.2.

Let $\rho\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}}(G_{\mathbb{Q}})$ be an irreducible representation and let
$d=d(\rho)$, $d^{+}=d^{+}(\rho)$ and $f=f(\rho)$. If $d^{+}=f=0$, then we have
$\delta^{\textbf{L}}_{\mathbb{Q}}(\rho)=\delta^{\textbf{G}}_{\mathbb{Q}}(\rho)=0$.
Otherwise, we have the following inequalities.

$\delta^{\textbf{L}}_{\mathbb{Q}}(\rho)\leq\dfrac{(d^{+})^{2}}{d+d^{+}},\qquad\delta^{\textbf{G}}_{\mathbb{Q}}(\rho)\leq\dfrac{f^{2}}{d^{+}+2f}.$

By Artin formalism, this yields the upper bound (e.g. for Leopoldt’s defect)
$\delta^{\textbf{L}}_{k}(\rho)\leq d^{+}(\rho)/2$ for an arbitrary
representation $\rho\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}}(G_{k})$.
We immediately recover Theorem 1.1 by choosing $\rho$ such that
$\mathrm{Ind}^{k}_{K}\mathds{1}_{K}=\rho\oplus\mathds{1}_{k}$.

The first bound in Theorem 1.2 is Laurent’s main theorem in [Lau89], but we
will provide a much shorter proof of this result via a lemma on local Galois
representations (Lemma 3.2.7). The second bound, however, cannot be deduced
from the classical methods employed by Laurent [Lau89] and Roy [Roy92] to
study the $p$-adic closure of $S$-units of $K$, for a given finite set of
places $S$. Indeed, they need to assume $p\notin S$, because $\log_{p}$ is not
injective on $\mathcal{O}_{K}[\tfrac{1}{p}]^{\times}$. We circumvent this
issue in Proposition 2.1.2 by determining the kernel of the $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu$-linear
extension of $\log_{p}$.

Theorem 1.2 easily implies Ax-Brumer’s theorem on the validity of Leopoldt’s
conjecture for abelian extensions of an imaginary quadratic field [Ax65,
Bru67]. In the same vein, we indicate two striking applications of Theorem
1.2.

######  Corollary 1.3.

Let $k$ be a totally real field and let $V$ be a totally odd Artin
representation of $G_{k}$. Then Gross’s $p$-adic regulator matrix $R_{p}(V)$
defined in [Gro81, (2.10)] has rank at least half of its size.

This corollary strengthens Gross’s classical result stating that the matrix
$R_{p}(V)$ has positive rank [Gro81, Prop. 2.13].

######  Corollary 1.4.

The Gross-Kuz’min conjecture holds for abelian extensions of imaginary
quadratic fields. It also holds for abelian extensions of real quadratic
fields having at least one real place.

Theorem 4.3.2 provides a more extensive list of number fields for which the
Gross-Kuz’min conjecture holds unconditionally.

The last part of this article is devoted to developing some tools to study
non-cyclotomic analogues of the Gross-Kuz’min conjecture. Given an arbitrary
${\mathbb{Z}_{p}}$-extension $K_{\infty}$ of $K$, we will define a map
$\mathscr{L}_{K_{\infty}/K}$ specializing to $\mathscr{L}_{K}$ if $K_{\infty}$
is the cyclotomic extension of $K$. Interestingly, there do exist examples of
${\mathbb{Z}_{p}}$-extensions $K_{\infty}/K$ for which
$\delta_{K_{\infty}/K}^{\textbf{G}}>0$ [Kis83, JS95], but a conjectural
description of all such $K_{\infty}/K$ is still missing.

######  Theorem 1.5.

Let $k$ be an imaginary quadratic field, and $K$ an abelian extension of $k$
in which $p$ splits completely. Let $K^{\textrm{ab}}/k$ be the maximal
subextension of $K/k$ that is abelian over $\mathbb{Q}$. Then there exist at
most $[K:\mathbb{Q}]-[K^{\textrm{ab}}:\mathbb{Q}]$ distinct
${\mathbb{Z}_{p}}$-extensions $k_{\infty}$ of $k$ for which
$\delta_{Kk_{\infty}/K}^{\textbf{G}}>0$.

In Proposition 3.1.1 we illustrate Theorem 1.5 with a classical application to
the semi-simplicity of Iwasawa modules attached to $Kk_{\infty}/K$. In
addition, we show that the exceptional ${\mathbb{Z}_{p}}$-extensions in
Theorem 1.5, for which $\delta_{Kk_{\infty}/K}^{\textbf{G}}>0$, necessarily
have a transcendental slope (see Example 5.2.3 for the meaning of slope). It
turns out that such ${\mathbb{Z}_{p}}$-extensions should not exist under the
$p$-adic Schanuel conjecture.

Theorem 1.5 can be generalized to arbitrary base fields $k$ having at most $r$
linearly disjoint ${\mathbb{Z}_{p}}$-extensions with $r\leq 2$ (Theorem
5.1.1). The main idea is that, under our assumption on $p$, one can
parameterize ${\mathbb{Z}_{p}}$-extensions of $k$ by points on a
$(r-1)$-dimensional linear subspace $L$ of
$\mathbb{P}^{n-1}({\mathbb{Q}_{p}})$, where $n=[k:\mathbb{Q}]$. The condition
$\delta_{Kk_{\infty}/K}^{\textbf{G}}>0$ then cuts out a closed subvariety
$\mathscr{C}$ of $L$ given by polynomial equations with coefficients in
$\Lambda:=\log_{p}(\mathcal{O}_{K}[\tfrac{1}{p}]^{\times})$. We then exploit
the fact that any linear (resp. algebraic) independence between elements of
$\Lambda$ implies strong conditions on the $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu$-points (resp.
the $\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}$-points) of $\mathscr{C}$.

Theorem 1.5 was inspired by Betina-Dimitrov’s work [BD21] where the authors
show the non-vanishing of a certain anticyclotomic $\mathcal{L}$-invariant for
Katz’s $p$-adic $L$-function. In fact, their result generalizes to any
${\mathbb{Z}_{p}}$-extension with non-transcendental slope. We expect that our
techniques can give further results on the non-vanishing of
$\mathcal{L}$-invariants in more general contexts.

The paper is structured as follows. In Section 2 we recall all the classical
results in $p$-adic transcendence theory which we make use of. In Section 3 we
describe Leopoldt’s and Gross’s defects via class field theory and we show
that they are compatible with Artin formalism. Our main results and
corollaries are proven in Section 4, except for Theorem 1.5 whose proof is
postponed to Section 5.

### Acknowledgments

The author would like to thank Dominik Bullach and Lassina Dembélé for reading
and providing comments on a preliminary version of this paper. This research
is supported by the Luxembourg National Research Fund, Luxembourg, INTER/
ANR/18/12589973 GALF.

##  2\. $p$-adic transcendence theory

Throughout this section we fix an embedding $\iota_{p}\colon\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu\hookrightarrow\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}$, allowing
us to view algebraic numbers as $p$-adic numbers. Let $\Lambda$ be the $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu$-linear
subspace of $\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}$ generated by $1$ and by $p$-adic logarithms of non-zero algebraic
numbers. The following very strong conjecture describes the algebraic
dependence between elements in $\Lambda$.

######  Conjecture ($p$-adic Schanuel conjecture).

If $\lambda_{1},\ldots,\lambda_{n}\in\Lambda$ are linearly independent over
$\mathbb{Q}$, then they are algebraically independent over $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu$.

We recall some classical results of Brumer, Waldschmidt and Roy and deduce
some consequences that turn out to be useful in the study of the Gross-Kuz’min
conjecture.

###  2.1. The Baker-Brumer theorem

Brumer [Bru67] extended Baker’s method to the $p$-adic setting and proved the
following theorem on linear independence of logarithms:

######  Theorem 2.1.1 (Baker-Brumer theorem).

If $\lambda_{1},\ldots,\lambda_{n}\in\Lambda$ are linearly independent over
$\mathbb{Q}$, then they are linearly independent over $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu$.

Recall that $\log_{p}$ is normalized so that we have $\log_{p}(p)=0$.

######  Proposition 2.1.2.

Let $H\subset\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu$ be a number field. The $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu$-linear
extension $\log_{p}\colon\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu\otimes_{\mathbb{Z}}H^{\times}\rightarrow\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}$, $c\otimes
x\mapsto c\log_{p}(\iota_{p}(x))$ of the $p$-adic logarithm has kernel the
line $p^{\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu}$ spanned by $1\otimes p$.

###### Proof.

Let $H_{p}$ be the completion of $\iota_{p}(H)$ inside $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}$, let
$\mathcal{O}^{\times}_{H_{p}}$ be its unit group and consider the abelian
group $\mathcal{T}=\\{x\in H^{\times}\ \colon\
\iota_{p}(x)\in\mathcal{O}_{H_{p}}^{\times}\\}$. Then we clearly have $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu\otimes
H^{\times}=\left(\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu\otimes\mathcal{T}\right)\bigoplus p^{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu}$. Moreover,
the $p$-adic logarithm map is injective on $\mathcal{O}_{H_{p}}^{\times}$, so
multiplicatively independent numbers
$\alpha_{1},\ldots,\alpha_{n}\in\mathcal{T}$ have $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu$-linearly
independent $p$-adic logarithms by the Baker-Brumer theorem. This shows that
the restriction of $\log_{p}$ to $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu\otimes\mathcal{T}$ is injective, hence $\ker(\log_{p})=p^{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu}$. ∎

###  2.2. Waldschmidt’s and Roy’s theorem

Extensions of Baker’s method due to Waldschmidt and Roy give a lower bound for
the rank of matrices with coefficients in $\Lambda$. To each matrix $M$ with
coefficients in $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}$, of size
$m\times\ell$, they assign a number $\theta(M)$ defined as the minimum of all
ratios $\frac{\ell^{\prime}}{m^{\prime}}$ where $(m^{\prime},\ell^{\prime})$
runs among the pairs of integers satisfying $0<m^{\prime}\leq m$ and
$0\leq\ell^{\prime}\leq\ell$, for which there exist matrices
$P\in\operatorname{GL}_{m}(\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu)$ and
$Q\in\operatorname{GL}_{\ell}(\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu)$ such that the
product $PMQ$ can be written as

$\begin{pmatrix}M^{\prime}&0\\\ N&M^{\prime\prime}\end{pmatrix}$

with $M^{\prime}$ of size $m^{\prime}\times\ell^{\prime}$. Note that
$\theta(M)\leq\tfrac{\ell}{m}$ with equality if all the entries of $M$ are
$\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu$-linearly independent. The following theorem is due to Waldschmidt
[Wal81, Théorème 2.1.p] and Roy [Roy92, Corollary 1].

######  Theorem 2.2.1.

Let $M$ be a matrix with coefficients in $\Lambda$, of size $m\times\ell$ with
$m,\ell>0$, and let $n$ be its rank. We have

$n\geq\frac{\theta(M)}{1+\theta(M)}\cdot m.$

Roy also deduced a useful corollary for $3\times 2$ matrices from Theorem
2.2.1 in [Roy92, Corollary 2].

######  Corollary 2.2.2 (Strong six exponentials theorem).

Let $M$ be a $(3\times 2)$-matrix with coefficients in $\Lambda$. If the rows
of $M$ are $\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu$-linearly independent, and if the columns of $M$ are also $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu$-linearly
independent, then $M$ has rank 2.

##  3\. Regulator maps and class groups

###  3.1. Galois cohomology

For all fields $L\subset\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu$ and all finite
sets $S$ of places of $L$ containing $S_{p}(L)$, we let $X(L)$ (resp.
$X^{\prime}_{S}(L)$) be the Galois group of the maximal abelian pro-$p$
extension of $L$ which is unramified everywhere (resp. unramified everywhere
and totally split at all $v\in S$). If $S=S_{p}(L)$, we simply put
$X^{\prime}(L)=X^{\prime}_{S}(L)$. Given a ${\mathbb{Z}_{p}}$-extension
$K_{\infty}=\bigcup_{n}K_{n}$ of $K$ with Galois group $\Gamma$, we have
$X(K_{\infty})=\varprojlim_{n}X(K_{n})$ and
$X_{S}^{\prime}(K_{\infty})=\varprojlim_{n}X_{S}^{\prime}(K_{n})$, the
transition maps being the restriction maps. Therefore, $X(K_{\infty})$ and
$X^{\prime}_{S}(K_{\infty})$ are modules over the Iwasawa algebra
${\mathbb{Z}_{p}}[[\Gamma]]$. They are finitely generated torsion as shown by
Iwasawa [Iwa73]. We let

$\delta^{\textbf{G}}_{K_{\infty}/K}:=\operatorname{rk}_{{\mathbb{Z}_{p}}}X^{\prime}(K_{\infty})_{\Gamma},$

where $(-)_{\Gamma}$ means $\Gamma$-coinvariants. If $K_{\infty}$ is the
cyclotomic ${\mathbb{Z}_{p}}$-extension $K_{\textrm{cyc}}$ of $K$ we simply
write $\delta^{\textbf{G}}_{K}$ for $\delta^{\textbf{G}}_{K_{\infty}/K}$. We
will later see that this definition is compatible with that of the
introduction. One motivation in classical Iwasawa theory to compute
$\delta^{\textbf{G}}_{K_{\infty}/K}$ originates in the following simple result
by Jaulent and Sands [JS95, Proposition 6].

######  Proposition 3.1.1.

Let $\gamma$ be a topological generator of $\Gamma$. If no $p$-adic prime of
$K$ splits completely in $K_{\infty}$ and if
$\delta^{\textbf{G}}_{K_{\infty}/K}=0$, then $\gamma-1$ acts semi-simply on
$X(K_{\infty})$. That is, $(\gamma-1)^{2}$ does not divide the elements
$P_{i}\in{\mathbb{Z}_{p}}[[\Gamma]]$ appearing in any elementary module
$\bigoplus_{i}{\mathbb{Z}_{p}}[[\Gamma]]/(P_{i})$ pseudo-isomorphic to
$X(K_{\infty})$.

