Recall that we denoted by $U_n$ (resp. $U_n'$) the $\cO_p$-span of the pro-$p$ completion of the group of units (resp. of $p$-units) of $H_n$ (Notation <ref>). We omit the index when $n=0$. Note that $U_n'$ is torsion-free because $p$ is unramified in $H$. The Rubin-Stark conjecture over $\bZ$ <cit.> implies the following conjecture.

One has

$$\varepsilon_n^\chi \in {\bigcap}^{d^+}_{\cO_p[\Gamma_n]} U_n' \quad (n\geq 1), \qquad \textrm{resp. } \ u^\chi \in {\bigwedge}^{d^++f}_{\cO_p} U' \quad (n=0).

Recall that, if $\varphi$ is a linear form on a $R$-module $M$ (for a commutative ring $R$), and if $t\geq 1$ is an integer, then $\varphi$ induces a $R$-linear map $\varphi : {\bigwedge}^t_R M \longrightarrow {\bigwedge}^{t-1}_R M$ which sends $m_1\wedge\ldots\wedge m_t$ to $\sum_{i=1}^t(-1)^{i-1}m_1\wedge\ldots\wedge m_{i-1}\wedge m_{i+1}\wedge\ldots\wedge m_t$. More generally, $s$ linear forms $\varphi_1,\ldots\varphi_s$ on $M$ with $s\leq t$ induce a $R$-linear map

$$\bigwedge_{1\leq i\leq s} \varphi_i : {\bigwedge}^t_R M \longrightarrow {\bigwedge}^{t-s}_R M$$

given by $m\mapsto \varphi_s \circ \ldots \circ \varphi_1 (m)$.

We take $n=0$ for the rest of this section and for $1\leq i \leq f$ we consider the $p$-adic valuation $\ord_{w_{p,i}} : \bC\cO_H[\tfrac{1}{p}]^\times \longrightarrow \bC$ induced by the place ${w_{p,i}}$. By <cit.>, the induced map

$$\bigwedge_{1 \leq i \leq f} \ord_{w_{p,i}} : {\bigwedge}_\bC^{d^++f} \bC\cO_H[\tfrac{1}{p}]^\times \longrightarrow {\bigwedge}_\bC^{d^+} \bC\cO_H[\tfrac{1}{p}]^\times$$

sends $u^\chi$ on the Rubin-Stark element

\begin{equation}\label{eq:xi^chi}
\xi^\chi\in {\bigwedge}_\bC^{d^+} \bC\cO_H[\tfrac{1}{p}]^\times
\end{equation}

defined as the inverse image under $\lambda_{L,S}$ of $L^*(\chi^{-1},0)e_\chi \cdot w_{\infty,1}\wedge \ldots \wedge w_{\infty,d^+}$. If Conjecture <ref> or Conjecture <ref> holds for $u^\chi$, then the corresponding statement for $\xi^\chi$ is also true.

§.§ Iwasawa-theoretic conjectures

We assume in this section that $\chi$ is non-trivial and of prime-to-$p$ order. The idempotent $e_\chi$ has coefficients in $\cO_p$ and the $\chi$-part $M^\chi$ and the $\chi$-quotient $M_\chi$ of an $\cO_p$-module $M$ (see <ref>) both coincide with $e_\chi M$. Moreover, $\chi$ is residually non-trivial, a fact which is used in (<ref>) and, therefore, in the formulation of Conjectures <ref>, <ref> and <ref>.

We let the integer $n\geq 0$ of last section vary and we assume Conjecture <ref> for every $n$. As explained in <cit.> the family $(\varepsilon_n^\chi)_{n\geq 1}$ is norm-compatible, so it defines an element

\begin{equation}\label{eq:family_RS}
\varepsilon_\infty^\chi \in \varprojlim_n {\bigcap}^{d^+}_{\cO_p[\Gamma_n]} (U_n')^\chi = {\bigcap}^{d^+}_\Lambda (U_\infty')^\chi = {\bigwedge}^{d^+}_\Lambda (U_\infty')^\chi.
\end{equation}

Here, the first identification follows from <cit.> and the second one from the fact that $(U_\infty')^\chi = \HH^1_{\Iw,\ff,p}(\bQ,\widecheck{T}_p)$ which is free (of rank $d^+$) over $\Lambda$ by the results of <ref>. The following conjecture is taken from <cit.> and should be thought as a cyclotomic Iwasawa main conjecture for $\chi$. Let us mention that this conjecture may also be formulated for other $\Zp$-extensions of $k$ than the cyclotomic one.

