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\newtheorem*{factB}{Fact B}
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\title[Leopoldt's conjecture for the Carlitz module]{A modular obstruction to the $P$-adic Leopoldt conjecture\\ for the Carlitz module}
\author{Shawn Krisman}
\email{krisman.shawn@gmail.com}
\date{\today}

\begin{document}

\begin{abstract}
Let $L$ be a finite extension of $K=\Fq(\theta)$ and $P$ a monic prime of $\Fq[\theta]$.
Angl\`es asked whether the $P$-adic Leopoldt defect of Taelman's module of units of the Carlitz module over $L$ always
vanishes, and Lucas conjectured this (together with the equivalence ``$L_P(E/\OL)\neq 0$ iff $\exp_E$ is injective'')
for Anderson modules. We show that both statements fail. For every odd prime $p$, the extension of $\Fp(\theta)$
generated by a root of $z^p-(-\theta)^{p-2}z-(-\theta)^{p-1}$ is real, has Galois group $\mathrm{AGL}(1,p)$, and has
positive $P$-adic Leopoldt defect and vanishing $P$-adic zeta value at $s=1$ for \emph{every} $P$. The
obstruction comes from modular representation theory at the infinite place: we attach to every separable $L/K$ an
integer $\dgen(L)$, depending only on the Galois group of the Galois closure, its decomposition group at $\infty$ and
the character by which this group acts on the Carlitz period, and prove that the $P$-adic defect is at least
$\dgen(L)$ for all $P$. We prove that the defect vanishes, for every $P$, for real abelian extensions and for abelian
extensions with cyclic Sylow $p$-subgroup, using Bosser's $P$-adic Baker--Brumer theorem, and we show that a $v$-adic
analogue of Papanikolas's algebraic independence theorem for Carlitz logarithms would imply that the defect equals
$\dgen(L)$ in general.
\end{abstract}

\maketitle

\section{Introduction}

Let $A=\Fq[\theta]$, $K=\Fq(\theta)$, $\Kinf=\Fq((1/\theta))$, and let $C$ be the Carlitz module. For a finite
extension $L/K$ with integral closure $\OL$ of $A$, Taelman \cite{Ta10,Ta12} introduced the module of units
$U(\OL)\subset C(\OL)$ and the class module $H(\OL)$, and proved the class formula
$\zeta_{\OL}(1)=[H(\OL)]_A\cdot R_{\infty,L}$, an analogue of the analytic class number formula.

Angl\`es \cite{An} proved a $P$-adic analogue for every monic prime $P$ of $A$ and every \emph{real} $L$ (no place of
$L$ above $\infty$ contains the Carlitz period $\pit$):
\[
\zeta_{P,\OL}(1)=[H(\OL)]_A\,\frac{[C(\OL/P\OL)]_A}{[\OL/P\OL]_A}\,R_{P,L},
\]
the analogue of Colmez's $p$-adic class number formula. He defined the $P$-adic Leopoldt defect
\[
\delta_P(U(\OL))=\rk_A U(\OL)-\rk_{A_P}\langle U(P\OL)\rangle_{A_P},
\]
observed that for real $L$ one has $\delta_P=0\iff R_{P,L}\neq 0\iff \zeta_{P,\OL}(1)\neq0$, and asked whether
$\delta_P(U(\OL))=0$ always holds \cite[\S6.3]{An}. Lucas \cite{Lu} extended the class formula to Anderson
$t$-modules $E$, stated the equality of ranks as a conjecture \cite[Conj.~A]{Lu}, and conjectured that the $P$-adic
$L$-series $L_P(E/\OL)$ is non-zero if and only if $\exp_E$ is injective on $\mathrm{Lie}_E(L_\infty)$
\cite[\S4.7]{Lu}. Besides $L=K$, the only known case is the $P$-th cyclotomic function field $L=K(\lambda_P)$ at the prime
$P$ itself, proved by Angl\`es and Taelman \cite[Thm.~9.8]{AT} using Bosser's $P$-adic Baker--Brumer theorem for
Carlitz logarithms \cite{Bo}. (Theorem numbers of \cite{Lu} refer to arXiv:2504.03430v2.)

\subsection*{Main results}
Our first result shows that the answer to Angl\`es's question is negative.

\begin{theorem}\label{thm:main}
Let $p$ be an odd prime, $q=p$, and let $L_p=K(z)$ with
\[
f_p(z)=z^p-(-\theta)^{p-2}z-(-\theta)^{p-1}=0 .
\]
Then $L_p/K$ is a real separable extension of degree $p$, and for every monic prime $P$ of $A$,
\[
\delta_P(U(\mathcal O_{L_p}))\ \ge\ 1,\qquad R_{P,L_p}=0,\qquad \zeta_{P,\mathcal O_{L_p}}(1)=0 .
\]
\end{theorem}

For $p=3$ this is the real cubic field $K(z)$, $z^3+\theta z-\theta^2=0$, whose Galois closure has group $S_3$. In
particular, for $E=C$ over $\mathcal O_{L_p}$, \cite[Conj.~A]{Lu} and the conjecture of \cite[\S4.7]{Lu} fail.

We give two independent proofs. The first (\S\ref{sec:family}) uses only Taelman's lattice theorem and the elementary
theory of the $P$-adic Carlitz logarithm, and shows directly that $\delta_P\ge1$. The second is analytic: in
characteristic $p$ the Goss zeta function of $L_p$ factors as $\zeta_A\cdot\zeta_{F'}$, where
$F'=K(\lambda_\theta)=K((-\theta)^{1/(p-1)})$ is the $\theta$-th Carlitz cyclotomic field, and $F'$ is not real, so
$\zeta_{P,\mathcal O_{F'}}(1)=0$ by \cite[Thm.~6.6]{An}.

The phenomenon behind Theorem~\ref{thm:main} is general. Let $L/K$ be separable with Galois closure $M$, put
$G=\Gal(M/K)$ and $H=\Gal(M/L)$, let $D\subset G$ be a decomposition group at $\infty$, and, if $\pit$ lies in the
completions of $M$ at $\infty$, let $\chi:D\to\Fq^\times$ be the character by which $D$ acts on $\pit$. Put
$\varepsilon_\chi=\sum_{d\in D}\chi(d)d^{-1}\in\Fq[G]$ ($\varepsilon_\chi=0$ if $\pit$ is not in these completions),
and let $r_L=\rk_AU(\OL)$.

\begin{definition}\label{def:dgen}
$\rgen(L)=\max\{\rk(F\mid \Fpb[G/H]) : F\in\Fpb[G],\ \varepsilon_\chi F=0\}$, where $F$ acts on the permutation module
$\Fpb[G/H]$ by left multiplication, and $\dgen(L)=r_L-\rgen(L)$.
\end{definition}

\begin{theorem}\label{thm:A}
For every finite separable $L/K$ and every monic prime $P$, $\delta_P(U(\OL))\ge \dgen(L)\ge0$. Moreover
$\dgen(L)=0$ if $M$ is real or if $p\nmid |D|$, and $\dgen(L_p)=1$.
\end{theorem}

Thus a positive defect requires an imaginary Galois closure in which $\infty$ is wildly ramified or has residue degree
divisible by $p$. There is no number-field analogue. Over $\overline{\mathbb Q}$ the archimedean logarithm
$\log|\cdot|$ is single-valued and gives the generic rank, and $\mathbb Q[G]$ is semisimple. Here the $\infty$-adic
Carlitz logarithm is defined only modulo $\pit A$, and the Galois module $\ker\exp_C\cong\mathrm{Ind}_D^G\chi$ need not
split off in characteristic $p$. Dually, the Goss zeta function of a real field sees only Brauer characters and can
contain an ``odd'' constituent.