Let $S_{0}\supseteq S_{p}(K)\bigcup S_{\infty}(K)$ be a finite set of places
of $K$. For any extension $L$ of $K$ and any discrete (resp. compact)
$G_{L}$-module $M$ which is unramified outside the places of $L$ above
$S_{0}$, we consider for all $i\geq 0$ the $S_{0}$-ramified $i$-th cohomology
group (resp. continuous cohomology group)
$\mathrm{H}^{i}_{S_{0}}(L,M)=\mathrm{H}^{i}(\operatorname{Gal}(L_{S_{0}}/L),M)$,
where $L_{S_{0}}/L$ is the largest extension of $L$ which is unramified
outside the places of $L$ above ${S_{0}}$. Given any subset $S\subset{S_{0}}$,
let

${}_{S}^{i}(L,M)=\ker\left[\mathrm{H}^{i}_{{S_{0}}}(L,M)\longrightarrow\prod_{v\in
S}\mathrm{H}^{i}(L_{v},M)\times\prod_{v\in
S_{0}-S}\mathrm{H}^{i}(L_{v}^{\mathrm{ur}},M)\right],$

where $L_{v}^{\mathrm{ur}}/L_{v}$ denotes the maximal unramified extension of
$L_{v}$ (so $\mathbb{R}^{\mathrm{ur}}=\mathbb{R}$ in particular) and the maps
above are the usual localization maps. Note that the definition of
${}_{S}^{i}(L,M)$ does not depend on the choice of $S_{0}$. We simply write
${}^{i}(L,M)$ instead of ${}_{S}^{i}(L,M)$ if $S=S_{p}(L)\bigcup
S_{\infty}(L)$. We also write $M^{*}$ for the Pontryagin dual
$\operatorname{Hom}_{{\mathbb{Z}_{p}}}(M,{\mathbb{Q}_{p}}/{\mathbb{Z}_{p}})$
of a ${\mathbb{Z}_{p}}$-module $M$.

######  Lemma 3.1.2.

Let $L\subset\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu$ be a number field and let $S\supset S_{p}(L)\bigcup S_{\infty}(L)$ be a
finite set of places of $L$. There are canonical isomorphisms
${}_{S}^{2}(L,{\mathbb{Z}_{p}}(1))\simeq{}_{S}^{1}(L,{\mathbb{Q}_{p}}/{\mathbb{Z}_{p}})^{*}\simeq
X^{\prime}_{S}(L)$.

###### Proof.

The first isomorphism is given by the Poitou-Tate duality theorem [Mil86,
Theorem 4.10 (a)]. Since
$\mathrm{H}^{1}_{S}(L,{\mathbb{Q}_{p}}/{\mathbb{Z}_{p}})=\operatorname{Hom}(\operatorname{Gal}(L_{S}/L),{\mathbb{Q}_{p}}/{\mathbb{Z}_{p}})$,
class field theory easily implies
${}^{1}(L,{\mathbb{Q}_{p}}/{\mathbb{Z}_{p}})=\operatorname{Hom}(X^{\prime}_{S}(L),{\mathbb{Q}_{p}}/{\mathbb{Z}_{p}})$.
∎

The isomorphisms provided by Lemma 3.1.2 are functorial in $L$ in the sense
that, given a finite extension $L^{\prime}/L$ of number fields, the norm map
$X(L^{\prime})\rightarrow X(L)$ corresponds to the corestriction map (resp. to
the Pontryagin dual of the restriction map)
${}^{2}(L^{\prime},\mathbb{Z}_{p}(1))\rightarrow{}^{2}(L,{\mathbb{Z}_{p}}(1))$
(resp.
${}^{1}(L^{\prime},{\mathbb{Q}_{p}}/{\mathbb{Z}_{p}})^{*}\rightarrow{}^{1}(L,{\mathbb{Q}_{p}}/{\mathbb{Z}_{p}})^{*}$).

Given any ${\mathbb{Z}_{p}}$-extension $K_{\infty}=\bigcup_{n}K_{n}$ of $K$
and any finite set $S\supset S_{p}(K)\bigcup S_{\infty}(K)$, Lemma 3.1.2
provides isomorphisms of ${\mathbb{Z}_{p}}[[\Gamma]]$-modules

$\varprojlim_{n}{}_{S}^{2}(K_{n},{\mathbb{Z}_{p}}(1))\simeq{}_{S}^{1}(K_{\infty},{\mathbb{Q}_{p}}/{\mathbb{Z}_{p}})^{*}\simeq
X^{\prime}_{S}(K_{\infty}).$

We now make use of the inflation-restriction exact sequence to study the
problem of Galois descent. We have a commutative diagram with exact rows

(3)
${0}$${\operatorname{Hom}(\Gamma,{\mathbb{Q}_{p}}/{\mathbb{Z}_{p}})}$${\mathrm{H}^{1}_{\\{p,\infty\\}}(K,{\mathbb{Q}_{p}}/{\mathbb{Z}_{p}})}$${\mathrm{H}^{1}_{\\{p,\infty\\}}(K_{\infty},{\mathbb{Q}_{p}}/{\mathbb{Z}_{p}})^{\Gamma}}$${0}$${0}$${\bigoplus_{v}\operatorname{Hom}(\Gamma_{v},{\mathbb{Q}_{p}}/{\mathbb{Z}_{p}})}$${\bigoplus_{v}\mathrm{H}^{1}(K_{v},{\mathbb{Q}_{p}}/{\mathbb{Z}_{p}})}$${\bigoplus_{w}\mathrm{H}^{1}(K_{\infty,w},{\mathbb{Q}_{p}}/{\mathbb{Z}_{p}}),}$

where the places $v$ (resp. $w$) of the second row run through all the
$p$-adic and archimedean places of $K$ (resp. of $K_{\infty}$). Here, we have
used the fact that $\Gamma$ has cohomological dimension one as it is pro-
cyclic.

######  Proposition 3.1.3.

We have
$\delta^{\textbf{G}}_{K_{\infty}/K}=\dim\ker(\mathrm{Loc}_{K_{\infty}/K})$,
where $\mathrm{Loc}_{K_{\infty}/K}$ is the localization map

$\mathrm{Loc}_{K_{\infty}/K}\colon\mathrm{H}^{1}_{\\{p\\}}(K,\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p})/\mathrm{H}^{1}(\Gamma,\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p})\longrightarrow\bigoplus_{\mathfrak{P}\in
S_{p}(K)}\mathrm{H}^{1}(K_{\mathfrak{P}},\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p})/\mathrm{H}^{1}(\Gamma_{\mathfrak{P}},\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}).$

###### Proof.

By Lemma 3.1.2, the kernel of the right vertical map of (3) is equal to the
Pontryagin dual of $X^{\prime}(K_{\infty})_{\Gamma}$. Since
$\mathrm{H}^{1}(K_{v},{\mathbb{Q}_{p}}/{\mathbb{Z}_{p}})$ for $v|\infty$ is
finite, this implies that $\delta_{K_{\infty}/K}^{\textbf{G}}$ is equal to the
rank of the Pontryagin dual of the kernel of the natural map

$\mathrm{H}^{1}_{\\{p\\}}(K,{\mathbb{Q}_{p}}/{\mathbb{Z}_{p}})/\mathrm{H}^{1}(\Gamma,{\mathbb{Q}_{p}}/{\mathbb{Z}_{p}})\rightarrow\bigoplus_{\mathfrak{P}\in
S_{p}(K)}\mathrm{H}^{1}(K_{\mathfrak{P}},{\mathbb{Q}_{p}}/{\mathbb{Z}_{p}})/\mathrm{H}^{1}(\Gamma_{\mathfrak{P}},{\mathbb{Q}_{p}}/{\mathbb{Z}_{p}}).$

To end the proof, it suffices to notice that for
$\mathcal{G}=\operatorname{Gal}(K_{S}/K)$, $\operatorname{Gal}(\mkern
1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern
1.5mu_{\mathfrak{P}}/K_{\mathfrak{P}})$, $\Gamma$ or $\Gamma_{\mathfrak{P}}$,
the natural map
$\mathrm{H}^{1}(\mathcal{G},{\mathbb{Z}_{p}})\otimes{\mathbb{Q}_{p}}/{\mathbb{Z}_{p}}\rightarrow\mathrm{H}^{1}(\mathcal{G},{\mathbb{Q}_{p}}/{\mathbb{Z}_{p}})$
(resp. $\mathrm{H}^{1}(\mathcal{G},{\mathbb{Z}_{p}})\otimes\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}\rightarrow\mathrm{H}^{1}(\mathcal{G},\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p})$) has
finite kernel and cokernel (resp. is an isomorphism), see [Rub00, Appendix B §
2]. ∎

######  Remark 3.1.4.

Since $\mathrm{H}^{1}_{\\{p\\}}(K,\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p})=\operatorname{Hom}(G_{K},\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p})$
parameterizes the ${\mathbb{Z}_{p}}$-extensions of $K$, the domain of
$\mathrm{Loc}_{K_{\infty}/K}$ has dimension $r_{2}+\delta^{\textbf{L}}_{K}$,
where $r_{2}$ is the number of complex places of $K$ ([Was97, Theorem 13.4]).
In particular, Proposition 3.1.3 yields an upper bound
$\delta^{\textbf{G}}_{K_{\infty}/K}\leq r_{2}+\delta^{\textbf{L}}_{K}$.
Therefore, Leopoldt’s conjecture for a totally real field $K$ implies the
Gross-Kuz’min conjecture for $K$, as already noticed by Kolster in [Kol91,
Corollary 1.3].

Given a prime $\mathfrak{P}\in S_{p}(K)$, let $\Gamma_{\mathfrak{P}}$ be the
decomposition subgroup of $\Gamma$ at $\mathfrak{P}$ and denote by
$\operatorname{rec}_{\Gamma_{\mathfrak{P}}}\colon
K_{\mathfrak{P}}^{\times}\to\Gamma_{\mathfrak{P}}$ the corresponding local
reciprocity map. Define also the ${\mathbb{Z}_{p}}$-module

$\mathcal{H}_{K_{\infty}/K}:=\ker\left(\bigoplus_{\mathfrak{P}\in
S_{p}(K)}\Gamma_{\mathfrak{P}}\longrightarrow\Gamma\right).$

By the usual product formula in class field theory the regulator map

(4)
$\mathscr{L}_{K_{\infty}/K}\colon\left\\{\begin{array}[]{ccl}\mathcal{O}_{K}[\tfrac{1}{p}]^{\times}&\longrightarrow&\mathcal{H}_{K_{\infty}/K}\\\
x&\mapsto&(\operatorname{rec}_{\Gamma_{\mathfrak{P}}}(x))_{\mathfrak{P}},\end{array}\right.$

is well-defined, and it extends to a $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}$-linear map
$\mathcal{O}_{K}[\tfrac{1}{p}]^{\times,\wedge}\to\mathcal{H}_{K_{\infty}/K}^{\wedge}$
which we still denote by $\mathscr{L}_{K_{\infty}/K}$. If
$K_{\infty}=K_{\textrm{cyc}}$, then the character
$\log_{p}\circ\chi_{\textrm{cyc}}\circ\operatorname{rec}_{\Gamma_{\mathfrak{P}}}\colon
K_{\mathfrak{P}}^{\times}\longrightarrow{\mathbb{Q}_{p}}$ coincides with
$-\log_{p}\circ\mathcal{N}_{\mathfrak{P}}$, where $\chi_{\textrm{cyc}}$ is the
cyclotomic character. Therefore, $\mathscr{L}_{K_{\textrm{cyc}}/K}$ is
essentially the same as the map $\mathscr{L}_{K}$ of the introduction, and we
easily see that
$\delta_{K_{\textrm{cyc}}/K}^{\textbf{G}}=\delta_{K}^{\textbf{G}}$.

######  Proposition 3.1.5.

We have
$\delta^{\textbf{G}}_{K_{\infty}/K}=\dim\operatorname{coker}(\mathscr{L}_{K_{\infty}/K})$.

###### Proof.

By Kummer theory and local class field theory, Tate’s local pairing
$\mathrm{H}^{1}(K_{\mathfrak{P}},\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p})\times\mathrm{H}^{1}(K_{\mathfrak{P}},\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}(1))\rightarrow\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}$ can be
identified with the evaluation map
$\operatorname{Hom}(K_{\mathfrak{P}}^{\times},\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p})\times
K_{\mathfrak{P}}^{\times,\wedge}\rightarrow\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}$.
Therefore, the orthogonal complement of
$\mathrm{H}^{1}(\Gamma_{\mathfrak{P}},\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p})\subseteq\mathrm{H}^{1}(K_{\mathfrak{P}},\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p})$ is simply
the kernel of $\operatorname{rec}_{\Gamma_{\mathfrak{P}}}\colon
K_{\mathfrak{P}}^{\times,\wedge}\twoheadrightarrow\Gamma^{\wedge}_{\mathfrak{P}}$.
Using the fact that ${}^{1}(K,{\mathbb{Q}_{p}})=0$, Poitou-Tate’s duality then
yields an isomorphism

$\ker\left(\mathrm{H}^{1}_{\\{p\\}}(K,\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p})\to\bigoplus_{\mathfrak{P}\in
S_{p}(K)}\frac{\mathrm{H}^{1}(K_{\mathfrak{P}},\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p})}{\mathrm{H}^{1}(\Gamma_{\mathfrak{P}},\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p})}\right)\simeq\operatorname{coker}\left(\oplus_{\mathfrak{P}}\operatorname{rec}_{\Gamma_{\mathfrak{P}}}:\
\mathcal{O}_{K}[\tfrac{1}{p}]^{\times,\wedge}\to\bigoplus_{\mathfrak{P}\in
S_{p}(K)}\Gamma^{\wedge}_{\mathfrak{P}}\right).$

In particular,
$\dim\ker(\mathrm{Loc}_{K_{\infty}/K})+1=\dim\operatorname{coker}(\mathscr{L}_{K_{\infty}/K})+1$,
so Proposition 3.1.3 yields the desired equality. ∎

###  3.2. Isotypic components

We consider in this paragraph the situation where the
${\mathbb{Z}_{p}}$-extension $K_{\infty}/K$ comes from the
${\mathbb{Z}_{p}}$-extension $k_{\infty}/k$ of a subfield $k$ of $K$, which
means that $K_{\infty}=Kk_{\infty}$. Assume that $K/k$ is Galois with Galois
group $G$. Given an algebraically closed field $\mathbf{Q}$ of characteristic
zero (typically, $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu$ or $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}$), the
Q-valued representations of $G$ are semi-simple and the regular representation
of $G$ splits as

$\textbf{Q}[G]=\bigoplus_{\rho}e(\rho)\cdot\textbf{Q}[G]=\bigoplus_{\rho}W^{\oplus\dim\rho},$

where $(W,\rho)$ runs through the set of all the Q-valued irreducible
representations of $G$ and $e(\rho)=\frac{\dim\rho}{|G|}\cdot\sum_{g\in
G}\operatorname{Tr}(\rho(g^{-1}))g\in\textbf{Q}[G]$ is the usual idempotent
attached to $\rho$.