We have

$$\char_\Lambda \left({\bigwedge}^{d^+} (U_\infty')^\chi\right) \big/\left(\Lambda\cdot \varepsilon_\infty^\chi\right) = \cA^f \cdot \char_\Lambda (A_\infty')_\chi,$$

where $\cA$ is the augmentation ideal of $\Lambda$ and  $A_\infty'$ is the inverse limit over $n\geq 0$ of the $p$-split ideal class groups of $H_n$ (see Notation <ref>).

Since ${\bigwedge}^{d^+} (U_\infty')^\chi$ is free of rank one over $\Lambda$, Conjecture <ref> implies immediately the non-vanishing of $\varepsilon_\infty^\chi$, as well as the following conjecture:

We have $\varepsilon_\infty^\chi \in \cA^f \cdot {\bigwedge}^{d^+} (U_\infty')^\chi$.

This is <cit.> for the cyclotomic extension (and a more general number field $k$), where it is referred to as the Exceptional Zero Conjecture for Rubin-Stark elements. Assume Conjecture <ref> and fix $\gamma$ a topological generator of $\Gamma$. Following loc. cit. we now reformulate the (cyclotomic) Iwasawa-theoretic Mazur-Rubin-Sano Conjecture for $(\chi,S,V')$ in terms of the element $\kappa_{\infty,\gamma}\in{\bigwedge}^{d^+} (U_\infty')^\chi$ which satisfies

$$\varepsilon_\infty^\chi = (\gamma-1)^f \cdot \kappa_{\infty,\gamma}.$$

For all $1\leq i \leq f$, let $\rec_{w_{p,i}} : L^\times \longrightarrow \Gal((\bQ_\infty L)_{w_{p,i}}/L_{w_{p,i}})\simeq \Gamma$ be the local reciprocity map for $L$ at ${w_{p,i}}$. We still denote by $\rec_{w_{p,i}}$ the induced $\cO_p$-homomorphism

$$\rec_{w_{p,i}} : (U')^\chi=e_\chi (\cO_p\otimes \cO_{L,S'}^\times) \longrightarrow \cO_p \otimes \Gamma \simeq \cA/\cA^2.$$

Conjecture <ref> holds true, and if we let $\kappa_\gamma \in {\bigwedge}^{d^+}_{\cO_p} (U')^\chi$ be the bottom layer of $\kappa_{\infty,\gamma}$, then the map

$$\bigwedge_{1 \leq i \leq f} \rec_{w_{p,i}} : {\bigwedge}^{d^++f}_{\cO_p} (U')^\chi \longrightarrow \cA^f/\cA^{f+1} \otimes_{\cO_p} {\bigwedge}^{d^+}_{\cO_p} (U')^\chi$$

sends $u^\chi$ to $(-1)^{d^+\cdot f} \cdot(\#\Delta)^{-f} \cdot (\gamma-1)^f \otimes \kappa_\gamma$.

As first noted in <cit.>, one may prove as in <cit.> that Conjecture <ref> is equivalent to <cit.>. As for Conjecture <ref>, taking $T=\emptyset$ (in the notations of [74, 11]) is allowed because the $\Zp$-module $U'_n$ is torsion-free for all $n\geq 0$. Lastly, note that, while the definition of  Rubin-Stark elements depends on how we ordered the places $v_{p,1},\ldots,v_{p,f}$ of $V'\backslash V$ and on the choice of $w_{p,i}$ above $v_{p,i}$, the validity of all the conjectures of this section does not depend on these choices.

§ MONOMIAL REPRESENTATIONS

§.§ Induced representations

Let $\rho$ be a monomial representation and fix an isomorphism $\rho \simeq \Ind_k^{\bQ}\chi$ over $E$, where $\chi : \Gal(H/k) \longrightarrow E^\times$ is a non-trivial character. We do not assume yet that $\chi$ has order prime to $p$. The underlying space of $\rho$ is then equal to $W=E[G] \otimes_{E[\Gal(H/k)]} E(\chi)$, where $E(\chi)$ is a $E$-line on which $\Gal(H/k)$ acts via $\chi$ and  the tensor product follows the rule $gh\otimes 1=g\otimes\chi(h)$ for all $g\in G,h\in\Gal(H/k)$. The (left) $G$-action on $W$ is given by $g\cdot(g'\otimes 1)=gg'\otimes 1$ for all $g,g'\in G$. By Frobenius reciprocity we have $\rho \otimes \eta \simeq \Ind_k^{\bQ}(\chi \otimes \eta)$ for any $\eta\in \widehat{\Gamma}$, where we still denoted by $\eta$ its restriction to $G_k$. We assume throughout <ref> that the $\cO_p$-lattice $T_p$ of $W_{p}$ is

so that the family $(g\otimes 1)_{g\in G}$ generates $T_p$ over $\cO_p$.