Bosser's theorem settles the abelian case under a cyclicity hypothesis.

\begin{theorem}\label{thm:C}
Let $L/K$ be a finite abelian extension. If $L$ is real, or if the Sylow $p$-subgroup of $\Gal(L/K)$ is cyclic (for
instance if $p\nmid[L:K]$), then $\delta_P(U(\OL))=0$ for every monic prime $P$.
\end{theorem}

This extends \cite[Thm.~9.8]{AT} from $L=K(\lambda_P)$ to all such $L$ and all $P$.

\begin{conjecture}\label{conj}
For every finite separable $L/K$ and every monic prime $P$, $\delta_P(U(\OL))=\dgen(L)$. In particular the defect does
not depend on $P$.
\end{conjecture}

\begin{theorem}[conditional]\label{thm:B}
Assume Hypothesis (VP$_P$) of \S\ref{sec:conditional}: $P$-adic Carlitz logarithms of algebraic points that are linearly
independent over $K$ are algebraically independent over $K$. Then $\delta_P(U(\OL))=\dgen(L)$ for every finite
separable $L/K$.
\end{theorem}

Hypothesis (VP$_P$) is the $P$-adic analogue of Papanikolas's theorem \cite{Pa}. It is open. The $v$-adic linear
independence criterion of Chen \cite{Ch} and its extension to all places by Tsui \cite{Ts} give the linear part; we
describe in Remark~\ref{rem:VP} what is missing. Theorem~\ref{thm:B} is the function-field form of the
classical observation that $p$-adic Schanuel-type statements imply Leopoldt's conjecture (cf.\ \cite[\S3]{CM}), with the
important difference that its conclusion is $\delta_P=\dgen$ and not $\delta_P=0$. By Theorem~\ref{thm:main}, no transcendence input can
give $\delta_P=0$ in general.

\subsection*{Organisation}
\S\ref{sec:prelim} collects facts on Taelman units, completions at $\infty$ and the $P$-adic logarithm, and quotes
the two results of \cite{An} that we use. \S\ref{sec:forced} proves Theorem~\ref{thm:A} except for the computation
$\dgen(L_p)=1$. \S\ref{sec:family} treats the family $L_p$ and proves Theorem~\ref{thm:main}. \S\ref{sec:abelian}
proves Theorem~\ref{thm:C}, and \S\ref{sec:conditional} proves Theorem~\ref{thm:B} and discusses (VP$_P$).

\section{Preliminaries}\label{sec:prelim}

Throughout, $L/K$ is a finite separable extension of degree $n$. We fix an algebraic closure of $\Kinf$ with
completion $\Cinf$, and, for a monic prime $P$ of degree $d$, an algebraic closure of $K_P$ with completion $\CP$. We
write $v_P$ for the valuation of $\CP$ with $v_P(P)=1$.

\subsection{The Carlitz module}
$C_\theta=\theta+\tau$, where $\tau$ is the $q$-power Frobenius. The exponential
$\exp_C(x)=\sum_{i\ge0}x^{q^i}/D_i$, with $D_i=\prod_{j=0}^{i-1}(\theta^{q^i}-\theta^{q^j})$, has coefficients in $K$, is
entire on $\Cinf$, satisfies $\exp_C(ax)=C_a(\exp_C x)$ for $a\in A$, and has kernel $\pit A$ \cite[Ch.~3]{Go}. Here
\[
\pit=(-\theta)^{q/(q-1)}\prod_{i\ge1}\bigl(1-\theta^{1-q^i}\bigr)^{-1}
\]
for a suitable choice of $(-\theta)^{1/(q-1)}$ (see \cite[Cor.~5.1.4]{ABP} and \cite[Rem.~3.1.2]{Ch}). The product lies
in $1+\theta^{-1}\Fq[[\theta^{-1}]]$ and $(-\theta)^{q/(q-1)}=(-\theta)\cdot(-\theta)^{1/(q-1)}$, so
$\pit^{q-1}\in\Kinf$ and $\Kinf(\pit)=\Kinf((-\theta)^{1/(q-1)})$. Since $v_\infty(-\theta)=-1$ is prime to $q-1$ and
$\mu_{q-1}=\Fq^\times\subset\Kinf$, the extension $\Kinf(\pit)/\Kinf$ is Galois and totally ramified of degree
$q-1$, and $\sigma\mapsto\sigma(\pit)/\pit$ identifies its Galois group with $\Fq^\times$.

\subsection{Taelman units}
Put $L_\infty=L\otimes_K\Kinf\cong\prod_{v\mid\infty}L_v$. The map $\exp_C$ acts on $L_\infty$ componentwise. A place
$v\mid\infty$ of $L$ is \emph{complex} if some (equivalently, since $\Kinf(\pit)/\Kinf$ is Galois, every)
$\Kinf$-embedding $L_v\hookrightarrow\Cinf$ has image containing $\pit$. It is \emph{real} otherwise, and $L$ is
\emph{real} if all its places above $\infty$ are real; this is the notion of \cite[\S2]{An}. Let $c_L$ be the number
of complex places.

\begin{lemma}\label{lem:ker}
$N_L:=\ker(\exp_C|_{L_\infty})$ is a free $A$-module of rank $c_L$, equal to $\bigoplus_{v\ \mathrm{complex}}A\varpi_v$,
where $\varpi_v\in L_v$ is the element mapped to $\pit$ by a fixed embedding $L_v\hookrightarrow\Cinf$.
\end{lemma}
\begin{proof}
Let $x\in L_v$ and $\iota:L_v\hookrightarrow\Cinf$. Then $\exp_C(x)=0$ iff $\iota(x)\in\pit A$. If $v$ is real, this forces
$x=0$. If $v$ is complex, the solutions are $A\varpi_v$.
\end{proof}

Let $\Lam_L=\{x\in L_\infty:\exp_C(x)\in\OL\}$. By Taelman \cite[Thm.~1]{Ta10} (see also \cite[\S1]{An}), $\Lam_L$ is a discrete and co-compact $A$-submodule of
$L_\infty$. Hence it is an $A$-lattice: it is free of rank $n$, and the natural map $\Lam_L\otimes_A\Kinf\to L_\infty$
is an isomorphism. The module of units is $U(\OL)=\exp_C(\Lam_L)=\exp_C(L_\infty)\cap\OL$. Since $N_L\subset\Lam_L$,
\begin{equation}\label{eq:UL}
U(\OL)\cong\Lam_L/N_L,\qquad r_L:=\rk_AU(\OL)=n-c_L .
\end{equation}
Write $U_L:=U(\OL)\otimes_AK$, a $K$-vector space of dimension $r_L$. By exactness of $\otimes_AK$,
$U_L=(\Lam_L\otimes_AK)/(N_L\otimes_AK)$.