For any finite set of places $S$ of $k$ containing $S_{\infty}(k)$, let
$\mathcal{O}_{K}[1/S]^{\times}$ be the group of $S$-units of $K$. Dirichlet’s
unit theorem implies that we have a decomposition of $\textbf{Q}[G]$-modules

$\textbf{Q}\otimes_{\mathbb{Z}}\mathcal{O}_{K}[1/S]^{\times}=\left(\textbf{Q}\otimes_{\mathbb{Z}}\mathcal{O}_{k}[1/S]^{\times}\right)\bigoplus\left(\bigoplus_{\mathds{1}\neq\rho}W^{\oplus
d^{+}_{S}(\rho)}\right),$

where $(W,\rho)$ runs through the set of all non-trivial irreducible
representations of $G$ and $d^{+}_{S}(\rho)=\sum_{v\in
S}\dim\mathrm{H}^{0}(k_{v},W)$. It will be convenient to introduce the
following invariants:

(5) $d(\rho):=[k:\mathbb{Q}]\cdot\dim\rho,\qquad
d^{+}(\rho):=\sum_{v|\infty}\dim\mathrm{H}^{0}(k_{v},W),\qquad
f(\rho):=\sum_{\mathfrak{p}|p}\dim\mathrm{H}^{0}(k_{\mathfrak{p}},W),$

so that $d^{+}(\rho)=d^{+}_{S_{\infty}(k)}(\rho)$ and
$f(\rho)=d^{+}_{S_{\infty}(k)\bigcup
S_{p}(k)}(\rho)-d^{+}_{S_{\infty}(k)}(\rho)$.

We record in the next lemma a list of useful properties satisfied by the
invariants introduced in (5) and which we make use of in Sections 4 and 5.
Recall that a rule $\rho\mapsto a(\rho)\in\mathbb{Z}$, where $\rho$ runs among
all the representations of Galois groups of finite extensions of number
fields, is said to be compatible with Artin formalism if, for all finite
Galois extensions $M/L/E$:

  1. (a)

$a(\tilde{\rho})=a({\rho})$ if $\tilde{\rho}\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}}(\operatorname{Gal}(M/E))$ is the inflation of
$\rho\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}}(\operatorname{Gal}(L/E))$,

  2. (b)

$a(\rho_{1}\oplus\rho_{2})=a(\rho_{1})+a(\rho_{2})$ for all
$\rho_{1},\rho_{2}\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}}(\operatorname{Gal}(M/E))$, and

  3. (c)

$a(\rho^{\prime})=a(\rho)$ if $\rho^{\prime}$ is the induction of $\rho$ from
$\operatorname{Gal}(M/L)$ to $\operatorname{Gal}(M/E)$.

######  Lemma 3.2.1.

Let $L/E$ be a finite extension of number fields and let
$a\in\\{d,d^{+},f\\}$.

  1. (1)

The rule $\rho\mapsto a(\rho)$ is compatible with Artin formalism.

  2. (2)

We have $d(\mathds{1}_{L})=[L:\mathbb{Q}]$,
$d^{+}(\mathds{1}_{L})=|S_{\infty}(L)|$ and $f(\mathds{1}_{L})=|S_{p}(L)|$.
For any representation $(W,\rho)\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}}(\operatorname{Gal}(M/L))$ with $\mathds{1}_{L}\not\subset\rho$, we
have
$\dim\operatorname{Hom}_{\operatorname{Gal}(M/L)}(W,\mathcal{O}_{M}^{\times,\wedge})=d^{+}(\rho)$
and
$\dim\operatorname{Hom}_{\operatorname{Gal}(M/L)}(W,\mathcal{O}_{M}[\tfrac{1}{p}]^{\times,\wedge})=d^{+}(\rho)+f(\rho)$.

  3. (3)

Let $\rho\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}}(G_{L})$
and let $M=\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu^{\ker\rho}$ be the field extension cut out by $\rho$. Then $M$ is
totally real (resp. $L$ is totally real and $M$ is a CM field) if and only if
$d^{+}(\rho)=d(\rho)$ (resp. $d^{+}(\rho)=0$). If $M$ has at least one real
place, then $d^{+}(\rho)\geq\dim\rho$ and any subrepresentation $\theta$ of
$\mathrm{Ind}_{L}^{E}\rho$ satisfies $d^{+}(\theta)\geq 1$. If $L$ has $r$
complex places, then $d^{+}(\rho)\geq r\cdot\dim\rho$.

  4. (4)

Let $\rho\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}}(G_{L})$.
Then we have $a(\rho)\leq(\dim\rho)\cdot a(\mathds{1}_{L})$.

  5. (5)

Let $M/L$ be a finite Galois extension, let $\theta\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}}(\operatorname{Gal}(M/L))$ be irreducible and let $\chi$ be a
multiplicative character of $G_{L}$. Then we have $(\dim\theta)\cdot
a(\theta\otimes\chi)\leq a(\chi_{|G_{M}})\leq a(\mathds{1}_{M})$. Moreover, if
$\theta\neq\mathds{1}_{L}$, then we have $(\dim\theta)\cdot
a(\theta\otimes\chi)\leq a(\chi_{|G_{M}})-a(\chi)\leq
a(\mathds{1}_{M})-a(\chi)$.

###### Proof.

The claims (1), (2) and (4) are easy to deduce from the definitions and
Dirichlet’s unit theorem. Let us justify claim (3). It is clear that
$d^{+}(\rho)\leq d(\rho)$, and we have $d^{+}(\rho)=d(\rho)$ (resp.
$d^{+}(\rho)=0$) if and only if all archimedean places of $L$ are real and
$\rho(\sigma)=\operatorname{id}$ (resp. $\rho(\sigma)=-\operatorname{id}$) for
all complex conjugations $\sigma\in\operatorname{Gal}(M/L)$. Since $\rho$ is
faithful on $\operatorname{Gal}(M/L)$, this is equivalent to $M$ being totally
real (resp. $M$ being CM and $L$ totally real). Assume now that $M$ has at
least one real place $w$ and let $v$ (resp. $v_{0}$) be the place of $L$
(resp. of $E$) lying below $w$. Then we clearly have
$d^{+}(\rho)\geq\dim\mathrm{H}^{0}(L_{v},\rho)=\dim\rho$. Moreover, if
$\theta\subset\mathrm{Ind}_{L}^{E}\rho$, then the Frobenius reciprocity
implies that there exists a subrepresentation $\rho^{\prime}\subset\rho$ such
that $\rho^{\prime}\subset\theta_{|G_{L}}$, yielding
$d^{+}(\theta)\geq\dim\mathrm{H}^{0}(E_{v_{0}},\theta)=\dim\mathrm{H}^{0}(L_{v},\theta)\geq\mathrm{H}^{0}(L_{v},\rho^{\prime})=\dim\rho^{\prime}\geq
1$ as claimed. Suppose finally that $L$ has $r$ complex places
$v_{1},\ldots,v_{r}$. Then
$d^{+}(\rho)\geq\sum_{i=1}^{r}\dim\mathrm{H}^{0}(L_{v_{i}},\rho)=r\cdot\dim\rho$,
so this ends the proof of claim (3).

We now prove claim (5). The upper bounds on $a(\chi_{|G_{M}})$ directly follow
from claim (4), so we only prove the lower bounds. Since $\theta$ is
irreducible, the representation $(\theta\otimes\chi)^{\oplus\dim\theta}$ (and
even $(\theta\otimes\chi)^{\oplus\dim\theta}\oplus\chi$ if
$\theta\neq\mathds{1}_{L}$) occurs as a subrepresentation of
$(\mathrm{Ind}_{M}^{L}\mathds{1}_{M})\otimes\chi=\mathrm{Ind}_{M}^{L}\chi_{|G_{M}}$.
Artin formalism then yields the lower bounds of claim (5), as
$a(\mathrm{Ind}_{M}^{L}\chi_{|G_{M}})=a(\chi_{|G_{M}})$ and $a$ takes non-
negative values. ∎

We now describe the isotypic components of the map
$\mathscr{L}_{K_{\infty}/K}$. For $g\in G$, $\mathfrak{P}\in S_{p}(K)$ and
$\eta$ a place of $K_{\infty}$ above $\mathfrak{P}$, the map
$K_{\mathfrak{P}}\to K_{g(\mathfrak{P})}$ (resp. $K_{\infty,\eta}\to
K_{\infty,\tilde{g}(\eta)})$ induced by $g$ (resp. by a lift
$\tilde{g}\in\operatorname{Gal}(K_{\infty}/k)$ of $g$) is a field isomorphism
which yields a left $G$-action $x\mapsto g(x)$ (resp.
$\gamma\mapsto\tilde{g}\gamma\tilde{g}^{-1}$) on
$\bigoplus_{\mathfrak{P}|p}K_{\mathfrak{P}}^{\times}$ and on
$\bigoplus_{\mathfrak{P}|p}\Gamma_{\mathfrak{P}}$ respectively. This action
also restricts to $\mathcal{H}_{K_{\infty}/K}$, and $G$ acts trivially on the
quotient
$(\bigoplus_{\mathfrak{P}|p}\Gamma_{\mathfrak{P}})/\mathcal{H}_{K_{\infty}/K}$.
Moreover, the map $\mathscr{L}_{K_{\infty}/K}$ is $G$-equivariant for the
natural $G$-action on $\mathcal{O}_{K}[\tfrac{1}{p}]^{\times}$ and the action
on $\mathcal{H}_{K_{\infty}/K}$ described above.

Fix any $(W,\rho)\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}}(G)$ and
let $\operatorname{Hom}_{G}(X,Y)$ be the $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}$-vector
space of all $G$-equivariant linear maps between two $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}[G]$-modules
$X$ and $Y$. By definition, the $\rho$-isotypic component of a $G$-equivariant
$\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}$-linear map $f:X\to Y$ is the linear map
$\operatorname{Hom}_{G}(W,X)\to\operatorname{Hom}_{G}(W,Y)$ obtained by post-
composing by $f$. Write $\mathscr{L}_{k_{\infty}/k}(\rho)$ for the
$\rho$-isotypic component of $\mathscr{L}_{K_{\textrm{cyc}}/K}$ and define

$\delta_{k_{\infty}/k}^{\textbf{G}}(\rho):=\dim\operatorname{coker}\left(\mathscr{L}_{k_{\infty}/k}(\rho)\right).$

If $k_{\infty}=k_{\textrm{cyc}}$, we abbreviate
$\mathscr{L}_{k_{\infty}/k}(\rho)$ and
$\delta_{k_{\infty}/k}^{\textbf{G}}(\rho)$ as $\mathscr{L}_{k}(\rho)$ and
$\delta^{\textbf{G}}_{k}(\rho)$ respectively.

For all $\mathfrak{p}\in S_{p}(k)$, fix a place $\mathfrak{P}_{0}$ of $K$
above $\mathfrak{p}$, let $G_{\mathfrak{p}}$ be the decomposition subgroup of
$G$ at $\mathfrak{P}_{0}$, let $W_{\mathfrak{p}}^{0}=W^{G_{\mathfrak{p}}}$ and
denote by $\mathrm{res}_{\mathfrak{p}}$ the restriction-
to-$W_{\mathfrak{p}}^{0}$ map.

######  Proposition 3.2.2.

The map $\mathscr{L}_{k_{\infty}/k}(\mathds{1})$ can be naturally identified
with $\mathscr{L}_{k_{\infty}/k}$. If $\mathds{1}\not\subset\rho$, then the
map $\mathscr{L}_{k_{\infty}/k}(\rho)$ can be naturally identified with the
composite map

(6)
${\operatorname{Hom}_{G}(W,\mathcal{O}_{K}[\tfrac{1}{p}]^{\times,\wedge})}$${\bigoplus_{\mathfrak{p}}\operatorname{Hom}_{G_{\mathfrak{p}}}(W,K_{\mathfrak{P}_{0}}^{\times,\wedge})}$${\bigoplus_{\mathfrak{p}}\operatorname{Hom}(W_{\mathfrak{p}}^{0},k_{\mathfrak{p}}^{\times,\wedge})}$${\bigoplus_{\mathfrak{p}}\operatorname{Hom}(W_{\mathfrak{p}}^{0},\Gamma_{\mathfrak{p}}^{\wedge}).}$$\scriptstyle{\oplus\mathrm{loc}_{\mathfrak{p}}}$$\scriptstyle{\oplus\mathrm{res}_{\mathfrak{p}}}$$\scriptstyle{\oplus\operatorname{rec}_{\Gamma_{\mathfrak{p}}}}$

Here, $\mathfrak{p}$ runs over $S_{p}(k)$ in each sum, and we implicitly used
the fact that
$\left(K_{\mathfrak{P}_{0}}^{\times,\wedge}\right)^{G_{\mathfrak{p}}}=k_{\mathfrak{p}}^{\times,\wedge}$.

###### Proof.

Let
$j\colon\mathcal{H}_{K_{\infty}/K}\hookrightarrow\bigoplus_{\mathfrak{P}|p}\Gamma_{\mathfrak{P}}$
be the inclusion map and let $j(\rho)$ be its $\rho$-isotypic component.