* Given any $\cO_p[G]$-module $M$, there is a canonical isomorphism

$$\Hom_G(T_p,M) \overset{\sim}{\longrightarrow} {M}^\chi, \qquad \psi \mapsto \psi(1\otimes 1),$$

where ${M}^\chi$ denotes the $\chi$-part of $M$, seen as a $\Gal(H/k)$-module. In the same fashion, for any $\eta\in\widehat{\Gamma}$ and for any $E_\eta[G_{\bQ}]$-module $M$, there is a canonical isomorphism $\Hom_{G_{\bQ}}(W_\eta,M)\simeq e_{\chi \otimes \eta} M = {M}^{\chi\otimes\eta}$.

* Given any $\cO_p[G]$-module $M$, the module $\Hom_G(M,D_p)$ is canonically isomorphic to the Pontryagin dual $\left(M_\chi\right)^\vee$ of the $\chi$-quotient of $M$.

The first statement is straightforward to check, so we only prove the second one. Once we have fixed a generator of the different of $\cO_p$ over $\Zp$, the $\cO_p$-module $\left(M_\chi\right)^\vee$ can be identified with $\Hom_{\cO_p}(M_\chi,E_p/\cO_p)$ as in <ref>. On the other hand, since induction and co-induction functors over finite groups coincide, $D_p$ can be described as the $\cO_p$-module of maps $f : G \longrightarrow E_p/\cO_p \otimes \cO_p(\chi)$ satisfying $f(hg)=h\cdot f(g)$ for all $h\in\Gal(H/k)$ and $g\in G$, the left $G$-action being $(g\cdot f)(g')=f(g'g)$. Therefore, the map $F \mapsto (m \mapsto F(m)(1\otimes 1))$ identifies $\Hom_G(M,D_p)$ with $\Hom_{\cO_p}(M_\chi,E_p/\cO_p)$.

For any $\eta\in\widehat{\Gamma}$, for any module $M$ as in Lemma <ref> (1) and for any $m\in M$, we  let $\psi_m$ be the element of $\Hom_G(T_p,M)$ (resp. of $\Hom_{G_\bQ}(W_\eta,M)$) which satisfies $\psi_m(1\otimes 1)=m$. More generally, for any $\omega\in {\bigwedge}^r M$ (and $r\geq 0$) we let $\psi_\omega$ be the element of $ {\bigwedge}_{\cO_p}^r \Hom_G(T_p,M)$ (resp. of ${\bigwedge}_{E_\eta}^r \Hom_{G_{\bQ}}(W_\eta,M)$) corresponding to $\omega$ under the induced isomorphism on exterior products.

§.§ Complex regulators

We first define a natural basis $\omega_\infty^+$ of $\det_E\HH^0(\bR,W)$ in which we will compute all the complex regulators. The embedding $\iota_\infty:\ob{\bQ}\subseteq \bC$ defines a place $w_\infty$ (resp. $v_\infty$) of $\ob{\bQ}$ (resp. of $k$) as well as a complex conjugation which is denoted by $\sigma_\infty$. As in <ref>, we denote by $V=\{v_{\infty,1},\ldots,v_{\infty,d^+}\}$ the set of archimedean places of $k$ which split completely in $L=H^{\ker \chi}$. We choose for each $i=1,\ldots,d^+$ an automorphism $\tau_{\infty,i}\in G_{\bQ}$ which sends $v_{\infty,i}$ onto $v_\infty$ and we put $w_{\infty,i}=\tau^{-1}_{\infty,i}(w_\infty)$. For simplicity we still write $\tau_{\infty,i}$ and $w_{\infty,i}$ for their restrictions to finite extensions of $L$. We obtain a basis $\omega_\infty^+=\{t_{\infty,1},\ldots,t_{\infty,d^+}\}$ of $\det_E H^0(\bR,W)=\det_E W^{\sigma_\infty=1}$ by letting

$$t_{\infty,i}= \left\{\begin{array}{ll}
\tau_{\infty,i}\otimes 1 &  \textrm{if } v_i \textrm{ is real,}\\
\tau_{\infty,i}\otimes 1 + \sigma_\infty\cdot\tau_{\infty,i}\otimes 1 &  \textrm{if } v_i \textrm{ is complex.}
\end{array}\right.$$

Note that it is moreover $T_p$-optimal for our fixed choice of $T_p$.