\subsection{The Galois closure}\label{sec:closure}
Let $M$ be the Galois closure of $L$, $G=\Gal(M/K)$ and $H=\Gal(M/L)$. Fix a $K$-embedding $\iota:M\hookrightarrow\Cinf$
and extend it $\Kinf$-linearly to $\iota:M_\infty\to\Cinf$. This map factors through the projection to one factor
$M_{w_0}$; let $D=\{g\in G: gw_0=w_0\}$ be the decomposition group. For $g\in G$ let $\iota_g=\iota\circ g$; it factors
through $M_{g^{-1}w_0}$. The group $G$ acts on $M_\infty$ and on $\Lam_M$ and commutes with $\exp_C$, so it acts on
$U(\OM)$, on $U_M$ and on $N_M$.

\begin{lemma}\label{lem:closure}
\begin{enumerate}[label=\textup{(\alph*)}]
\item The map $M_\infty\otimes_{\Kinf}\Cinf\to\Cinf^{G}$, $x\otimes c\mapsto(c\,\iota_g(x))_{g\in G}$, is an isomorphism.
Similarly, for $a=gH\in G/H$ the map $\rho_a=\iota_g|_{L_\infty}$ is well defined, and
$L_\infty\otimes_{\Kinf}\Cinf\to\Cinf^{G/H}$, $x\otimes c\mapsto(c\rho_a(x))_a$, is an isomorphism.
\item Either every place of $M$ above $\infty$ is real (we say $M$ is real), or every such place is complex.
\item Suppose $M$ is not real. Let $\varpi\in M_\infty$ be the element supported on $M_{w_0}$ with $\iota(\varpi)=\pit$.
There is a homomorphism $\chi:D\to\Fq^\times$ with $d(\varpi)=\chi(d)\varpi$ for $d\in D$. The elements
$z_h=h^{-1}(\varpi)$, $h\in D\backslash G$, form an $A$-basis of $N_M$, and
\[
\iota_g(z_h)=\begin{cases}\chi(gh^{-1})\,\pit& gh^{-1}\in D,\\ 0&\text{otherwise.}\end{cases}
\]
\item The places of $L$ above $\infty$ correspond to the orbits of $D$ on $G/H$. The place belonging to the orbit of
$gH$ is complex if and only if $M$ is not real and $\chi$ is trivial on $D\cap gHg^{-1}$, which is the stabiliser of
$gH$ in $D$.
\end{enumerate}
\end{lemma}
\begin{proof}
(a) The $\iota_g$ are the $|G|=[M:K]$ distinct $K$-embeddings $M\to\Cinf$, and $M/K$ is separable. The second
statement follows in the same way, since $\rho_{gH}$ runs over the $n$ embeddings of $L$.

(b) For any $h$, the element $h$ induces an isomorphism $M_{h^{-1}w_0}\cong M_{w_0}$ over $\Kinf$, and whether a
completion contains $\pit$ is invariant under $\Kinf$-isomorphism.

(c) Let $e_{w_0}$ be the idempotent of $M_{w_0}$. Then $\varpi^{q-1}=\pit^{q-1}e_{w_0}$ with $\pit^{q-1}\in\Kinf$. An element
$d\in D$ fixes $e_{w_0}$ and is $\Kinf$-linear, so $(d\varpi)^{q-1}=\varpi^{q-1}$. Hence $d\varpi=\zeta\varpi$ with
$\zeta\in\mu_{q-1}(M_{w_0})=\Fq^\times$, and $\chi(d):=\zeta$ is multiplicative. Next,
$\exp_C(\varpi)$ is supported on $w_0$ with $\iota(\exp_C\varpi)=\exp_C(\pit)=0$, so $\exp_C(\varpi)=0$ and
$\exp_C(z_h)=h^{-1}\exp_C(\varpi)=0$. The element $z_h$ is supported on the factor $M_{h^{-1}w_0}$, and these factors
are distinct for distinct cosets $Dh$. By Lemma~\ref{lem:ker} applied to $M$, the $z_h$ form a basis. Finally,
$\iota_g(z_h)=\iota(gh^{-1}\varpi)$. This is $\chi(gh^{-1})\pit$ if $gh^{-1}\in D$. Otherwise $gh^{-1}\varpi$ is
supported away from $w_0$, and the value is $0$.

(d) The embeddings $\rho_{gH}$ and $\rho_{g'H}$ define the same place of $L$ iff they differ by a continuous
$\Kinf$-automorphism of $\iota(M_{w_0})$, that is, iff $g'H\in D\,gH$. The completion of $L$ at this place is
identified by $g$ with the closure of $gL$ in $M_{w_0}$, whose Galois group over itself inside
$\Gal(M_{w_0}/\Kinf)=D$ is $D\cap\Gal(M/gL)=D\cap gHg^{-1}$. It contains $\pit$, i.e.\ $\varpi$, iff this group fixes
$\varpi$, i.e.\ iff $\chi$ is trivial on it.
\end{proof}

If $M$ is real we put $N_M=0$ and $\varepsilon_\chi=0$. Otherwise $\varepsilon_\chi=\sum_{d\in D}\chi(d)d^{-1}\in\Fq[G]$. A
direct computation gives
\begin{equation}\label{eq:eps}
d\,\varepsilon_\chi=\chi(d)\varepsilon_\chi,\qquad \varepsilon_\chi d=\chi(d)\varepsilon_\chi\qquad(d\in D).
\end{equation}

\subsection{The $P$-adic logarithm and the defect}
Let $\mathcal D_P=\{x\in\CP: v_P(x)>1/(q^d-1)\}$. On $\mathcal D_P$ the series $\exp_C$ and its inverse
$\Log=\log_C=\sum_i x^{q^i}/\ell_i$, with
$\ell_i=\prod_{j=1}^{i}(\theta-\theta^{q^j})$, converge and are mutually inverse isometric bijections of $\mathcal D_P$, with
$\Log(C_a x)=a\Log(x)$ for $a\in A$ \cite[\S1]{Bo}. Put $\mathcal O_{L,P}=\OL\otimes_AA_P=\prod_{\mathfrak P\mid P}
\mathcal O_{L_{\mathfrak P}}$. Since $P\mathcal O_{L_\mathfrak P}\subset\mathcal D_P$, the map $\Log$ is a bijection
$C(P\mathcal O_{L,P})\to P\mathcal O_{L,P}$. By continuity, the $A$-module structure of $C(P\mathcal O_{L,P})$
extends uniquely to an $A_P$-module structure for which $\Log$ is $A_P$-linear (\cite[\S6.2]{An}).