Given a prime $\mathfrak{p}\in S_{p}(k)$ and a fixed prime
$\mathfrak{P}_{0}|\mathfrak{p}$ of $K$ as before, we have
$\bigoplus_{\mathfrak{P}|\mathfrak{p}}K_{\mathfrak{P}}^{\times}=\mathrm{Ind}_{G_{\mathfrak{p}}}^{G}K_{\mathfrak{P}_{0}}^{\times}$
and
$\bigoplus_{\mathfrak{P}|\mathfrak{p}}\Gamma_{\mathfrak{P}}=\mathrm{Ind}_{G_{\mathfrak{p}}}^{G}\Gamma_{\mathfrak{P}_{0}}$
as $G$-modules and Frobenius reciprocity shows that
$\operatorname{Hom}_{G}(W,\bigoplus_{\mathfrak{P}|\mathfrak{p}}K_{\mathfrak{P}}^{\times,\wedge})\simeq\operatorname{Hom}_{G_{\mathfrak{p}}}(W,K_{\mathfrak{P}_{0}}^{\times,\wedge})$
and
$\operatorname{Hom}_{G}(W,\bigoplus_{\mathfrak{P}|\mathfrak{p}}\Gamma_{\mathfrak{P}}^{\times,\wedge})\simeq\operatorname{Hom}_{G_{\mathfrak{p}}}(W,\Gamma_{\mathfrak{P}_{0}}^{\times,\wedge})$,
the isomorphisms being the natural projection maps. Therefore,
$j(\rho)\circ\mathscr{L}_{k_{\infty}/k}(\rho)$ can be identified with the
composite map

${\operatorname{Hom}_{G}(W,\mathcal{O}_{K}[\tfrac{1}{p}]^{\times,\wedge})}$${\bigoplus_{\mathfrak{p}|p}\operatorname{Hom}_{G_{\mathfrak{p}}}(W,K_{\mathfrak{P}_{0}}^{\times,\wedge})}$${\bigoplus_{\mathfrak{p}|p}\operatorname{Hom}_{G_{\mathfrak{p}}}(W,\Gamma_{\mathfrak{P}_{0}}^{\wedge})=\bigoplus_{\mathfrak{p}|p}\operatorname{Hom}(W_{\mathfrak{p}}^{0},\Gamma_{\mathfrak{P}_{0}}^{\wedge}),}$$\scriptstyle{\oplus\mathrm{loc}_{\mathfrak{p}}}$$\scriptstyle{\oplus\operatorname{rec}_{\Gamma_{\mathfrak{P}_{0}}}}$

where the last identification is induced by
$\oplus\mathrm{res}_{\mathfrak{p}}$. Note that $\mathrm{res}_{\mathfrak{p}}$
and $\operatorname{rec}_{\Gamma_{\mathfrak{P}_{0}}}$ commute. Furthermore,
letting
$[n_{\mathfrak{p}}]:\Gamma_{\mathfrak{P}_{0}}^{\wedge}\rightarrow\Gamma_{\mathfrak{p}}^{\wedge}$
be the multiplication by
$n_{\mathfrak{p}}=[K_{\mathfrak{P}_{0}}:k_{\mathfrak{p}}]$ map, the
functoriality of Artin’s reciprocity law shows that
$\operatorname{rec}_{\mathfrak{p}}$ coincides with
$[n_{\mathfrak{p}}]\circ\operatorname{rec}_{\mathfrak{P}_{0}}$ on
$k_{\mathfrak{p}}^{\times,\wedge}$. Hence, if $\mathds{1}\not\subseteq\rho$,
then the map $j(\rho)\circ\mathscr{L}_{k_{\infty}/k}(\rho)$ coincides with the
map of (6) under the identification
$[n_{\mathfrak{p}}]:\Gamma_{\mathfrak{P}_{0}}^{\wedge}\simeq\Gamma_{\mathfrak{p}}^{\wedge}$.
Since $j(\rho)$ is an isomorphism in this case, we obtain the desired
description of $\mathscr{L}_{k_{\infty}/k}(\rho)$. In the case where
$\rho=\mathds{1}$, the map $\mathscr{L}_{K_{\infty}/K}(\mathds{1})$ is nothing
but the restriction
$(\mathscr{L}_{K_{\infty}/K})^{G}\colon\mathcal{O}_{k}[\tfrac{1}{p}]^{\times,\wedge}\to(\mathcal{H}_{K_{\infty}/K}^{\wedge})^{G}$
of $\mathscr{L}_{K_{\infty}/K}$ to the $G$-invariants. Under the
identifications

$(\bigoplus_{\mathfrak{P}|p}\Gamma_{\mathfrak{P}}^{\wedge})^{G}=(\bigoplus_{\mathfrak{p}|p}\mathrm{Ind}_{G_{\mathfrak{p}}}^{G}\Gamma_{\mathfrak{P}_{0}}^{\wedge})^{G}=\bigoplus_{\mathfrak{p}|p}\Gamma_{\mathfrak{P}_{0}}^{\wedge}\simeq\bigoplus_{\mathfrak{p}|p}\Gamma_{\mathfrak{p}}^{\wedge}$

induced by by Frobenius reciprocity and by
$\oplus_{\mathfrak{p}}[n_{\mathfrak{p}}]$, it is clear that
$(\mathcal{H}_{K_{\infty}/K}^{\wedge})^{G}$ is mapped onto
$\mathcal{H}_{k_{\infty}/k}^{\wedge}$, and that
$\mathscr{L}_{K_{\textrm{cyc}}/K}(\mathds{1})$ can be naturally identified
with $\mathscr{L}_{k_{\infty}/k}$.

∎

######  Remark 3.2.3.

The map $\mathscr{L}_{k_{\infty}/k}(\rho)$ admits a more intrinsic description
in terms of Bloch-Kato Selmer groups. Namely, let
$\mathrm{H}^{1}_{\mathrm{f},p}(k,\widecheck{W})\subset\mathrm{H}^{1}(k,\widecheck{W})$
be the Selmer group of $\widecheck{W}$ defined by the Bloch-Kato condition at
all places not dividing $p$ (see [BK90, § 3]), and let
$\mathrm{H}^{1}_{\Gamma}(k_{\mathfrak{p}},\widecheck{W}^{0}_{\mathfrak{p}})$
be the orthogonal complement of
$\mathrm{H}^{1}(\Gamma_{\mathfrak{p}},W^{0}_{\mathfrak{p}})\subset\mathrm{H}^{1}(k_{\mathfrak{p}},W)$
under Tate’s local pairing. Then Kummer theory and the Inflation-Restriction
exact sequence provide natural isomorphisms
$\mathrm{H}^{1}_{\mathrm{f},p}(k,\widecheck{W})\simeq\operatorname{Hom}_{G_{k}}(W,\mathcal{O}_{K}[\tfrac{1}{p}]^{\times,\wedge})$
and
$\mathrm{H}^{1}(k_{\mathfrak{p}},\widecheck{W})/\mathrm{H}^{1}_{\Gamma}(k_{\mathfrak{p}},\widecheck{W}^{0}_{\mathfrak{p}})\simeq\operatorname{Hom}(W_{\mathfrak{p}}^{0},\Gamma_{\mathfrak{p}})$.
An easy adaptation of the proof of Proposition 3.1.5 then shows that
$\mathscr{L}_{k_{\infty}/k}(\rho)$ coincides with the localization map
$\mathrm{H}^{1}_{f,p}(k,\widecheck{W})\rightarrow\bigoplus_{\mathfrak{p}|p}\mathrm{H}^{1}(k_{\mathfrak{p}},\widecheck{W})/\mathrm{H}^{1}_{\Gamma}(k_{\mathfrak{p}},\widecheck{W}^{0}_{\mathfrak{p}})$
under these identifications.

######  Corollary 3.2.4.

  1. (1)

If $\rho=\mathds{1}$ then we have
$\delta^{\textbf{G}}_{k_{\infty}/k}(\mathds{1})=\delta^{\textbf{G}}_{k_{\infty}/k}$.

  2. (2)

Assume $k=\mathbb{Q}$, $k_{\infty}=\mathbb{Q}_{\textrm{cyc}}$ and
$\mathds{1}\not\subset\rho$. Fix an embedding $\iota_{p}\colon\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu\hookrightarrow\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}$ and let
$W_{p}^{0}=W^{G_{{\mathbb{Q}_{p}}}}$. Then,

$\delta^{\textbf{G}}_{\mathbb{Q}}(\rho)=\dim\operatorname{coker}\left[\operatorname{Hom}_{G}(W,\mathcal{O}_{K}[\tfrac{1}{p}]^{\times,\wedge})\longrightarrow\operatorname{Hom}(W_{p}^{0},\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p})\right],$

the map being the restriction-to-$W_{p}^{0}$ map followed by the post-
composition by $\log_{p}\circ\iota_{p}$.

###### Proof.

The first claim directly follows from Propositions 3.1.5 and 3.2.2. The second
claim follows from from Proposition 3.2.2 and from the fact that, if
$\Gamma_{p}=\operatorname{Gal}(\mathbb{Q}_{p,\textrm{cyc}}/{\mathbb{Q}_{p}})$,
then the composite map
$\log_{p}\circ\chi_{\textrm{cyc}}\circ\operatorname{rec}_{\Gamma_{p}}\colon\mathbb{Q}_{p}^{\times,\wedge}\to\Gamma_{p}^{\wedge}\simeq\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}\otimes_{{\mathbb{Z}_{p}}}(1+p{\mathbb{Z}_{p}})\simeq\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}$ coincides
with $-\log_{p}$. ∎

######  Corollary 3.2.5.

The assignment $\rho\mapsto\delta^{\textbf{G}}_{k_{\infty}/k}(\rho)$ is
compatible with Artin formalism. More precisely,

  1. (a)

$\delta^{\textbf{G}}_{k_{\infty}/k}(\rho)$ does not depend on the choice of
the splitting field $K$ of $\rho\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}}(G_{k})$.

  2. (b)

$\delta^{\textbf{G}}_{k_{\infty}/k}(\rho_{1}\oplus\rho_{2})=\delta^{\textbf{G}}_{k_{\infty}/k}(\rho_{1})+\delta^{\textbf{G}}_{k_{\infty}/k}(\rho_{2})$
for any $\rho_{1},\rho_{2}\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}}(G_{k})$.

  3. (c)

If $k_{\infty}=kk^{\prime}_{\infty}$ for some subfield $k^{\prime}\subset k$
and ${\mathbb{Z}_{p}}$-extension $k^{\prime}_{\infty}/k^{\prime}$ and if
$\rho^{\prime}=\mathrm{Ind}_{k}^{k^{\prime}}\rho$ is induced from
$\rho\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}}(G_{k})$,
then
$\delta^{\textbf{G}}_{k^{\prime}_{\infty}/k^{\prime}}(\rho^{\prime})=\delta^{\textbf{G}}_{k_{\infty}/k}(\rho)$.

###### Proof.

Part (b) is obvious from the definition of $\mathscr{L}_{k_{\infty}/k}(\rho)$.
Part (a) is true if $\rho$ is trivial by Corollary 3.2.4 (1). Let
$K^{\prime}/K/k$ be Galois extensions, take
$\mathds{1}\not\subseteq{\rho}\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}}(\operatorname{Gal}(K/k))$ and denote by $\tilde{\rho}$ its
inflation to $\operatorname{Gal}(K^{\prime}/k)$. Then the maps of (6) attached
to $\rho$ and $\tilde{\rho}$ coincide on
$\operatorname{Hom}_{\operatorname{Gal}(K/k)}(W,\mathcal{O}_{K}[\tfrac{1}{p}]^{\times,\wedge})=\operatorname{Hom}_{\operatorname{Gal}(K^{\prime}/k)}(W,\mathcal{O}_{K^{\prime}}[\tfrac{1}{p}]^{\times,\wedge})$,
so
$\delta^{\textbf{G}}_{k_{\infty}/k}(\rho)=\delta^{\textbf{G}}_{k_{\infty}/k}(\tilde{\rho})$.
Hence, (a) holds for every $\rho\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}}(G_{k})$.
For (c), take any splitting field $K$ of $\rho$ which is Galois over
$k^{\prime}$ and put $G=\operatorname{Gal}(K/k)$,
$G^{\prime}=\operatorname{Gal}(K/k^{\prime})$. Then Frobenius reciprocity
identifies the $\rho^{\prime}$-isotypic component of
$\mathscr{L}_{K_{\infty}/K}$ (seen as a $G^{\prime}$-equivariant map) with the
$\rho$-isotypic component of $\mathscr{L}_{K_{\infty}/K}$ (seen as a
$G$-equivariant map). Hence, the last property follows from Proposition 3.2.2.
∎

We next define an analogous invariant for Leopoldt’s conjecture. For any
Galois extension $K/k$ with Galois group $G$, the localization map

$\iota_{K}\colon\mathcal{O}_{K}^{\times,\wedge}\longrightarrow\bigoplus_{\mathfrak{P}\in
S_{p}(K)}\mathcal{O}_{K_{\mathfrak{P}}}^{\times,\wedge}$

is clearly $G$-equivariant. For any $(W,\rho)\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}}(G)$ we let
$\delta^{\textbf{L}}_{k}(\rho)$ be the dimension of the kernel of the
$\rho$-isotypic component

(7) $\iota_{k}(\rho)\ \colon\
\operatorname{Hom}_{G}(W,\mathcal{O}_{K}^{\times,\wedge})\longrightarrow\bigoplus_{\mathfrak{p}\in
S_{p}(k)}\operatorname{Hom}_{G}(W,\bigoplus_{\mathfrak{P}|\mathfrak{p}}\mathcal{O}_{K_{\mathfrak{P}}}^{\times,\wedge})\simeq\bigoplus_{\mathfrak{p}\in
S_{p}(k)}\operatorname{Hom}_{G_{\mathfrak{p}}}(W,\mathcal{O}_{K_{\mathfrak{P}_{0}}}^{\times,\wedge})$

of $\iota_{K}$. Here (and as in the definition of
$\mathscr{L}_{k_{\infty}/k}(\rho)$), $\mathfrak{P}_{0}$ is a fixed place of
$K$ above $\mathfrak{p}$ for every place $\mathfrak{p}$ of $k$. The last
isomorphism is given by Frobenius reciprocity and is induced by the natural
projection map. As in Corollary 3.2.5, it is easy to see that the rule
$\rho\mapsto\delta^{\textbf{L}}_{k}(\rho)$ is compatible with Artin formalism.