Assume Conjecture <ref>. Let $\eta \in \widehat{\Gamma}$ be a character of order $p^{n}$. Put $\omega_{\ff,\eta}=\psi_{e_\eta\cdot \varepsilon_n^\chi}$ if $\eta\neq\mathds{1}$ and $\omega_{\ff,\eta}=\psi_{\xi^\chi}$ if $\eta=\mathds{1}$ (see Notation <ref>). The complex regulator of $\rho\otimes\eta$ computed in the bases $\omega_\infty^+$ and $\omega_{\ff,\eta}$ is equal to

$$\Reg_{\omega_\infty^+}(\rho\otimes \eta)= p^{-n\cdot d^+}\cdot L^*((\rho\otimes \eta)^*,0).$$

For $a \in L_n$ which is seen in $L_{n,w_{\infty,i}}$, put

\textrm{sgn}(a)a &  \textrm{if } v_i \textrm{ is real,}\\
a\cdot \ob{a} &  \textrm{if } v_i \textrm{ is complex,}
\end{array}\right.$$

where sgn is the sign function when $L_{n,w_{\infty,i}}=\bR$ and $a\mapsto \ob{a}$ is the complex conjugation when $L_{n,w_{\infty,i}}=\bC$. Write $e_\eta \cdot \varepsilon_n^{\chi}$ (or $\xi^\chi$ if $\eta$ is trivial) as $\mu_1\wedge\ldots\wedge\mu_{d^+}$ and write $\omega_{\ff,\eta}=\psi_{\mu_1}\wedge\ldots\wedge\psi_{\mu_{d^+}}$ accordingly. Then by construction of the $t_{\infty,i}$'s, one has $1\otimes (\iota_\infty)(\psi_{\mu_{j}}(t_{\infty,i}))=|\mu_j|_{w_{\infty,i}}\in E_\eta \otimes \bR^\times$ for all $1\leq i,j\leq d^+$, so we have

$$\Reg_{\omega_\infty^+}(\rho\otimes \eta)= \det\left(\log_\infty|\mu_j|_{w_{\infty,i}}\right)_{1\leq i,j\leq d^+}.$$

Since $L(\rho\otimes\eta)^*,s)=L((\chi\otimes\eta)^{-1},s)$ and since $e_\eta \cdot \varepsilon_n^{\chi}=p^{-n} \sum_{g\in\Gamma_n}\eta^{-1}(g)g(\varepsilon_n^{\chi})$ by definition, the result follows directly from <cit.> and Remark <ref>.

§.§ Iwasawa main conjectures

In the two last sections we explore the relation between Conjecture <ref> and the various conjectures on Rubin-Stark elements of <ref>. We henceforth assume that $\chi$ is of prime-to-$p$ order. In what follows, the basis of a given $p$-stabilization $W_p^+$ of $W_p$ is always assumed to be a $T_p$-optimal eigenbasis for $\sigma_p$ as in Lemma <ref>.

Assume Conjectures <ref> and <ref> and pick any $p$-stabilization $W_p^+$ of $W_p$.

* The statement EX$_{\rho,\rho^+}$ in Conjecture <ref> is true, and the element $\theta_{\rho,\rho^+}'$ of Proposition <ref> coincides with $\sC^\str_{\omega_p^+}(\psi_{\varepsilon_\infty^\chi})$, where $\sC_{\omega_p^+}^\str$ is the operator introduced in <ref>.

* Conjecture <ref> implies that $\theta_{\rho,\rho^+}$ has an order of vanishing at $\mathds{1}$ greater than or equal to $e$. The converse implication also holds if we moreover assume that $W_p^+$ is admissible and that $\cL(\rho,\rho^+)$ does not vanish.