Let $U(P\OL)=U(\OL)\cap P\OL$. It has finite index in $U(\OL)$, since $U(\OL)/U(P\OL)$ embeds in the finite
module $C(\OL/P\OL)$. Angl\`es \cite[\S6.3]{An} defines
\[
\delta_P(U(\OL))=\rk_AU(\OL)-\rk_{A_P}\langle U(P\OL)\rangle_{A_P},
\]
where $\langle U(P\OL)\rangle_{A_P}$ is the $A_P$-submodule of $C(P\mathcal O_{L,P})$ generated by $U(P\OL)$.

\begin{lemma}\label{lem:ellP}
Fix a $K$-embedding $\iota_P:M\hookrightarrow\CP$. There is a unique $K$-linear map $\ell_P:U_M\to\CP$ with
$\ell_P(u)=\Log(\iota_Pu)$ for $u\in U(P\OM)$. It is injective.
\end{lemma}
\begin{proof}
The map $u\mapsto\Log(\iota_Pu)$ is $A$-linear on $U(P\OM)$, and $U(P\OM)\otimes_AK=U_M$. Hence it extends uniquely.
For injectivity: if $\ell_P(\sum a_k u_k)=0$ with $a_k\in A$ and $u_k\in U(P\OM)$, then $x=\sum C_{a_k}(u_k)\in P\OM$
satisfies $\Log(\iota_Px)=0$. Since $\iota_Px\in\mathcal D_P$ and $\Log$ is injective there, $x=0$.
\end{proof}

\begin{lemma}\label{lem:rank}
Let $u_1,\dots,u_{r_L}\in U(P\OL)$ be a $K$-basis of $U_L$, and $g_1H,\dots,g_nH$ the cosets of $H$. Then
\[
\delta_P(U(\OL))=r_L-\rk_{\CP}\bigl(\ell_P(g_ju_i)\bigr)_{i\le r_L,\,j\le n}.
\]
\end{lemma}
\begin{proof}
$\langle U(P\OL)\rangle_{A_P}$ is a finitely generated submodule of the free $A_P$-module
$C(P\mathcal O_{L,P})\cong P\mathcal O_{L,P}$. Its rank is the $K_P$-dimension of the span of $\Log(U(P\OL))$ in
$L\otimes_KK_P$, which is the span of the $\Log(u_i)$. Now $L\otimes_K\CP\cong\CP^n$ via the embeddings
$\iota_P\circ g_j|_L$. The logarithm has coefficients in $K$ and commutes with these continuous embeddings, so
$\Log(u_i)$ maps to the row $(\Log(\iota_Pg_ju_i))_j=(\ell_P(g_ju_i))_j$, using $g_ju_i\in U(P\OM)$. Finally
$\rk_AU(\OL)=r_L$.
\end{proof}

\begin{remark}\label{rem:R}
For real $L$, Angl\`es's regulator $R_{P,L}$ is the determinant of the $K_P$-linear map $L_P\to L_P$ sending an
$A$-basis $(b_i)$ of $\OL$ to the Iwasawa logarithms of an $A$-basis of $U(\OL)$ \cite[\S6.2]{An}. The Iwasawa
logarithm is $A$-linear and agrees with $\Log$ on $C(P\mathcal O_{L,P})$, so these logarithms span the same
$K_P$-space as $\Log(U(P\OL))$. Hence $R_{P,L}=0$ if and only if $\delta_P(U(\OL))\ge1$.
\end{remark}

\subsection{$P$-adic zeta values}
For a non-zero ideal $I$ of $\OL$ let $n(I)\in A$ be the monic generator of the Fitting ideal of $\OL/I$; it is
multiplicative in $I$. Following \cite[\S4]{An}, put
\[
Z_{P,\OL}(z)=\sum_{d\ge0}U_{P,d}\,z^d,\qquad U_{P,d}=\sum_{\substack{I\ \text{prime to }P\\ \deg n(I)=d}}n(I)^{-1}\in K .
\]
By \cite[Lemma~4.2]{An}, $v_P(U_{P,d})\to\infty$, and $\zeta_{P,\OL}(1):=\sum_dU_{P,d}\in K_P$
\cite[Def.~4.3]{An}. Unique factorisation of ideals gives, in $K[[z]]$,
\begin{equation}\label{eq:euler}
Z_{P,\OL}(z)=\prod_{\mathfrak p\nmid P}\bigl(1-n(\mathfrak p)^{-1}z^{\deg n(\mathfrak p)}\bigr)^{-1},\qquad
n(\mathfrak p)=a^{f(\mathfrak p)}\ \text{if}\ \mathfrak p\cap A=aA .
\end{equation}
We use two results of Angl\`es.

\begin{factA}[{\cite[Thm.~6.6]{An}}]
If $L/K$ is not real, then $\zeta_{P,\OL}(1)=0$. (See also \cite[Thm.~4.23]{Lu}.)
\end{factA}

\begin{factB}[{\cite[Thm.~6.7]{An}}]
If $L/K$ is real, then $\zeta_{P,\OL}(1)=c_{P,L}\,R_{P,L}$ with $c_{P,L}=[H(\OL)]_A[C(\OL/P\OL)]_A/[\OL/P\OL]_A\in
K^\times$. (See also \cite[Thm.~B]{Lu}.)
\end{factB}

\begin{remark}
The isomorphism $C(\OL/P\OL)\cong\prod_{\mathfrak P\mid P}A/(n(\mathfrak P)-1)$ stated in \cite[\S6.2]{An} (and its
$z$-twisted form) is incorrect for ramified $P$. The correct value of $[\tilde C(\OL/\mathfrak P^e)]$ is
$n^{e-1}(n-z^{\deg n})$ with $n=n(\mathfrak P)$. This only changes a non-zero normalising factor, so Fact~B holds as
stated. Our proof of $\delta_P\ge1$ in Theorem~\ref{thm:main} uses neither Fact~A nor Fact~B.
\end{remark}

\section{The forced defect}\label{sec:forced}

Keep the notation of \S\ref{sec:closure}. Let $u_1,\dots,u_r$ ($r=r_L$) be a $K$-basis of $U_L$ and
$e_1,\dots,e_s$ a $K$-basis of $U_M$. Write $g_ju_i=\sum_kc_{ijk}e_k$ with $c_{ijk}\in K$, and let
$\mathcal N(X)=(\sum_kc_{ijk}X_k)_{i,j}\in M_{r\times n}(K[X_1,\dots,X_s])$. For a field $\Omega\supset K$ and a
$K$-linear map $\ell:U_M\to\Omega$, put $\mathcal N(\ell)=\mathcal N(\ell(e_1),\dots,\ell(e_s))=(\ell(g_ju_i))_{i,j}$.

Let $\rho$ be the rank of $\mathcal N(X)$ over $K(X)$. Then $\rk\mathcal N(\ell)\le\rho$ for every $\ell$, since every
minor of size $>\rho$ vanishes identically. Equality holds for some $\Cinf$-valued $\ell$, since a non-zero
$\rho$-minor does not vanish on all of $\Cinf^s$. The number $\rho$ does not depend on the choice of bases.

\begin{proposition}\label{prop:rank}
For every $K$-linear $\ell:U_M\to\Cinf$ there is $F\in\Cinf[G]$ with $\varepsilon_\chi F=0$ and
$\rk\mathcal N(\ell)=\rk(F\mid\Cinf[G/H])$, and every such $F$ arises in this way. Consequently
$\rho=\rgen(L)$.
\end{proposition}

\begin{proof}
\emph{Step 1.} Since $U_M=(\Lam_M\otimes_AK)/(N_M\otimes_AK)$ and $\Lam_M\otimes_A\Cinf=(\Lam_M\otimes_A\Kinf)\otimes_{\Kinf}\Cinf
\cong M_\infty\otimes_{\Kinf}\Cinf$, we get
\[
U_M\otimes_K\Cinf\cong(M_\infty\otimes_{\Kinf}\Cinf)/\langle z_h\rangle_{\Cinf}.
\]
Thus $\Cinf$-valued $K$-linear maps $\ell$ on $U_M$ correspond bijectively to $\Cinf$-linear forms $\lambda$ on
$M_\infty\otimes\Cinf$ vanishing on all $z_h$.