######  Remark 3.2.6.

In terms of Bloch-Kato Selmer groups for $\widecheck{W}$, the injectivity of
$\iota_{k}(\rho)$ is equivalent to that of the localization map
$\mathrm{H}^{1}_{\mathrm{f}}(k,\widecheck{W})\rightarrow\prod_{\mathfrak{p}|p}\mathrm{H}^{1}_{\mathrm{f}}(k_{\mathfrak{p}},\widecheck{W})$.
This last statement is Jannsen’s conjecture for $\widecheck{W}$ ([Jan89]).

The following lemma on local Galois representations with finite image will
help us describe $\delta^{\textbf{L}}_{k}(\rho)$ in another way if
$k=\mathbb{Q}$.

######  Lemma 3.2.7.

Let $(W,\rho)\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}}(G_{{\mathbb{Q}_{p}}})$ be a local representation factoring through
the Galois group of a finite extension $E\subset\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}$ of
${\mathbb{Q}_{p}}$. Then the internal multiplication map $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}\otimes_{{\mathbb{Q}_{p}}}E\rightarrow\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}$ induces an
isomorphism

$m\ \colon\ \operatorname{Hom}_{G_{{\mathbb{Q}_{p}}}}(W,\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}\otimes_{{\mathbb{Q}_{p}}}E)\simeq\operatorname{Hom}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}}(W,\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}).$

Here, we let $G_{{\mathbb{Q}_{p}}}$ act on $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}\otimes_{{\mathbb{Q}_{p}}}E$ via $g(a\otimes x)=a\otimes g(x)$.

###### Proof.

Since $E\simeq{\mathbb{Q}_{p}}[\operatorname{Gal}(E/{\mathbb{Q}_{p}})]$ as a
Galois module, it is enough to show that $m$ is injective. Choose any finite
Galois extension $L/{\mathbb{Q}_{p}}$ which contains $E$ and over which $\rho$
is realizable, i.e., there exist a $L[G_{{\mathbb{Q}_{p}}}]$-module $W_{L}$
and an isomorphism $W_{L}\otimes_{L}\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}\simeq W$.
Then we have to show that the map
$\operatorname{Hom}_{L[G_{{\mathbb{Q}_{p}}}]}(W_{L},L\otimes_{{\mathbb{Q}_{p}}}E)\to\operatorname{Hom}_{L}(W_{L},L)$
is injective. By considering ${\mathbb{Q}_{p}}$-linear homomorphisms instead
of $L$-linear ones in the previous map, it suffices to prove that the map
$(V\otimes_{{\mathbb{Q}_{p}}}\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p})^{G_{{\mathbb{Q}_{p}}}}\otimes_{{\mathbb{Q}_{p}}}\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}\to
V\otimes_{{\mathbb{Q}_{p}}}\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}$ given by
$(v\otimes a)\otimes b\mapsto v\otimes(ab)$ is injective, where
$V=\operatorname{Hom}_{{\mathbb{Q}_{p}}}(W_{L},{\mathbb{Q}_{p}})$ and
$G_{{\mathbb{Q}_{p}}}$ acts on both factors of
$V\otimes_{{\mathbb{Q}_{p}}}\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}$. But $V$
is finite dimensional over ${\mathbb{Q}_{p}}$, so this follows from the
$\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}$-admissibility in the sense of Fontaine of Galois representations of
finite image (see e.g. [FO08, Proposition 2.7]). ∎

######  Proposition 3.2.8.

Let $(W,\rho)\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}}(G_{\mathbb{Q}})$, let $K\subset\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu$ be a splitting
field of $\rho$ and let $\mathfrak{P}_{0}$ be the $p$-adic place of $K$
defined by a fixed embedding $\iota_{p}\colon\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu\hookrightarrow\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}$. We have

$\delta^{\textbf{L}}_{\mathbb{Q}}(\rho)=\dim\ker\left[\mathcal{L}\ \colon\
\operatorname{Hom}_{G_{\mathbb{Q}}}(W,\mathcal{O}_{K}^{\times,\wedge})\rightarrow\operatorname{Hom}(W,\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p})\right],$

the map $\mathcal{L}$ being the post-composition by
$\log_{p}\colon\mathcal{O}_{K}^{\times,\wedge}\to\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}\otimes_{{\mathbb{Q}_{p}}}K_{\mathfrak{P}_{0}}$ followed by the
internal multiplication $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}\otimes_{{\mathbb{Q}_{p}}}K_{\mathfrak{P}_{0}}\to\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}$.

###### Proof.

First note that the $p$-adic logarithm
$\log_{p}\colon\mathcal{O}_{E}^{\times}\rightarrow E$ over a finite extension
$E$ of ${\mathbb{Q}_{p}}$ induces an isomorphism
$\mathcal{O}_{E}^{\times,\wedge}\simeq\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}\otimes_{{\mathbb{Q}_{p}}}E$. Applying Lemma 3.2.7 to
$E=K_{\mathfrak{P}_{0}}$ we obtain isomorphisms
$\operatorname{Hom}_{G_{\mathbb{Q}_{p}}}(W,\mathcal{O}_{K_{\mathfrak{P}_{0}}}^{\times,\wedge})\simeq\operatorname{Hom}_{G_{\mathbb{Q}_{p}}}(W,\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}\otimes_{{\mathbb{Q}_{p}}}K_{\mathfrak{P}_{0}})\simeq\operatorname{Hom}(W,\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p})$. The map
$\mathcal{L}$ is simply the map $\iota_{\mathbb{Q}}(\rho)$ composed with these
isomorphisms, so the claim follows. ∎

##  4\. Bounds on Leopoldt’s and Gross’s defects

Throughout this section we fix an embedding $\iota_{p}\colon\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu\hookrightarrow\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}$.

###  4.1. Bounds on Leopoldt’s defect

######  Theorem 4.1.1.

Let $\rho\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}}(G_{\mathbb{Q}})$ be irreducible and let $d=d(\rho)$,
$d^{+}=d^{+}(\rho)$. We have
$\delta^{\textbf{L}}_{\mathbb{Q}}(\rho)\leq\frac{(d^{+})^{2}}{d+d^{+}}$.

###### Proof.

Since
$\delta_{\mathbb{Q}}^{\textbf{L}}(\mathds{1})=\delta_{\mathbb{Q}}^{\textbf{L}}=0$,
we may assume that $\rho\neq\mathds{1}$. Let $K$ be a splitting field of
$\rho$ and let $G=\operatorname{Gal}(K/\mathbb{Q})$. Recall from Lemma 3.2.1
that $\dim\operatorname{Hom}_{G}(W,\mathcal{O}_{K}^{\times,\wedge})=d^{+}$. By
Proposition 3.2.8 it is enough to show that the rank of the map
$\operatorname{Hom}_{G}(W,\mathcal{O}_{K}^{\times,\wedge})\rightarrow\operatorname{Hom}(W,\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p})$ induced
by $a\otimes x\mapsto a\log_{p}(\iota_{p}(x))$ on
$\mathcal{O}_{K}^{\times,\wedge}$ is at least $\frac{d\cdot d^{+}}{d+d^{+}}$.

Consider a $\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu$-structure on $W$, that is, a $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu$-linear
representation $W_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu}$ of $G$ such
that $W_{\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu}\otimes_{\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu}\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}\simeq W$. Let also $w_{1},\ldots,w_{d}$ be a $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu$-basis of
$W_{\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu}$.
Using an isomorphism $e(\rho)\cdot\left(\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu\otimes\mathcal{O}_{K}^{\times}\right)\simeq W_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu}^{\oplus
d^{+}}$ we may define $G$-equivariant morphisms
$\Psi_{1},\ldots,\Psi_{d^{+}}\colon W_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu}\longrightarrow\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu\otimes\mathcal{O}_{K}^{\times}$ which form a basis of
$\operatorname{Hom}_{G}(W,\mathcal{O}_{K}^{\times,\wedge})$. Moreover, the
elements $\Psi_{j}(w_{i})\in\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu\otimes\mathcal{O}_{K}^{\times}$ for $1\leq i\leq d$, $1\leq j\leq d^{+}$
are $\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu$-linearly independent by construction, as well as their $p$-adic
logarithms by Proposition 2.1.2. Hence, the matrix
$M=(\log_{p}(\iota_{p}(\Psi_{j}(w_{i}))))_{i,j}$ of size $d\times d^{+}$
satisfies $\theta(M)=\frac{d^{+}}{d}$ and Theorem 2.2.1 implies
$\operatorname{rk}M\geq\frac{d\cdot d^{+}}{d+d^{+}}$ as claimed. ∎

######  Corollary 4.1.2.

  1. (1)

Let $\rho\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}}(G_{k})$
and let $d^{+}=d^{+}(\rho)$. We have $\delta_{k}^{\textbf{L}}(\rho)\leq
d^{+}/2$.

  2. (2)

For every finite extensions $K/k$ of number fields, we have
$\delta^{\textbf{L}}_{K}\leq\delta^{\textbf{L}}_{k}+\left(\operatorname{rk}\mathcal{O}_{K}^{\times}-\operatorname{rk}\mathcal{O}_{k}^{\times}\right)/2$.

###### Proof.

Since $d^{+}\leq d$, the first inequality obviously follows from Theorem 4.1.1
for irreducible $\rho\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}}(G_{\mathbb{Q}})$, hence it follows for general $\rho$ from Artin
formalism. By Lemma 3.2.1, the unique representation $\rho_{0}$ of $G_{k}$
such that $\mathrm{Ind}_{K}^{k}\mathds{1}=\mathds{1}\oplus\rho_{0}$ satisfies
$d^{+}(\rho_{0})=\operatorname{rk}\mathcal{O}_{K}^{\times}-\operatorname{rk}\mathcal{O}_{k}^{\times}$,
so the second inequality follows from the first one applied to $\rho_{0}$. ∎

###  4.2. Bounds on Gross’s defect

######  Theorem 4.2.1.

Let $\rho\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}}(G_{\mathbb{Q}})$ be irreducible and let $f=f(\rho)$,
$d^{+}=d^{+}(\rho)$. If $d^{+}=f=0$, then
$\delta_{\mathbb{Q}}^{\textbf{G}}(\rho)=0$. Otherwise, we have
$\delta_{\mathbb{Q}}^{\textbf{G}}(\rho)\leq\frac{f^{2}}{d^{+}+2f}$.

###### Proof.

Recall from Section 3.2 that
$\dim\operatorname{Hom}_{G}(W,\mathcal{O}_{K}[\tfrac{1}{p}]^{\times,\wedge})=d^{+}+f$
and that $\dim W_{p}^{0}=f$. If $\rho=\mathds{1}$ or $f=0$, then the codomain
of $\mathscr{L}_{\mathbb{Q}}(\rho)$ is $\\{0\\}$, yielding
$\delta_{\mathbb{Q}}^{\textbf{G}}(\rho)=0$. We assume henceforth that
$\rho\neq\mathds{1}$ and $f>0$. Let $K$ be a splitting field of $\rho$ and let
$G=\operatorname{Gal}(K/\mathbb{Q})$. By Corollary 3.2.4 (2) and Proposition
3.2.2 it is enough to show that the rank of the map
$\operatorname{Hom}_{G}(W,\mathcal{O}_{K}[\tfrac{1}{p}]^{\times,\wedge})\rightarrow\operatorname{Hom}(W_{p}^{0},\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p})$ induced
by $a\otimes x\mapsto a\log_{p}(\iota_{p}(x))$ on
$\mathcal{O}_{K}[\tfrac{1}{p}]^{\times,\wedge}$ is at least
$\frac{(d^{+}+f)\cdot f}{d^{+}+2f}$.

As in the proof of Theorem 4.1.1, fix a $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu$-structure
$W_{\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu}$ of
$W$. Fix also a basis $w_{1},\ldots,w_{f}$ of the subspace $W_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu,p}^{0}$ of
$G_{{\mathbb{Q}_{p}}}$-invariants of $W_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu}$ and an
isomorphism $e(\rho)\cdot\left(\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu\otimes\mathcal{O}_{K}[\tfrac{1}{p}]^{\times}\right)\simeq W_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu}^{\oplus(d^{+}+f)}$. These choices yield a basis
$\Psi_{1},\ldots,\Psi_{d^{+}+f}$ of
$\operatorname{Hom}_{G}(W,\mathcal{O}_{K}[\tfrac{1}{p}]^{\times,\wedge})$ such
that the elements $\Psi_{j}(w_{i})\in\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu\otimes\mathcal{O}_{K}^{\times}$ for $1\leq i\leq f$, $1\leq j\leq
d^{+}+f$ are $\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu$-linearly independent. Since $e(\rho)$ kills $p^{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu}$, we deduce
from Proposition 2.1.2 that the entries of the matrix
$M^{\prime}=(\log_{p}(\iota_{p}(\Psi_{j}(w_{i}))))_{i,j}$ of size
$f\times(d^{+}+f)$ are $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu$-linearly
independent as well. Therefore, $\theta(M^{\prime})=\tfrac{d^{+}+f}{f}$ and
Theorem 2.2.1 implies $\operatorname{rk}M^{\prime}\geq\tfrac{(d^{+}+f)\cdot
f}{d^{+}+2f}$. ∎

######  Corollary 4.2.2.

  1. (1)

Let $\rho\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}}(G_{k})$,
let $K$ be the field cut out by $\rho$ and let $f=f(\rho)$. Then the following
inequalities hold true:

$\delta_{k}^{\textbf{G}}(\rho)\ \left\\{\begin{array}[]{ccl}\leq&f/2&\\\
<&f/2&\mbox{if $f\neq 0$ and $K$ contains at least one real place}\\\
\leq&f/3&\mbox{if $K$ is totally real}\end{array}\right.$

  2. (2)

Let $K/k$ be a finite extension of number fields. Then the following
inequalities hold true:

$\delta_{K}^{\textbf{G}}\
\left\\{\begin{array}[]{ccl}\leq&\delta_{k}^{\textbf{G}}+\left(|S_{p}(K)|-|S_{p}(k)|\right))/2&\\\
<&\delta_{k}^{\textbf{G}}+\left(|S_{p}(K)|-|S_{p}(k)|\right))/2&\mbox{if
$|S_{p}(K)|\neq|S_{p}(k)|$ and $K$ contains at least one real place}\\\
\leq&\delta_{k}^{\textbf{G}}+\left(|S_{p}(K)|-|S_{p}(k)|\right))/3&\mbox{if
$K$ is totally real}\end{array}\right.$

###### Proof.

We know that $\rho\mapsto\delta_{k}^{\textbf{G}}(\rho)$ is compatible with
Artin formalism by Corollary 3.2.5. We now explain how to prove (1). Again by
Artin formalism, it suffices to prove (1) with $\rho$ replaced by any
irreducible subrepresentation $\theta\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}}(G_{\mathbb{Q}})$ of $\mathrm{Ind}_{k}^{\mathbb{Q}}\rho$. For such a
$\theta$, Lemma 3.2.1 implies that $d^{+}(\theta)\geq 1$ if $K$ contains at
least one real place, and $d^{+}(\theta)=d(\theta)$ if $K$ is totally real.
Therefore, the inequalities in (1) for $\theta$ directly follow from Theorem
4.2.1. Finally, (2) follows from (1) as in the proof of Corollary 4.1.2. ∎

######  Remark 4.2.3.

The matrices $M$ and $M^{\prime}$ appearing in the course of the proof of
Theorems 4.1.1 and 4.2.1 have full rank under the $p$-adic Schanuel
conjecture. Therefore, Artin formalism shows that Leopoldt’s and Gross-
Kuz’min’s conjectures hold in great generality under the $p$-adic Schanuel
conjecture.