Put $\theta=\sC^\str_{\omega_p^+}(\psi_{\varepsilon_\infty^\chi})$ and fix a non-trivial character $\eta\in\widehat{\Gamma}$ of conductor $p^n$. By Lemma <ref> and Lemma <ref>, we have

\begin{align*}
\eta(\theta) &= p^{(1-n)\cdot d^+}\cdot\dfrac{\det(\rho^+)(\sigma_p^n)}{\g(\eta^{-1})^{d^+}}\Reg_{\omega_p^+}(\rho\otimes\eta)\\
&=\dfrac{\det(\rho^+)(\sigma_p^n)}{\g(\eta^{-1})^{d^+}} \Reg_{\omega_p^+}(\rho\otimes\eta)\cdot j\left(\dfrac{L^*((\rho\otimes\eta)^*,0)}{\Reg_{\omega_\infty^+}(\rho\otimes\eta)}\right),
\end{align*}

where the regulators are computed with respect to the basis $\omega_{\ff_\eta}$ defined in Lemma <ref>. A comparison with the interpolation property of $\theta_{\rho,\rho^+}'$ and Weierstrass' preparation theorem then shows that $\theta_{\rho,\rho^+}'=\theta$; hence, EX$_{\rho,\rho^+}$ is true by Proposition <ref>, and (1) follows.

The first implication of (2) is obvious, since $\sC^\str_{\omega_p^+}$ is $\Lambda$-linear and since $\theta$ has at most a pole of order $f-e$ at $\mathds{1}$ by construction. For the converse implication, assume that $\theta_{\rho,\rho^+}'\in\cA^e$ (so $\theta$ is also in $\cA^e$), that $W_p^+$ is admissible and that $\cL(\rho,\rho^+)\neq 0$. Then, we know by Theorem <ref> (1), by Lemma <ref> and by the exactness of the last row of (<ref>) that the cokernel of $\sC^\str_{\omega_p^+}$ has finite $\Gamma$-coinvariants, so its image is generated over $\Lambda$ by an element in $\cI^{f-e}\backslash\cI^{f-e-1}$. We may then write $\theta$ as $(\gamma-1)^f\cdot \theta_\gamma$ for some topological generator $\gamma$ of $\Gamma$ and some $\theta_\gamma\in \im(\sC^\str_{\omega_p^+})$; hence, Conjecture <ref> follows from the injectivity of $\sC^\str_{\omega_p^+}$.

Assume Conjectures <ref> and <ref>. Let $W_p^+$ be a $p$-stabilization of $W_p$ such that $X_\infty(\rho,\rho^+)$ is of $\Lambda$-torsion.

* If either IMC$_{\rho,\rho^+}$ or IMC$_\chi$ are true, then Conjecture <ref> is also true.

* IMC$_{\rho,\rho^+}$ and IMC$_\chi$ are equivalent.

We have already seen that IMC$_\chi$ implies Conjecture <ref> because ${\bigwedge}^{d^+}(U_\infty')^\chi$ is free of rank one over $\Lambda$. Thus, the first claim is implied by the second one, which we prove now. By Theorem <ref> and by Lemma <ref> we know that the map $\Loc_+^\str$ of <ref> is injective. Since its domain and codomain are free, $\coker(\Loc_+^\str)$ and $\coker(\wedge^{d^+}\Loc_+^\str)$ have the same characteristic ideal. On the other hand, recall that we may identify $(U_\infty')^\chi$ with $\Hom_G(T_p,U'_\infty)$ via Lemma <ref> and $\Lambda \cdot \sC_{\omega_p^+}^\str({\varepsilon_\infty^\chi})$ with $\Lambda \cdot \theta_{\rho,\rho^+}$ via Theorem <ref>. By Lemma <ref> and by (<ref>) we have exact three short exact sequences:

\begin{equation*}
\begin{tikzcd}
0 \rar & \underbrace{\left({\bigwedge}^{d^+}(U'_\infty)^\chi\right) \big/\left(\Lambda\cdot \varepsilon_\infty^\chi\right)}_B \ar[r, "\sC_{\omega_p^+}^\str"] & \underbrace{\cI^{e-f} \big/\left(\Lambda\cdot \theta_{\rho,\rho^+}\right)}_C \rar & \underbrace{\coker(\wedge^{d^+}\Loc_+^\str)}_D \rar & 0, \\
0 \rar & \HH^0(\Qp,T^-_p) \rar & X_\infty(\rho,\rho^+) \rar & X_\infty^\str(\rho,\rho^+) \rar & 0,  \\
0 \rar & \coker(\Loc_+^\str) \rar & X_\infty^\str(\rho,\rho^+) \rar &  \Sha_\infty^1(D_p)^\vee \rar &  0.
\end{tikzcd}
\end{equation*}