\emph{Step 2.} By Lemma~\ref{lem:closure}(a), every linear form is $\lambda=\sum_gf(g)\iota_g$ for a unique
$f:G\to\Cinf$. Put $F=\sum_gf(g)g$. By Lemma~\ref{lem:closure}(c),
$\lambda(z_h)=\pit\sum_{d\in D}\chi(d)f(dh)$. The sum is the coefficient of $h$ in $\varepsilon_\chi F$. So
$\lambda$ kills every $z_h$ iff $\varepsilon_\chi F=0$. If $M$ is real there is no condition.

\emph{Step 3.} Choose $\lambda_i\in\Lam_L\otimes_AK$ lifting $u_i$, and $\nu_1,\dots,\nu_{c_L}$ a $K$-basis of
$N_L\otimes_AK$. Then $\beta=(\lambda_1,\dots,\lambda_r,\nu_1,\dots,\nu_{c_L})$ is a $K$-basis of $\Lam_L\otimes_AK$,
hence a $\Kinf$-basis of $L_\infty$, and the matrix $\hat R=(\rho_a(\beta_i))_{i,a}$ is invertible by
Lemma~\ref{lem:closure}(a). Since $g_j\lambda_i\in\Lam_M\otimes K$ lifts $g_ju_i$,
\[
\ell(g_ju_i)=\lambda(g_j\lambda_i)=\sum_gf(g)\,\iota(gg_j\lambda_i)=\sum_{a\in G/H}m_{a j}\,\rho_a(\lambda_i),\qquad
m_{aj}=\sum_{g:\ gg_jH=a}f(g).
\]
The matrix $m=(m_{aj})$ is the matrix of left multiplication by $F$ on $\Cinf[G/H]$ in the basis $([g_jH])_j$, since
$F[g_jH]=\sum_gf(g)[gg_jH]$. The rows of $\hat Rm$ are the rows of $\mathcal N(\ell)$, followed by the rows
$(\lambda(g_j\nu_k))_j$, which vanish because $g_jN_L\subset N_M$. Hence
$\rk\mathcal N(\ell)=\rk(\hat Rm)=\rk(m)=\rk(F\mid\Cinf[G/H])$.

\emph{Step 4.} The space $S=\{F:\varepsilon_\chi F=0\}$ is defined over $\Fq$, and the condition
$\rk(F\mid\Cinf[G/H])\ge k$ is the non-vanishing of some $k$-minor, a polynomial with coefficients in $\Fq$ in the
coordinates of $S$. Such a polynomial vanishes on $S(\Cinf)$ iff it vanishes on $S(\Fpb)$, because $\Fpb$ is
infinite. Hence $\rho=\max_{F\in S(\Cinf)}\rk=\max_{F\in S(\Fpb)}\rk=\rgen(L)$.
\end{proof}

\begin{theorem}\label{thm:Aproof}
For every monic prime $P$, $\delta_P(U(\OL))\ge\dgen(L)\ge0$.
\end{theorem}
\begin{proof}
Choose the $u_i$ in $U(P\OL)$. By Lemma~\ref{lem:rank},
$\delta_P=r_L-\rk\mathcal N(\ell_P)$ with $\ell_P$ as in Lemma~\ref{lem:ellP}. Hence
$\delta_P\ge r_L-\rho=\dgen(L)$ by Proposition~\ref{prop:rank}. Finally, $\rho\le r_L$ because $\mathcal N$ has
$r_L$ rows.
\end{proof}

\begin{proposition}\label{prop:zero}
\begin{enumerate}[label=\textup{(\alph*)}]
\item If $M$ is real, then $\dgen(L)=0$.
\item If $p\nmid|D|$, then $\dgen(L)=0$.
\end{enumerate}
\end{proposition}
\begin{proof}
(a) Here $L$ is real, so $r_L=n$, and $F=1$ has rank $n$.

(b) We may assume $M$ is not real. Then $e_\chi=|D|^{-1}\varepsilon_\chi$ is an idempotent by \eqref{eq:eps}, and
$\varepsilon_\chi(1-e_\chi)=0$. The image of $e_\chi$ on a $D$-module $V$ is the $\chi$-eigenspace
$\{v: dv=\chi(d)v\}$. For $V=\Fpb[G/H]=\bigoplus_{\text{orbits}}\Fpb[D/(D\cap gHg^{-1})]$, the $\chi$-eigenspace of an orbit
summand has dimension $1$ if $\chi|_{D\cap gHg^{-1}}=1$ and $0$ otherwise. By Lemma~\ref{lem:closure}(d) its total
dimension is $c_L$. So $\rk(1-e_\chi)=n-c_L=r_L$.
\end{proof}

\begin{lemma}\label{lem:compfactor}
Suppose $M$ is not real and $D=G$. Then $\varepsilon_\chi F=0$ iff $\chi(F):=\sum_gf(g)\chi(g)=0$. If $\Fpb[G/H]$ has a
filtration by $G$-submodules with a one-dimensional subquotient on which $G$ acts through $\chi$, then
$\det(F\mid\Fpb[G/H])=0$ for every such $F$, and $\rgen(L)\le n-1$.
\end{lemma}
\begin{proof}
By \eqref{eq:eps}, $\varepsilon_\chi F=\chi(F)\varepsilon_\chi$, and $\varepsilon_\chi\neq0$. The determinant of $F$
is the product of its determinants on the subquotients of the filtration, and on the given one $F$ acts by $\chi(F)=0$.
\end{proof}

\section{The family $L_p$}\label{sec:family}

Let $p$ be an odd prime and $q=p$. Let $s$ be a root of $x^{p-1}+\theta$, put $F'=K(s)$, let $y$ be a root of
$y^p-y-s$, put $M=F'(y)$, and set $z=ys^{p-2}$.