###  4.3. Applications

######  Theorem 4.3.1.

Let $k$ be a totally real number field and let
$(V,\rho)\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}}(G_{k})$ be
such that $d^{+}(\rho)=0$. Then Gross’s $p$-adic regulator matrix $R_{p}(V)$
[Gro81, (2.10)] is of size $f(\rho)$ and of rank at least $f(\rho)/2$.

###### Proof.

Let $K$ be the CM field cut out by $\rho$ and let $(\operatorname{Hom}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}}(V,\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}),\rho^{*})$
be the contragredient representation of $\rho$. Gross’s regulator map
$\lambda_{p}$ defined in [Gro81, (1.18)] can be identified with the “minus
part” of $\mathscr{L}_{K}$, which is, by definition, the restriction of
$\mathscr{L}_{K}$ to the subspace where the complex conjugation acts by $-1$.
This means that $\lambda_{p}$ and $\mathscr{L}_{K}$ share the same
$\theta$-isotypic component for every representation
$\theta\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}}(\operatorname{Gal}(K/k))$ such that $d^{+}(\theta)=0$. Since taking
$(V\otimes-)^{G_{k}}$ amounts to taking $\rho^{*}$-isotypic components, we
conclude that
$\operatorname{rk}R_{p}(V)=\operatorname{rk}\mathscr{L}_{k}(\rho^{*})=f(\rho^{*})-\delta^{\textbf{G}}_{k}(\rho^{*})$,
so $\operatorname{rk}R_{p}(V)\geq f(\rho^{*})/2=f(\rho)/2$ by Theorem 4.2.1. ∎

In the next theorem, we write $k^{+}$ for the maximal totally real subfield of
a number field $k$, and $\mathbb{Q}^{\textrm{ab}}$ for the maximal abelian
extension of $\mathbb{Q}$.

######  Theorem 4.3.2.

Let $K/k$ be an abelian extension of number fields. The Gross-Kuz’min
conjecture holds for $K$ in each of the following cases.

  1. (a)

Either $|S_{p}(K)|\leq 2$, or $|S_{p}(K)|\leq 3$ and $K$ has at least one real
place, or $|S_{p}(K)|\leq 4$ and $K/\mathbb{Q}$ is Galois, or $|S_{p}(K)|\leq
6$ and $K/\mathbb{Q}$ is a real Galois extension.

  2. (b)

$|S_{p}(k)|=1$, or $|S_{p}(k)|\leq 2$ and $K$ has at least one real place.

  3. (c)

$K\subset k\cdot\mathbb{Q}^{\textrm{ab}}$, $k/\mathbb{Q}$ is Galois, and
either $|S_{p}(k)|\leq 3$, or $|S_{p}(k)|\leq 5$ and $K$ is real.

  4. (d)

$k$ is an imaginary quadratic field, or $k$ is a real quadratic field and $K$
has at least one real place.

  5. (e)

$k/\mathbb{Q}$ is Galois, $|S_{p}(k)|\leq 2$, $|S_{p}(k^{+})|=1$ and $[K:k]$
and $[k:\mathbb{Q}]$ are coprime.

###### Proof.

Recall that $\delta^{\textbf{G}}(-)$ is compatible with Artin formalism by
Corollary 3.2.5. We shall often appeal to Lemma 3.2.1 and to the following
consequence of Theorem 4.2.1 without further notice. For any irreducible
representation $\theta\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}}(G_{\mathbb{Q}})$, we have
$\delta_{\mathbb{Q}}^{\textbf{G}}(\theta)=0$ if $f(\theta)\leq 1$, or if
$f(\theta)=2$ and $d^{+}(\theta)\geq 1$. In particular,
$\delta_{\mathbb{Q}}^{\textbf{G}}(\theta)=0$ if $\theta$ is a multiplicative
character of $G_{\mathbb{Q}}$, so $\delta_{M}^{\textbf{G}}=0$ for any abelian
extension $M/\mathbb{Q}$ by Artin formalism.

Since $\delta_{\mathbb{Q}}^{\textbf{G}}=0$, it follows from Corollary 4.2.2
that $\delta_{K}^{\textbf{G}}=0$ for $K$ satisfying one of the two first
assumptions in case (a). Consider the two last assumptions in (a) and assume
that $K/\mathbb{Q}$ is Galois. We claim that
$\delta_{\mathbb{Q}}^{\textbf{G}}(\theta)=0$ for all irreducible
$\theta\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}}(\operatorname{Gal}(K/\mathbb{Q}))$. We may assume that
$\dim\theta\geq 2$, so
$f(\theta)\leq(f(\mathds{1}_{K})-f(\mathds{1}_{\mathbb{Q}}))/(\dim\theta)\leq(|S_{p}(K)|-1)/2$.
The two last assumptions in (a) ensure that we either have $f(\theta)\leq 1$,
or $f(\theta)\leq 2$ and $d^{+}(\theta)=d(\theta)\geq 1$, so we indeed have
$\delta_{\mathbb{Q}}^{\textbf{G}}(\theta)=0$. Therefore,
$\delta_{K}^{\textbf{G}}=0$ in case (a).

Let $G=\operatorname{Gal}(K/k)$ and let $\hat{G}=\operatorname{Hom}(G,\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}^{\times})$
be the group of characters of $G$. We place ourselves in cases (b), (c) and
(d), we fix ${\chi\in\hat{G}}$ and we show that
$\delta_{k}^{\textbf{G}}(\chi)=0$. Since $f(\chi)\leq|S_{p}(k)|$, Corollary
4.2.2 (1) implies $\delta_{k}^{\textbf{G}}(\chi)=0$ in case (b). Suppose now
we are in case (c). Then $\chi$ descends to a character $\chi_{\mathbb{Q}}$ of
$G_{\mathbb{Q}}$. Moreover, as $k/\mathbb{Q}$ is Galois, any irreducible
subrepresentation $\rho$ of
$\mathrm{Ind}_{k}^{\mathbb{Q}}\chi\simeq(\mathrm{Ind}_{k}^{\mathbb{Q}}\mathds{1}_{k})\otimes\chi_{\mathbb{Q}}$
occurs $(\dim\rho)$ times, so it satisfies $f(\rho)\leq|S_{p}(k)|/(\dim\rho)$.
Moreover, if $K$ is totally real, then any such $\rho$ satisfies
$d^{+}(\rho)=d(\rho)\geq 1$, so we can conclude
$\delta_{\mathbb{Q}}^{\textbf{G}}(\rho)=0$. Therefore,
$\delta_{k}^{\textbf{G}}(\chi)=0$ in case (c). We now assume to be in case
(d). Then $\mathrm{Ind}_{k}^{\mathbb{Q}}\chi$ has dimension $2$, so it is
either irreducible or it is the sum of two characters of $G_{\mathbb{Q}}$, say
$\eta_{1}$ and $\eta_{2}$. In the latter case, we already know that
$\delta_{\mathbb{Q}}^{\textbf{G}}(\eta_{i})=0$ for $i=1,2$, so
$\delta_{k}^{\textbf{G}}(\chi)=\delta_{\mathbb{Q}}^{\textbf{G}}(\mathrm{Ind}_{k}^{\mathbb{Q}}\chi)=0$.
If $\mathrm{Ind}_{k}^{\mathbb{Q}}\chi$ is irreducible, then the assumptions on
$K$ and $k$ imply that $f(\chi)\leq 2$ and $d^{+}(\chi)\geq 1$, yielding
$\delta_{k}^{\textbf{G}}(\chi)=0$. Therefore, $\delta_{K}^{\textbf{G}}=0$ in
the cases (b), (c) and (d).

We now make the assumptions in (e) and we assume without loss of generality
that $K$ is Galois over $\mathbb{Q}$ with Galois group $\mathcal{G}$. By the
Schur-Zassenhaus theorem, $\mathcal{G}$ is the semi-direct product of
$H:=\operatorname{Gal}(k/\mathbb{Q})$ acting on $G$. Let
$\rho\in\textrm{Art}_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}}(\mathcal{G})$ be irreducible and let us prove that
$\delta_{\mathbb{Q}}^{\textbf{G}}(\rho)=0$. By [Ser78, Chap II § 8.2], $\rho$
can be written as $\mathrm{Ind}_{k^{\prime}}^{\mathbb{Q}}(\theta\otimes\chi)$,
where $k^{\prime}/\mathbb{Q}$ is a subextension of $k/\mathbb{Q}$, $\theta$ an
irreducible representation of $\operatorname{Gal}(k/k^{\prime})$ and $\chi$ a
character of $\operatorname{Gal}(K/k^{\prime})$. Note that
$f(\rho)\leq|S_{p}(k)|/(\dim\theta)\leq 2/(\dim\theta)$, so we may assume
without loss of generality that $\dim\theta=1$. If $k^{\prime}$ is totally
real, then $f(\rho)\leq|S_{p}(k^{\prime})|\leq|S_{p}(k^{+})|=1$, and otherwise
we have $d^{+}(\rho)\geq 1$. In any case,
$\delta_{\mathbb{Q}}^{\textbf{G}}(\rho)=0$ so we may infer
$\delta_{K}^{\textbf{G}}=0$ in case (e) as well. ∎

##  5\. Vanishing locus of Gross’s defect

###  5.1. Preliminaries

This section is devoted to the proof of the following theorem which, in turn,
implies Theorem 1.5 stated in the introduction.

######  Theorem 5.1.1.

Let $k$ be a number field and let $\varphi\colon G_{k}\to\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}^{\times}$
be a finite-order character. Assume also that $p$ completely splits in the
number field cut out by $\varphi$.

  1. (1)

Assume that $r_{2}+\delta^{\textbf{L}}_{k}\leq 1$, where $r_{2}$ is the number
of complex places of $k$. If there exists at least one
${\mathbb{Z}_{p}}$-extension $k_{\infty}$ of $k$ such that
$\delta_{k_{\infty}/k}^{\textbf{G}}(\varphi)=0$, then there are at most
$[k:\mathbb{Q}]$ (and at most $[k:\mathbb{Q}]-1$ if $\varphi=\mathds{1}$)
${\mathbb{Z}_{p}}$-extensions $k_{\infty}/k$ such that
$\delta_{k_{\infty}/k}^{\textbf{G}}(\varphi)\neq 0$.

  2. (2)

Assume that $k$ is an imaginary quadratic field. There is at most one
${\mathbb{Z}_{p}}$-extension $k_{\infty}/k$ for which
$\delta_{k_{\infty}/k}^{\textbf{G}}(\varphi)\neq 0$, and it has a
transcendental slope (see Example 5.2.3 for a definition of the slope of
$k_{\infty}/k$). Moreover, if $\varphi$ cuts out an abelian extension of
$\mathbb{Q}$ or if the $p$-adic Schanuel conjecture holds, then
$\delta^{\textbf{G}}_{k_{\infty}/k}(\varphi)=0$ for any
${\mathbb{Z}_{p}}$-extension $k_{\infty}$ of $k$.

###### Proof of Theorem 1.5, assuming Theorem 5.1.1.

Let $K$ be an abelian extension of an imaginary quadratic field $k$ and let
$K^{\textrm{ab}}\subset K$ be its maximal absolutely abelian subfield. By
Artin formalism (Corollary 3.2.5) and by Theorem 5.1.1, we have
$\delta^{\textbf{G}}_{Kk_{\infty}/K}>0$ for a given
${\mathbb{Z}_{p}}$-extension $k_{\infty}/k$ if and only if there exists a
character $\varphi$ of $\operatorname{Gal}(K/k)$ such that
$\delta_{k_{\infty}/k}^{\textbf{G}}(\varphi)>0$. Such a character cannot be a
character of $\operatorname{Gal}(K^{\textrm{ab}}/k)$, and moreover,
$k_{\infty}$ is uniquely determined by $\varphi$. Therefore, we have
$\delta^{\textbf{G}}_{Kk_{\infty}/K}>0$ for at most
$[K:k]-[K^{\textrm{ab}}:k]$ distinct ${\mathbb{Z}_{p}}$-extensions of $k$. ∎

In the rest of Section 5, we fix once and for all an abelian extension $K/k$
with Galois group $G$ such that $p$ totally splits in $K$. We let $n$ be the
degree of $k$ and $(r_{1},r_{2})$ its signature, and we put
$r=r_{1}+r_{2}-1-\delta^{\textbf{L}}_{k}$ so that the maximal multiple
${\mathbb{Z}_{p}}$-extension of $k$ has rank $n-r$. Write
$\mathfrak{p}_{1},\ldots,\mathfrak{p}_{n}$ for the $p$-adic primes of $k$.
Finally, denote by $\mathscr{Z}(k)$ the set of all
${\mathbb{Z}_{p}}$-extensions of $k$.

Instead of working with the map $\mathscr{L}_{k_{\infty}/k}(\varphi)$, it will
be more convenient to consider the following alternative description of
$\delta_{k_{\infty}/k}^{\textbf{G}}(\varphi)$.