\begin{proposition}\label{prop:structure}
\begin{enumerate}[label=\textup{(\roman*)}]
\item $[F':K]=p-1$ and $[M:F']=p$. $M/K$ is Galois, with group
$G=\{\tau_{c,b}:\ s\mapsto cs,\ y\mapsto cy+b\ \ (c\in\Fp^\times, b\in\Fp)\}$. Its normal subgroup
$N=\{\tau_{1,b}\}\cong\Fp$ and the subgroup $H=\{\tau_{c,0}\}\cong\Fp^\times$ satisfy $G=N\rtimes H$, with
$\tau_{c,0}\tau_{1,b}\tau_{c,0}^{-1}=\tau_{1,c^{-1}b}$. Thus $G\cong\mathrm{AGL}(1,p)$.
\item $z$ is a root of $f_p$, $L_p:=K(z)=M^H$ has degree $p$, and $M$ is the Galois closure of $L_p$.
\item The finite primes of $A$ ramified in $M$ are exactly $\theta$. One has $\theta\mathcal O_{L_p}=\mathfrak p_1
\mathfrak p_2^{\,p-1}$ with $f(\mathfrak p_1)=f(\mathfrak p_2)=1$, and $\theta$ is totally ramified in $F'$.
\item $\infty$ has a single place in $M$, which is totally ramified, so $D=G$. $M$ is not real, and
$\chi(\tau_{c,b})=c$.
\item $L_p$ is real and $F'$ is not real.
\end{enumerate}
\end{proposition}
\begin{proof}
(i) $x^{p-1}+\theta$ is Eisenstein at $\theta$. Suppose $s=\beta^p-\beta$ with $\beta\in F'$. At the unique place of
$F'$ above $\infty$ we have $v(s)=-1$. If $v(\beta)<0$, then $v(\beta^p-\beta)=pv(\beta)\neq-1$; if $v(\beta)\ge0$,
then $v(\beta^p-\beta)\ge0$. This is a contradiction, so the Artin--Schreier polynomial $y^p-y-s$ is irreducible over
$F'$. Next, $(cy+b)^p-(cy+b)=c(y^p-y)=cs$, so each $\tau_{c,b}$ defines an automorphism of $M$. These give $p(p-1)=[M:K]$
automorphisms, so $M/K$ is Galois with group $G$. The conjugation formula is a direct computation on $y$.

(ii) We have $z^p=(y+s)s^{p(p-2)}=z\,s^{(p-1)(p-2)}+s^{(p-1)^2}=(-\theta)^{p-2}z+(-\theta)^{p-1}$. The element
$\tau_{c,b}$ sends $z$ to $(cy+b)c^{p-2}s^{p-2}=z+bc^{-1}s^{p-2}$, using $c^{p-1}=1$. So the stabiliser of $z$ is $H$,
$[K(z):K]=p$, and $f_p$ is the minimal polynomial of $z$. It is separable, since $f_p'=-(-\theta)^{p-2}\neq0$. The
core of $H$ in $G$ is trivial, so $M$ is the Galois closure.

(iii) The discriminant of $x^{p-1}+\theta$ is a unit times a power of $\theta$, so $F'/K$ is unramified at finite
$a\neq\theta$ and totally ramified at $\theta$. Since $s$ is integral and $\frac{d}{dy}(y^p-y-s)=-1$, $M/F'$ is
unramified at all finite places. Hence the inertia group $I_\theta$ at a prime above $\theta$ has order $p-1$ and meets
$N$ trivially, so it is a complement of $N$. In $\mathrm{AGL}(1,p)$ every complement is the stabiliser of a point and
is self-normalising. Thus the decomposition group $D_\theta$, which normalises $I_\theta$, equals $I_\theta$. As a
$G$-set, $G/H$ is the affine line on which $G$ acts. A point stabiliser has two orbits on it, of sizes $1$ and $p-1$.
These give the primes of $L_p$ above $\theta$, with residue degrees $1$ (because $D_\theta=I_\theta$) and ramification
indices $1$ and $p-1$.

(iv) Let $v_\infty(\theta)=-1$. The Newton polygon of $f_p$ at $\infty$ has vertices $(0,-(p-1))$ and $(p,0)$. The
middle point $(1,-(p-2))$ lies strictly above the segment, because $(p-1)^2/p>p-2$. So all roots of $f_p$ have valuation
$-(p-1)/p$, and $\infty$ is totally ramified in $L_p$, with $e=p$. It is also totally ramified in $F'$, with $e=p-1$.
Hence $e(M/\infty)=p(p-1)=|G|$: there is one place and $D=G$. The completion $F'_\infty=\Kinf(s)$ contains
$(-\theta)^{1/(p-1)}=\zeta s$ for some $\zeta\in\Fp^\times$, so it contains $\pit$ (\S2.1). Hence $M$ is not real.
Finally $\pit=\zeta s w$ with $\zeta,w\in\Kinf$, so $\tau_{c,b}(\pit)/\pit=\tau_{c,b}(s)/s=c$.

(v) $L_p$ has one place above $\infty$, and its completion has degree $p$ over $\Kinf$. Since $(p-1)\nmid p$, it cannot
contain the degree-$(p-1)$ field $\Kinf(\pit)$. On the other hand $F'_\infty\ni\pit$.
\end{proof}

\begin{theorem}\label{thm:dgen}
$\dgen(L_p)=1$. Consequently, for every monic prime $P$, $\delta_P(U(\mathcal O_{L_p}))\ge1$, $R_{P,L_p}=0$ and
$\zeta_{P,\mathcal O_{L_p}}(1)=0$.
\end{theorem}
\begin{proof}
Since $L_p$ is real, $r_{L_p}=p$. Since $D=G$, Lemma~\ref{lem:compfactor} applies with constraint $\chi(F)=0$.

\emph{The $G$-set.} Identify $G/H$ with $N\cong\Fp$ via $\tau_{1,b}H\mapsto b$. By
Proposition~\ref{prop:structure}(i), $\tau_{1,b'}$ acts by translation $x\mapsto x+b'$ and $\tau_{c,0}$ acts by
$x\mapsto c^{-1}x$.

\emph{A filtration.} Let $V=\Fpb[\Fp]$ with basis $[x]$. Let $t$ be translation by $1$, put $\mathsf N=t-1$, and
$V_k=\mathsf N^kV$ for $0\le k\le p$. Since $\mathsf N^p=t^p-1=0$ and $V=\Fpb[t]\cdot[0]$, we have $\dim V_k=p-k$, and
$V_k/V_{k+1}$ is spanned by $\mathsf N^k[0]$. Translations act trivially on $V_k/V_{k+1}$, because $t=1+\mathsf N$.
For $u\in\Fp^\times$ let $\mu_u[x]=[ux]$. Then $\mu_ut\mu_u^{-1}=t^u$ and $\mu_u\mathsf N\mu_u^{-1}=(1+\mathsf N)^u-1=
u\mathsf N+O(\mathsf N^2)$. As $\mu_u[0]=[0]$, we get $\mu_u\mathsf N^k[0]\equiv u^k\mathsf N^k[0]\pmod{V_{k+1}}$, and
each $V_k$ is $G$-stable. So $\tau_{c,b}$ acts on $V_k/V_{k+1}$ by $c^{-k}$. For $k=p-2$ this is $c=\chi(\tau_{c,b})$.

\emph{The rank.} By Lemma~\ref{lem:compfactor}, $\rgen(L_p)\le p-1$. Conversely, $F=1-t$ satisfies
$\chi(F)=1-\chi(t)=0$, and its kernel on $V$ is the line of translation invariants, so $\rk F=p-1$. Hence
$\rgen(L_p)=p-1$ and $\dgen(L_p)=1$.