######  Lemma 5.1.2.

Let $k_{\infty}/k$ be a ${\mathbb{Z}_{p}}$-extension with Galois group
$\Gamma$. The quantity $\delta_{k_{\infty}/k}^{\textbf{G}}(\varphi)$ is the
dimension of the kernel of the $\varphi$-isotypic component of the map
$\mathrm{Loc}_{K_{\infty}/K}$ of Proposition 3.1.3.

###### Proof.

This follows from Poitou-Tate duality as in the proof of Proposition 3.1.5
where one replaces the $G_{K}$-module ${\mathbb{Q}_{p}}$ by the module $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}(\varphi)$
on which $G_{k}$ acts by $\varphi$. ∎

Since $\varphi$ is a multiplicative character, the $\varphi$-isotypic
component of a $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}[G]$-module
$X$ is canonically isomorphic to the linear subspace $X[\varphi]$ of $X$
consisting of elements $x\in X$ such that $g\cdot x=\varphi(g)x$ for all $g\in
G$. Hence Lemma 5.1.2 asserts that, if $\varphi\neq\mathds{1}_{k}$, then
$\delta_{k_{\infty}/k}^{\textbf{G}}(\varphi)$ is equal to the dimension of the
kernel of the localization map

(8) $\operatorname{Hom}(G_{K},\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p})[\varphi]\longrightarrow\bigoplus_{i=1}^{n}\left(\bigoplus_{\mathfrak{P}|\mathfrak{p}_{i}}\mathrm{H}^{1}(K_{\mathfrak{P}},\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p})/\operatorname{Hom}(\Gamma_{\mathfrak{P}},\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p})\right)[\varphi].$

###  5.2. Matrices in logarithms of algebraic numbers

For any $\mathfrak{P}\in S_{p}(K)$, we identify
$\mathrm{H}^{1}(K_{\mathfrak{P}},\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p})$ with
$\operatorname{Hom}(K_{\mathfrak{P}}^{\times},\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p})\simeq\operatorname{Hom}(\mathbb{Q}_{p}^{\times},\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p})$ via local
class field theory. We also see $\log_{p}$ and the $p$-adic valuation map
$\mathrm{ord}_{p}$ as additive characters
$K_{\mathfrak{P}}^{\times}\simeq\mathbb{Q}_{p}^{\times}\to\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}$. In order
to describe elements in the domain of the map (8) we make use of the short
exact sequence of $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}[G]$-modules

(9) ${0}$${\operatorname{Hom}(G_{K},\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p})}$${\bigoplus_{i=1}^{n}\operatorname{Hom}(\prod_{\mathfrak{P}|\mathfrak{p}_{i}}K^{\times}_{\mathfrak{P}},\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p})}$${\operatorname{Hom}(\mathcal{O}_{K}[\tfrac{1}{p}]^{\times},\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}),}$$\scriptstyle{A}$

where $A$ is induced by the Artin map.

Let $1\leq i\leq n$ and fix a prime $\mathfrak{P}_{i}$ of $K$ above
$\mathfrak{p}_{i}$. We define a basis
$\\{\eta_{i,\varphi},\tilde{\eta}_{i,\varphi}\\}$ of the $\varphi$-component
of
$\operatorname{Hom}(\prod_{\mathfrak{P}|\mathfrak{p}_{i}}K^{\times}_{\mathfrak{P}},{\mathbb{Q}_{p}})$
as follows. First define characters $\eta_{i}$ and $\tilde{\eta}_{i}$ of
$\prod_{\mathfrak{P}|\mathfrak{p}_{i}}K^{\times}_{\mathfrak{P}}$ by imposing
that they are supported on $K_{\mathfrak{P}_{i}}^{\times}$ and that
$\eta_{i|K_{\mathfrak{P}_{i}}^{\times}}=-\log_{p}$ and
$\tilde{\eta}_{i|K_{\mathfrak{P}_{i}}^{\times}}=\mathrm{ord}_{p}$. We then
define

$\eta_{i,\varphi}=\sum_{\sigma\in
G}\varphi(\sigma)\cdot\eta_{i}\circ\sigma,\qquad\tilde{\eta}_{i,\varphi}=\sum_{\sigma\in
G}\varphi(\sigma)\cdot\tilde{\eta}_{i}\circ\sigma.$

Let $u_{i}$ be any $\mathfrak{P}_{i}$-unit of $K$ which is not a unit (take
for example a generator of $\mathfrak{P}_{i}^{h}$, where $h$ is the class
number of $K$). The choice of $u_{i}$ with a given
$\mathfrak{P}_{i}$-valuation is unique, up to multiplication by a unit of $K$.
Consider

$u_{i,\varphi}=\prod_{\sigma\in
G}\varphi(\sigma)\otimes\sigma^{-1}(u_{i})\in\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu\otimes
K^{\times}.$

It is clear that $u_{i,\varphi}$ is a unit away from the primes above
$\mathfrak{p}_{i}$, and $u_{1,\varphi},\ldots,u_{n,\varphi}$ form a basis of
$(\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu\otimes\mathcal{O}_{K}[1/\mathfrak{p}_{i}]^{\times})[\varphi]$ modulo
$(\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu\otimes\mathcal{O}_{K}^{\times})[\varphi]$. We also fix a basis
$\\{\varepsilon_{1,\varphi},\ldots,\varepsilon_{r(\varphi),\varphi}\\}$ of
$(\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu\otimes\mathcal{O}_{K}^{\times})[\varphi]$ modulo the kernel of
Leopoldt’s map $\iota_{k}(\varphi)$ of (7), where
$r(\varphi)=d^{+}(\varphi)-\delta^{\textbf{L}}_{k}(\varphi)$. For all
$j=1,\ldots,n$, one can see via $\iota_{\mathfrak{P}_{j}}\colon
K\hookrightarrow K_{\mathfrak{P}_{j}}=\mathbb{Q}_{p}$ the elements
$u_{i,\varphi}$ and $\varepsilon_{i,\varphi}$ inside $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu\otimes\mathbb{Q}_{p}^{\times}$. We then define two matrices
$L_{\varphi}=(L_{i,j,\varphi})$ and $M_{\varphi}=(M_{i,j,\varphi})$ of
respective sizes $n\times n$ and $r(\varphi)\times n$ by letting

$L_{i,j,\varphi}=\dfrac{\log_{p}(\iota_{\mathfrak{P}_{j}}(u_{i,\varphi}))}{\mathrm{ord}_{p}(\iota_{\mathfrak{P}_{i}}(u_{i}))},\qquad
M_{i,j,\varphi}={\log_{p}(\iota_{\mathfrak{P}_{j}}(\varepsilon_{i,\varphi}))},$

where we extended $\log_{p}$ to $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu\otimes\mathbb{Q}_{p}^{\times}$ by linearity. Notice that $M_{\varphi}$
has full rank by construction.

Let $\eta^{\prime}$ be an element in the $\varphi$-component of
$\bigoplus_{i=1}^{n}\operatorname{Hom}(\prod_{\mathfrak{P}|\mathfrak{p}_{i}}K^{\times}_{\mathfrak{P}},\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p})$, which we
write as $\sum t_{i}\eta_{i,\varphi}+\tilde{t}_{i}\tilde{\eta}_{i,\varphi}$ in
the basis $\\{\eta_{i,\varphi},\tilde{\eta}_{i,\varphi}\ \colon\ 1\leq i\leq
n\\}$. Denote by $T$ and $\tilde{T}$ the column matrices of respective
coordinates $(t_{1},\ldots,t_{n})$ and $(\tilde{t}_{1},\ldots,\tilde{t}_{n})$.

######  Lemma 5.2.1.

$\eta^{\prime}$ belongs to the image of the map $A$ of (9) if and only if
$\tilde{T}=L_{\varphi}T$ and $M_{\varphi}T=0$.

###### Proof.

By the exactness of (9) such an $\eta^{\prime}$ is characterized by its
vanishing at all the $u_{i}$’s and the $\varepsilon_{i,\varphi}$’s. The lemma
then follows from a straightforward computation, using that
$\eta_{j,\varphi}(\iota_{\mathfrak{P}_{j}}(u_{i}))=-\log_{p}(\iota_{\mathfrak{P}_{j}}(u_{i,\varphi}))$
and
$\tilde{\eta}_{j,\varphi}(\iota_{\mathfrak{P}_{j}}(u_{i}))=\mathrm{ord}_{p}(\iota_{\mathfrak{P}_{j}}(u_{i}))$
for all $1\leq i,j\leq n$. ∎

In what follows we repeatedly use the following elementary fact. For all
compact topological groups $\mathcal{G}$, any non-trivial continuous group
homomorphism $\eta\colon\mathcal{G}\to{\mathbb{Q}_{p}}$ factors through a
quotient $Z_{\eta}$ isomorphic to ${\mathbb{Z}_{p}}$, and two such
homomorphisms $\eta$ and $\eta^{\prime}$ are proportional if and only if
$Z_{\eta}=Z_{\eta^{\prime}}$. Conversely, any topological group $Z$ isomorphic
to ${\mathbb{Z}_{p}}$ which arises as a quotient of $\mathcal{G}$ defines a
continuous homomorphism $\eta\colon\mathcal{G}\to{\mathbb{Q}_{p}}$, which is
unique up to scaling.

Fix $k_{\infty}\in\mathscr{Z}(k)$ and put
$\Gamma=\operatorname{Gal}(k_{\infty}/k)$. The above argument attaches to
$\Gamma$ a non-zero element
$\eta\in\operatorname{Hom}(G_{k},{\mathbb{Q}_{p}})$, unique up to scaling.
Since the restriction map induces an isomorphism
$\operatorname{Hom}(G_{k},{\mathbb{Q}_{p}})\simeq\operatorname{Hom}(G_{K},{\mathbb{Q}_{p}})[\mathds{1}]$,
one can write $A(\eta)$ as
$\sum_{i}(s_{i}\eta_{i,\mathds{1}}+\tilde{s}_{i}\tilde{\eta}_{i,\mathds{1}})$.
We shall refer to the column matrices
$S=(s_{1},\ldots,s_{n})^{\textbf{t}}\in\mathbb{P}^{n-1}({\mathbb{Q}_{p}})$ and
$\tilde{S}=(\tilde{s}_{1},\ldots,\tilde{s}_{n})^{\textbf{t}}\in\mathbb{P}^{n-1}({\mathbb{Q}_{p}})$
as the coordinates of $k_{\infty}$.

######  Proposition 5.2.2.

  1. (1)

The map sending a ${\mathbb{Z}_{p}}$-extension $k_{\infty}/k$ to its
coordinates $S$ defines a bijection between $\mathscr{Z}(k)$ and
$\\{S\in\mathbb{P}^{n-1}({\mathbb{Q}_{p}})\ \colon\ M_{\mathds{1}}S=0\\}$.

  2. (2)

Let $k_{\infty}\in\mathscr{Z}(k)$ with coordinates $S\in\ker(M_{\mathds{1}})$
and let $N_{\varphi}(S)$ be the matrix of size $(n+r(\varphi))\times n$ given
in block notation by:

$\begin{bmatrix}\operatorname{Diag}(S)L_{\varphi}-\operatorname{Diag}(L_{\mathds{1}}S)\\\
M_{\varphi}\end{bmatrix},$

where $\operatorname{Diag}(U)$ denotes the diagonal matrix associated with a
column matrix $U$. Then

$\delta_{k_{\infty}/k}^{\textbf{G}}(\varphi)=\dim\ker N_{\varphi}(S)-\gamma,$

where $\gamma=1$ if $\varphi=\mathds{1}$ and $\gamma=0$ otherwise.

###### Proof.

The first point follows from Lemma 5.2.1 applied to $\varphi=\mathds{1}$. More
precisely, $S$ and $\tilde{S}$ are uniquely determined by the relations
$\tilde{S}=L_{\mathds{1}}S$ and $M_{\mathds{1}}S=0$. Conversely, any
$S,\tilde{S}\in\mathbb{P}^{n-1}({\mathbb{Q}_{p}})$ satisfying these relations
define a non-zero element $\eta\in\operatorname{Hom}(G_{k},{\mathbb{Q}_{p}})$
which cuts out the Galois group of a ${\mathbb{Z}_{p}}$-extension
$k_{\infty}/k$.

Let us prove the second claim. Fix $k_{\infty}\in\mathscr{Z}(k)$ and denote by
$\eta\in\operatorname{Hom}(G_{K},{\mathbb{Q}_{p}})[\mathds{1}]$ the
corresponding continuous homomorphism. By Lemma 5.1.2 and by (8),
$\delta_{k_{\infty}/k}^{\textbf{G}}(\varphi)+\gamma$ is the dimension of the
space consisting of all elements
$\eta^{\prime}\in\bigoplus_{i=1}^{n}\left(\operatorname{Hom}(\prod_{\mathfrak{P}|\mathfrak{p}_{i}}K^{\times}_{\mathfrak{P}},\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p})\right)[\varphi]$ satisfying the conditions of Lemma 5.2.1 and which
are proportional to $A(\eta)$. This last condition means that for all $1\leq
i\leq n$, the restriction to $K_{\mathfrak{P}_{i}}^{\times}$ of
$\eta^{\prime}$ and $A(\eta)$ are proportional, i.e.,

$\eta^{\prime}(\varpi_{i})\cdot
A(\eta)_{|\mathcal{O}^{\times}_{K_{\mathfrak{P}_{i}}}}=A(\eta)(\varpi_{i})\cdot\eta^{\prime}_{|\mathcal{O}^{\times}_{K_{\mathfrak{P}_{i}}}},$

where $\varpi_{i}$ is a uniformizer of $K_{\mathfrak{P}_{i}}$. In terms of
coordinates $S,\tilde{S}$ and $T,\tilde{T}$, this last equality is equivalent
to $\tilde{t}_{i}s_{i}=\tilde{s}_{i}t_{i}$, an equality for all $i$ which can
be rephrased as
$\operatorname{Diag}(S)L_{\varphi}T=\operatorname{Diag}(L_{\mathds{1}}S)T$.
Therefore, $\delta_{k_{\infty}/k}^{\textbf{G}}(\varphi)+\gamma$ is the
dimension of the space of all $T\in\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}^{n}$ in the
kernel of both
$\operatorname{Diag}(S)L_{\varphi}-\operatorname{Diag}(L_{\mathds{1}}S)$ and
$M_{\varphi}$, as claimed. ∎

######  Example 5.2.3.

If $k$ is an imaginary quadratic field, then $M_{\mathds{1}}$ is of size $0$
and Proposition 5.2.2 (1) provides a bijection between $\mathscr{Z}(k)$ and
$\mathbb{P}^{1}({\mathbb{Q}_{p}})$. The ratio
${s_{1}}/{s_{2}}\in{\mathbb{Q}_{p}}\cup\\{\infty\\}$ attached to any
$k_{\infty}\in\mathscr{Z}(k)$ of coordinates
$S=(s_{1},s_{2})\in\mathbb{P}^{1}({\mathbb{Q}_{p}})$ is referred to as the
_slope_ of $k_{\infty}$. For instance, the cyclotomic extension of $k$ has
slope $1$, whereas its anticyclotomic extension has slope $-1$.