\emph{Consequences.} Theorem~\ref{thm:Aproof} gives $\delta_P\ge1$. Remark~\ref{rem:R} gives $R_{P,L_p}=0$, and Fact~B
then gives $\zeta_{P,\mathcal O_{L_p}}(1)=0$.
\end{proof}

The vanishing of $\zeta_{P,\mathcal O_{L_p}}(1)$ also has a direct analytic proof, independent of Taelman units.

\begin{proposition}\label{prop:euler}
In $K[[z]]$, $Z_{P,\mathcal O_{L_p}}(z)=Z_{P,A}(z)\,Z_{P,\mathcal O_{F'}}(z)$ for every monic prime $P$.
\end{proposition}
\begin{proof}
We compare the Euler factors \eqref{eq:euler} prime by prime. Write $X=z^{\deg a}/a$.

For $a\neq\theta$ the prime $a$ is unramified in $M$, by Proposition~\ref{prop:structure}(iii). Let $\phi\in G$ be a
Frobenius at a prime above $a$. The primes of $L_p$ (resp.\ $F'$) above $a$ correspond to the $\langle\phi\rangle$-orbits
on $G/H\cong\Fp$ (resp.\ $G/N\cong\Fp^\times$), with residue degree equal to the orbit length. Write $\phi=\tau_{c,b}$.
\begin{itemize}
\item If $c=1$ and $b=0$: $L_p$ gives $(1-X)^{-p}$, and $F'$ gives $(1-X)^{-(p-1)}$.
\item If $c=1$ and $b\neq0$: $\phi$ is a $p$-cycle on $\Fp$, so $L_p$ gives $(1-X^p)^{-1}=(1-X)^{-p}$ in characteristic
$p$; $F'$ gives $(1-X)^{-(p-1)}$.
\item If $c$ has order $m>1$: $\phi$ has one fixed point and $(p-1)/m$ cycles of length $m$, so $L_p$ gives
$(1-X)^{-1}(1-X^m)^{-(p-1)/m}$; $F'$ gives $(1-X^m)^{-(p-1)/m}$.
\end{itemize}
In each case the factor of $L_p$ equals $(1-X)^{-1}$ times that of $F'$, and $(1-X)^{-1}$ is the factor of $A$.

For $a=\theta$, Proposition~\ref{prop:structure}(iii) gives $(1-X)^{-2}$ for $L_p$ and $(1-X)^{-1}$ for $F'$.

Since the factors agree prime by prime, removing the factor at $P$ on both sides preserves the identity.
\end{proof}

\begin{corollary}\label{cor:zeta}
$\zeta_{P,\mathcal O_{L_p}}(1)=\zeta_{P,A}(1)\cdot\zeta_{P,\mathcal O_{F'}}(1)=0$ for every $P$.
\end{corollary}
\begin{proof}
Let $\sum a_iz^i$ and $\sum b_jz^j$ be the two factors. By \cite[Lemma~4.2]{An}, $a_i\to0$ and $b_j\to0$ in $K_P$.
Then $|a_ib_j|_P\to0$ as $i+j\to\infty$, so the double series $\sum_{i,j}a_ib_j$ converges in the non-archimedean field
$K_P$ and may be summed in any order. Summing along $i+j=m$ shows that the coefficients of the product sum to
$(\sum a_i)(\sum b_j)$. By Proposition~\ref{prop:euler} the product is $Z_{P,\mathcal O_{L_p}}$. Finally, $F'$ is not
real, so $\zeta_{P,\mathcal O_{F'}}(1)=0$ by Fact~A.
\end{proof}

\begin{remark}\label{rem:numerics}
These identities have been checked by exact computation in $\Fp[[P]]$, using the factorisation of $f_p$ and of
$x^{p-1}+\theta$ modulo every prime of degree $\le6$ ($p=3,5$) or $\le4$ ($p=7$). For $p=3$ and $P=\theta\pm1$, the
partial sums of $L_P(\chi,1)=\sum_{a\ \mathrm{monic},\,P\nmid a}a(0)/a$, where $\zeta_{P,\mathcal O_{F'}}=\zeta_{P,A}
L_P(\chi,\cdot)$, have $P$-adic valuation exactly $3^m-(m+1)$ through degree $m\le6$. In particular they tend to $0$,
while $\zeta_{P,A}(1)$ is a unit.
\end{remark}

\section{Abelian extensions}\label{sec:abelian}

\begin{factC}[{Bosser \cite[Thm.~1]{Bo}}]
Let $\overline K\subset\CP$ be the algebraic closure of $K$, and $\alpha_1,\dots,\alpha_m\in\mathcal D_P\cap\overline K$. If
$\Log\alpha_1,\dots,\Log\alpha_m$ are linearly independent over $K$, then they are linearly independent over
$\overline K$.
\end{factC}

\begin{proof}[Proof of Theorem~\ref{thm:C}]
Let $G=\Gal(L/K)$, so $M=L$, $H=1$ and $r=r_L$, and write $\mathbf C=\CP$. Choose a $K$-basis of $U_L$ in $U(P\OL)$.
By Lemma~\ref{lem:ellP}, the logarithms $\ell_P$ of these basis elements are $K$-linearly independent. They are
logarithms of points of $\mathcal D_P\cap\overline K$, so by Fact~C the $\overline K$-linear extension of $\ell_P$ to
$U_L\otimes_K\overline K$ is injective. Let $\tilde\ell$ be the $\mathbf C$-linear extension to $U_L\otimes_K\mathbf C$.
Make $\mathbf C[G]$ act on linear forms by $(x\cdot\phi)(u)=\phi(xu)$; since $G$ is abelian this is a module structure.
The columns of the matrix of Lemma~\ref{lem:rank} are the forms $g\cdot\tilde\ell$, so
\[
\rk\bigl(\ell_P(gu_i)\bigr)=\dim\mathbf C[G]\cdot\tilde\ell=|G|-\dim\Ann(\tilde\ell),
\]
where $\Ann(\tilde\ell)=\{x: \tilde\ell(xU_L)=0\}$ is an ideal of $\mathbf C[G]$.

\emph{Cyclicity.} By the normal basis theorem $L_\infty\cong\Kinf[G]$. Hence $\Lam_L\otimes_AK$ and $K[G]$ are
$K[G]$-modules that become isomorphic over $\Kinf$, and by the Noether--Deuring theorem \cite[(29.7)]{CR} they are
isomorphic. So $U_L$ is a quotient of $K[G]$, i.e.\ a cyclic $K[G]$-module.

(i) $L$ real. Then $N_L=0$ and $U_L=K[G]e$ is free, with $e\in U(P\OL)$. Put $y_g=\ell_P(ge)$. The matrix of
Lemma~\ref{lem:rank} for the basis $(he)_h$ is $(y_{gh})_{h,g}$. After replacing $h$ by $h^{-1}$ and transposing, it is the
matrix of multiplication by $Y=\sum_gy_gg$ on $\mathbf C[G]$ in the basis $G$, so the two determinants agree up to sign. Write $G=G_p\times G_{p'}$. Then
$\mathbf C[G]=\bigoplus_\psi e_\psi\mathbf C[G]$ over the characters $\psi:G_{p'}\to\Fpb^\times$, and
$e_\psi\mathbf C[G]\cong\mathbf C[G_p]$ is a local ring with residue map $\sum x_kk\mapsto\sum x_k\psi(k_{p'})$.
Multiplication by an element $w$ of a local ring $R$ with residue field $\mathbf C$ has determinant
$\bar w^{\dim R}$, since $w-\bar w$ is nilpotent. Hence
\[
\det\bigl(y_{gh}\bigr)=\pm\prod_\psi\Bigl(\sum_{g\in G}\psi(g_{p'})\,y_g\Bigr)^{|G_p|}.
\]
Each factor equals $\tilde\ell\bigl(\sum_g\psi(g_{p'})\,ge\bigr)$. This is the value of the $\overline K$-linear
extension of $\ell_P$ at a non-zero element of $U_L\otimes\overline K$, hence non-zero. So the rank is $|G|=r_L$, and
$\delta_P=0$ by Lemma~\ref{lem:rank}.

(ii) $G_p=\langle\gamma\rangle$ cyclic of order $p^m$. Each block $e_\psi\mathbf C[G]\cong\mathbf C[x]/(x^{p^m})$ with
$x=\gamma-1$ has only the ideals $(x^j)$. So every ideal of $\mathbf C[G]$, in particular $\Ann(\tilde\ell)$, is
generated by elements $e_\psi x^j$ of $\Fpb[G]$. For $y\in\Fpb[G]$ and $u\in U_L$ we have $yu\in U_L\otimes\overline K$,
and $\tilde\ell(yu)=0$ forces $yu=0$ by Fact~C. Hence $\Ann(\tilde\ell)=\Ann(U_L\otimes\mathbf C)$. Since
$U_L\otimes\mathbf C$ is cyclic, it is isomorphic to $\mathbf C[G]/\Ann(U_L\otimes\mathbf C)$, and so
\[
\rk=\dim\mathbf C[G]/\Ann(U_L\otimes\mathbf C)=\dim U_L=r_L .
\]
Lemma~\ref{lem:rank} gives $\delta_P=0$.
\end{proof}

\begin{remark}
Only the homogeneous form of Bosser's theorem is used. When $G_p$ is not cyclic, the ideals of $\mathbf C[G_p]$ form
continuous families, and linear independence over $\overline K$ no longer controls $\Ann(\tilde\ell)$. Indeed, for
imaginary abelian $L$ with non-cyclic Sylow $p$-subgroup contained in $D$, $\dgen(L)$ can be positive: numerically
one finds $\dgen=2$ for $G=D=(\mathbb Z/3)^2\times\mathbb Z/2$ ($q=3$) and $\dgen=1$ for $G=D=(\mathbb Z/2)^2$
($q=2$). We have not constructed explicit fields with these data.
\end{remark}

\section{A corrected conjecture and a conditional theorem}\label{sec:conditional}

\begin{VP}
Let $\alpha_1,\dots,\alpha_m\in\mathcal D_P\cap\overline K$ (with $\overline K\subset\CP$). If
$\Log\alpha_1,\dots,\Log\alpha_m$ are linearly independent over $K$, then they are algebraically independent over $K$.
\end{VP}

\begin{proof}[Proof of Theorem~\ref{thm:B}]
Choose the basis $e_k$ of $U_M$ in $U(P\OM)$. The values $\ell_P(e_k)=\Log(\iota_Pe_k)$ are logarithms of points of
$\mathcal D_P\cap\overline K$, and they are $K$-linearly independent by Lemma~\ref{lem:ellP}. By (VP$_P$) they are
algebraically independent over $K$. Every minor of $\mathcal N(X)$ is a polynomial over $K$, so it vanishes at
$X=\ell_P(e)$ only if it vanishes identically. Hence $\rk\mathcal N(\ell_P)=\rho=\rgen(L)$, and Lemma~\ref{lem:rank}
gives $\delta_P=\dgen(L)$.
\end{proof}

By Proposition~\ref{prop:zero}, (VP$_P$) implies $\delta_P(U(\OL))=0$ whenever the Galois closure of $L$ is real or
$\infty$ has decomposition group of order prime to $p$ in it, for abelian and non-abelian $L$ alike. By
Theorem~\ref{thm:dgen} it cannot imply more.

\begin{remark}[status of (VP$_P$)]\label{rem:VP}
At $\infty$ the analogue is Papanikolas's theorem \cite[Thm.~1.2.6]{Pa}. Its proof combines the ABP criterion
\cite{ABP} with a Tannakian (difference Galois) computation.

At a finite place $v=P$, three inputs are available.
\begin{itemize}
\item Chen \cite{Ch} proved a $v$-adic ABP criterion for systems $\psi^{(-1)}=\Phi\psi$, with
$\Phi\in M(\overline K[t])$ and $\psi$ $v$-adically entire with algebraic coefficients, specialised at
$\xi\in\overline K$. For $\deg v=1$ he constructed such systems whose specialisations are $v$-adic Carlitz multiple
polylogarithms. His Theorem~1.2.2 shows that the algebra they span is graded by weight and that
$\overline K$-linear relations come from $K$-linear ones.
\item Tsui \cite{Ts} extended this to all $v$.
\item Mishiba \cite{Mi} built a $v$-adic Tannakian theory (\emph{op.\ cit.}, Thm.~1.1) for \emph{formal} periods in
$K^{\rm sep}((t-\zeta))$. Their specialisation requires his $v$-adic ABP conjecture (\emph{op.\ cit.}, Conj.~6.3), which
is proved only in rank one, and his formal polylogarithms are not identified with the convergent $v$-adic ones.
\end{itemize}
These results give the linear case of (VP$_P$), which in the homogeneous form needed here is already Fact~C. They do
not give (VP$_P$) itself: the grading reduces $\overline K$-algebraic relations among logarithms to $K$-linear relations
among monomials of one weight, but does not exclude such relations.

What seems to be missing is a transfer of Papanikolas's \S\S4--6 to Chen's systems. For $\deg P=1$:
\begin{enumerate}[label=(\roman*)]
\item the difference Galois group of the system attached to $(\Omega_\alpha,\log_C u_1,\dots,\log_Cu_m)$ over
$\overline K(t)$, inside the fraction field of the $v$-adic Tate algebra, whose $\sigma$-invariants are $\Fq(t)$;
\item the equality of the transcendence degrees of values and functions, via Chen's criterion applied to tensor powers;
\item the computation of this group as $\mathbb G_m\ltimes\mathbb G_a^m$.
\end{enumerate}
The last step needs the $\overline K$-linear independence of $1$, Chen's $v$-adic period $\tilde\Omega$ and the
$\tilde\Omega\log_Cu_i$, which does not seem to follow formally from the grading.
\end{remark}

\begin{remark}
Lucas \cite[Conj.~B]{Lu}, after Caruso--Gazda \cite{CG}, conjectures that $\mathrm{ord}_{z=1}L_P$ is independent of $P$.
Conjecture~\ref{conj} predicts the same for the defect. We have not computed $\mathrm{ord}_{z=1}L_P$ for $L_p$, and do
not claim that it equals $\dgen(L_p)$.
\end{remark}

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