###  5.3. Proof of Theorem 5.1.1

We keep the notations of the preceding sections and we abbreviate
$L_{\mathds{1}},M_{\mathds{1}}$ and $N_{\mathds{1}}(S)$ as $L,M$ and $N(S)$
respectively. Note that, by Proposition 5.2.2 (1), the set $\mathscr{Z}(k)$
can be identified with a closed linear subvariety of
$\mathbb{P}^{n-1}({\mathbb{Q}_{p}})$ of dimension
$n-r-1=r_{2}+\delta^{\textbf{L}}_{k}$.

######  Proposition 5.3.1.

Assume that $n-r\leq 2$.

  1. (1)

Let $\mathscr{C}(k)=\\{k_{\infty}\in\mathscr{Z}(k)\ \colon\
\delta_{k_{\infty}/k}^{\textbf{G}}\neq 0\\}$. If
$\mathscr{C}(k)\neq\mathscr{Z}(k)$, then $\mathscr{C}(k)$ is finite with
$|\mathscr{C}(k)|\leq n-1$. If, moreover, $k$ is imaginary quadratic, then
$\mathscr{C}(k)=\emptyset$.

  2. (2)

Let $\mathscr{C}_{\varphi}(k)=\\{k_{\infty}\in\mathscr{Z}(k)\ \colon\
\delta_{k_{\infty}/k}^{\textbf{G}}(\varphi)\neq 0\\}$. If
$\mathscr{C}_{\varphi}(k)\neq\mathscr{Z}(k)$, then $\mathscr{C}_{\varphi}(k)$
is finite with $|\mathscr{C}(k)|\leq n$.

###### Proof.

Let us prove (1). Notice first that all $S\in\ker M$ lie in the kernel of
$N(S)$. By Proposition 5.2.2 with $\varphi=\mathds{1}$ and $K=k$,
$\mathscr{C}(k)$ is in bijection with the set of all
$S\in\mathbb{P}^{n-1}({\mathbb{Q}_{p}})$ such that $MS=0$ and
$\operatorname{rk}N(S)<n-1$.

Assume first that $k$ is quadratic, so $n=2$ and $r=0$. Given any
$S=(s_{1},s_{2})\in\mathbb{P}^{1}({\mathbb{Q}_{p}})$, the matrix
$N(S)=\operatorname{Diag}(S)L-\operatorname{Diag}(LS)$ has the form

$\begin{pmatrix}-s_{2}L_{1,2}&s_{1}L_{1,2}\\\
s_{2}L_{2,1}&-s_{1}L_{2,1}\end{pmatrix}.$

As $\log_{p}$ is injective on $\mathbb{Z}_{p}^{\times}$,
$\log_{p}(\iota_{\mathfrak{p}_{1}}(u_{2}))$ and
$\log_{p}(\iota_{\mathfrak{p}_{2}}(u_{1}))$ both are non-zero, so at least one
of the two non-diagonal entries of $N(S)$ is non-zero. Therefore, this matrix
has rank one for any $S\in\mathbb{P}^{1}({\mathbb{Q}_{p}})$, and
$\mathscr{C}(k)=\emptyset$.

We no longer assume that $k$ is imaginary quadratic, but we still assume that
$n-r\leq 2$. The case where $n-r=1$ is trivial, because it forces
$\mathscr{Z}(k)=\\{k_{\textrm{cyc}}\\}$. We may then assume that $n-r=2$.
Since $M$ has full rank $r$, there exist invertible matrices $P,Q$ such that
$PMQ=\left(I_{r}\ |\ 0\right)$, where $I_{r}$ is the identity matrix of size
$r$. The change of variables $S^{\prime}=Q^{-1}S$ induces a linear bijection
between $\ker M\subset\mathbb{P}^{n-1}(\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p})$ and the
projective line $\\{0\\}\times\mathbb{P}^{1}(\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p})\subset\mathbb{P}^{n-1}(\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p})$. Now
consider the list $P_{1}(S^{\prime}),\ldots,P_{t}(S^{\prime})$ of all
$(n-1)\times(n-1)$-minors of the matrix

$N(S)=N(QS^{\prime})=\begin{bmatrix}\operatorname{Diag}(QS^{\prime})L-\operatorname{Diag}(LQS^{\prime})\\\
M\end{bmatrix}.$

All the $P_{k}$’s are two-variable homogeneous polynomials of degree $\leq
n-1$. In particular, if $\mathscr{C}(k)\neq\mathscr{Z}(k)$, then at least one
of the $P_{k}$’s is not the zero polynomial and hence, it has at most $n-1$
zeros in $\\{0\\}\times\mathbb{P}^{1}(\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p})$, so we
can conclude that $|\mathscr{C}(k)|\leq n-1$.

The proof of point (2) very similar to the previous one. Indeed, by
Proposition 5.2.2, $\mathscr{C}_{\varphi}(k)$ is in bijection with the set of
all $S\in\mathbb{P}^{n-1}({\mathbb{Q}_{p}})$ such that $MS=0$ and
$\operatorname{rk}N_{\varphi}(S)<n$. The same argument with the
$(n-1)\times(n-1)$ minors of $N(S)$ replaced by the $n\times n$ minors of
$N_{\varphi}(S)$ shows that, if $\mathscr{C}_{\varphi}(k)\neq\mathscr{Z}(k)$,
then $|\mathscr{C}(k)|\leq n$. ∎

We end the proof of Theorem 5.1.1 with the case where $k$ is imaginary
quadratic. We let $\tau$ be the complex conjugation of $k$. Recall that $\tau$
acts on $\varphi$ via $\varphi^{\tau}(g)=\varphi(\tau g\tau)$ and that
$\varphi^{\tau}=\varphi$ if and only if $\varphi$ cuts out an extension of $k$
which is abelian over $\mathbb{Q}$.

######  Proposition 5.3.2.

Assume that $k$ is imaginary quadratic and that $\varphi\neq\mathds{1}$.

  1. (1)

If $\varphi^{\tau}=\varphi$, then $\mathscr{C}_{\varphi}(k)=\emptyset$.

  2. (2)

If $\varphi^{\tau}\neq\varphi$, then any
$k_{\infty}\in\mathscr{C}_{\varphi}(k)$ has transcendental slope.

  3. (3)

If $\varphi^{\tau}\neq\varphi$, then $|\mathscr{C}_{\varphi}(k)|\leq 1$.
Moreover, if the $p$-adic Schanuel conjecture holds, then
$\mathscr{C}_{\varphi}(k)=\emptyset$.

###### Proof.

Let $k_{\infty}\in\mathscr{Z}(k)$ of coordinates $S=(s_{1},s_{2})$. Take $K$
to be the Galois closure over $\mathbb{Q}$ of the field cut out by $\varphi$.
Note that $K\neq k$ and that $p$ totally splits in $K$. By Proposition 5.2.2,
$k_{\infty}$ belongs to $\mathscr{C}_{\varphi}(k)$ if and only if the matrix

$N_{\varphi}(S)=\begin{pmatrix}s_{1}(L_{1,1,\varphi}-L_{1,1})-s_{2}L_{1,2}&s_{1}L_{1,2,\varphi}\\\
s_{2}L_{2,1,\varphi}&s_{2}(L_{2,2,\varphi}-L_{2,2})-s_{1}L_{2,1}\\\
M_{1,1,\varphi}&M_{1,2,\varphi}\end{pmatrix}$

has rank $1$. The definition of $L_{\varphi}$ (and also $L_{\varphi^{\tau}}$)
involves the choice, for $i=1,2$, of a prime $\mathfrak{P}_{i}$ of $K$ above
$\mathfrak{p}_{i}$ and a $\mathfrak{P}_{i}$-unit $u_{i}$ of non-zero
valuation. Since $\tau(\mathfrak{p}_{1})=\mathfrak{p}_{2}$, one may take
$\tau(\mathfrak{P}_{1})=\mathfrak{P}_{2}$ and $\tau(u_{1})=u_{2}$, so that
$L_{1,1}=L_{2,2}$, $L_{1,2}=L_{2,1}$, $L_{1,1,\phi}=L_{2,2,\phi^{\tau}}$ and
$L_{1,2,\phi}=L_{2,1,\phi^{\tau}}$ for $\phi\in\\{\varphi,\varphi^{\tau}\\}$.
We may also take $\tau(\varepsilon_{1,\varphi})$ to be
$\varepsilon_{1,\varphi^{\tau}}$, so that
$M_{1,2,\varphi}=M_{1,1,\varphi^{\tau}}$. Moreover, the elements
$u_{1,\varphi},u_{1,\mathds{1}},u_{2,\varphi},u_{2,\mathds{1}},\varepsilon_{1,\varphi}$
and, under the additional condition $\varphi\neq\varphi^{\tau}$, the elements
$u_{1,\varphi},u_{1,\varphi^{\tau}},u_{1,\mathds{1}},u_{2,\varphi},u_{2,\varphi^{\tau}},u_{2,\mathds{1}},\varepsilon_{1,\varphi},\varepsilon_{1,\varphi^{\tau}}$
form two sets of $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu$-linearly
independent $p$-units. To see this, it suffices to consider their valuations
at $\mathfrak{P}_{1}$ and $\mathfrak{P}_{2}$, and use the simple fact that
units belonging to distinct isotypic components are linearly independent.

Now, consider first the case where $\varphi=\varphi^{\tau}$. Then
$N_{\varphi}(S)$ has rank $1$ if and only if its two columns are equal. This
last condition easily implies that $s_{1}=\pm s_{2}$ and that
$u_{1,\varphi}/u_{1,\mathds{1}}$, $u_{2,\mathds{1}}$ and $u_{2,\varphi}$ have
$\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu$-linearly dependent $\mathfrak{P}_{1}$-adic logarithms. But all of these
units have trivial $\mathfrak{P}_{1}$-valuation, so they must be linearly
dependent by Proposition 2.1.2. We have already justified that this is not the
case, so $N_{\varphi}(S)$ has rank $2$ and
$\mathscr{C}_{\varphi}(k)=\emptyset$.

Assume now that $\varphi\neq\varphi^{\tau}$ and that $k_{\infty}$ has an
algebraic slope, i.e., $s_{1}/s_{2}\in\mathbb{P}^{1}(\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu)$. We may
assume that both $s_{1}$ and $s_{2}$ are algebraic numbers, hence
$N_{\varphi}(S)$ has coefficients in the $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu$-linear
subspace $\Lambda$ of $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}$ introduced
in Section 2. Moreover, $N_{\varphi}(S)$ has $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu$-linearly
independent rows and columns. Indeed, the sets
$\\{\varepsilon_{1,\varphi},\varepsilon_{1,\varphi^{\tau}}\\}$ and
$\\{u_{1,\varphi}/u_{1,\mathds{1}},u_{2,\mathds{1}},u_{2,\varphi},\varepsilon_{1,\varphi}\\}$
are two sets of independent units with trivial $\mathfrak{P}_{1}$-valuation.
Therefore, their images under $\log_{p}\circ\iota_{\mathfrak{P}_{1}}$ are
again linearly independent by Proposition 2.1.2. We thus may apply Corollary
2.2.2 and conclude that $\operatorname{rk}N_{\varphi}(S)=2$.

Finally, assume that $\varphi\neq\varphi^{\tau}$ and that
$k_{\infty}\in\mathscr{C}_{\varphi}(k)$, i.e., $N_{\varphi}(S)$ has rank $1$.
We already know that
$s_{1}/s_{2}\in\mathbb{P}^{1}({\mathbb{Q}_{p}})-\mathbb{P}^{1}(\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu)$ and in
particular, both $s_{1}$ and $s_{2}$ are non-zero. Since $L_{1,2}\cdot
M_{1,2,\varphi}\neq 0$, the vanishing of the minor obtained by removing the
second row uniquely determines the slope $s_{1}/s_{2}$, so
$|\mathscr{C}_{\varphi}(k)|\leq 1$. Using the vanishing of the other minors,
an elementary computation yields the polynomial relation

$M_{1,1,\varphi}\cdot M_{1,1,\varphi^{\tau}}\cdot
L_{2,1}^{2}=\left(M_{1,1,\varphi}\cdot(L_{1,1,\varphi^{\tau}}-L_{1,1})-M_{1,1,\varphi^{\tau}}\cdot
L_{2,1,\varphi}\right)\left(M_{1,1,\varphi^{\tau}}\cdot(L_{1,1,\varphi}-L_{1,1})-M_{1,1,\varphi}\cdot
L_{2,1,\varphi^{\tau}}\right).$

The elements of the set
$\\{u_{1,\varphi}/u_{1,\mathds{1}},u_{1,\varphi^{\tau}}/u_{1,\mathds{1}},u_{2,\varphi}/u_{2,\mathds{1}},u_{2,\varphi^{\tau}}/u_{2,\mathds{1}},\varepsilon_{1,\varphi},\varepsilon_{1,\varphi^{\tau}}\\}$
are linearly independent, and they all have a trivial $\mathfrak{P}_{1}$-adic
valuation, so their images under $\log_{p}\circ\iota_{\mathfrak{P}_{1}}$ are
also $\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu$-linearly independent. Therefore, the above polynomial identity
contradicts the $p$-adic Schanuel conjecture. This shows that
$\mathscr{C}_{\varphi}(k)=\emptyset$ under the $p$-adic Schanuel conjecture. ∎

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