The $\Gamma$-action on $\HH^0(\Qp,T^-_p)$ being trivial, its characteristic ideal is $\cA^e$. Moreover, $\Sha_\infty^1(D_p)^\vee$ and $(A'_\infty)_\chi$ are pseudo-isomorphic $\Lambda$-modules by Lemmas <ref> (3) and <ref> (2). Therefore, by multiplicativity of characteristic ideals, we may conclude that

\begin{align*}
\textbf{IMC}_\chi &\Longleftrightarrow \char_\Lambda(B)=\cA^f \cdot \char_\Lambda (A'_\infty)_\chi\\
&\Longleftrightarrow \char_\Lambda(C) = \cA^f \cdot \char_\Lambda X_\infty^\str(\rho,\rho^+)\\
&\Longleftrightarrow \cA^{f-e}\cdot (\Lambda\cdot \theta_{\rho,\rho^+}) = \cA^{f-e}\cdot \char_\Lambda X_\infty(\rho,\rho^+)\\
&\Longleftrightarrow (\Lambda\cdot \theta_{\rho,\rho^+}) = \char_\Lambda X_\infty(\rho,\rho^+)\\
&\Longleftrightarrow \textbf{IMC}_{\rho,\rho^+}.
\end{align*}

§.§ Extra zeros at the trivial character

We first construct an $\cO_p$-basis of $\HH^0(\Qp,T_p)$ as follows. Let $w_p$ be the $p$-adic place of $H$ defined by $\iota_p$, and denote by $v_{p,1}, \ldots, v_{p,f}$ the $p$-adic places of $k$ which totally split in $L$ as in <ref>. Fix also a place $w_{p,i}$ of $H$ above $v_{p,i}$, and let

\begin{equation}\label{eq:def_t_p,i}
t_{p,i}= [H_{w_{p,i}} : k_{v_{p,i}}]^{-1} \cdot \sum_{\mathclap{\substack{g\in G, \\ g(w_{p,i})=w_p}}} g \otimes 1 \quad \in T_p, \qquad (1\leq i \leq f).
\end{equation}

This defines an $\cO_p$-basis $t_{p,1},\ldots,t_{p,f}$ of $\HH^0(\Qp,T_p)$. For any $t\in T_p$, consider the following two composite maps

\begin{equation*}
\begin{tikzcd}[column sep = 12ex]
\ord_p^{(t)} \ :\ \Hom_G(T_p,U') \ar[r, "\iota_p \circ \textrm{ev}_{t}"] & \cO_p \otimes \widehat{K}^\times \ar[r, "\ord_p"] & \cO_p,
\end{tikzcd}  \end{equation*}

\begin{equation*}\begin{tikzcd}[column sep = 12ex]
\log_p^{(t)} \ :\ \Hom_G(T_p,U') \ar[r, "\iota_p \circ \textrm{ev}_{t}"] & \cO_p \otimes \widehat{K}^\times \ar[r, "\log_p"] & \cO_p,
\end{tikzcd}
\end{equation*}

where $\textrm{ev}_{t}$ is the evaluation map at $t$ and  $K=H_{w_p}$ as in <ref>.

For all $1\leq i \leq f$ and $u\in (U')^\chi$, we have

$$\ord_p^{(t_{p,i})}(\psi_u) = [k_{v_{p,i}} : \Qp] \cdot \ord_{w_{p,i}}(u), \qquad \log_p^{(t_{p,i})}(\psi_u) = -\log_p \circ \chi_{\cyc} \circ \rec_{w_{p,i}}(u). $$

Fix $1\leq i \leq f$ and $u\in (U')^\chi=(\cO_p \otimes \cO_L[\tfrac{1}{p}]^\times)^\chi$. Since $\psi_u$ is $G$-equivariant, we have

\begin{align*}
\iota_p(\psi_u(t_{p,i})) &= [H_{w_{p,i}} : k_{v_{p,i}}]^{-1} \cdot \iota_p \left( \prod_{{\substack{g\in G, \\ g(w_{p,i})=w_p}}} g(u)\right) \\
&= N_i(\iota_{p,i}(u)),
\end{align*}

where $N_i$ is the norm map of the extension $L_{w_{p,i}}=k_{v_{p,i}}$ over $\Qp$, and  $\iota_{p,i}$ is the $p$-adic embedding of $L$ defined by $w_{p,i}$. Since $\ord_p(N_i(\iota_{p,i}(u)))= [k_{v_{p,i}} : \Qp] \cdot \ord_{w_{p,i}}(u)$ and since $\chi_{\cyc} \circ \rec_{w_{p,i}}(u)=N_i(\iota_{p,i}(u))^{-1}$, the lemma follows easily.

* Let $W_p^0=\HH^0(\Qp,W_p)$. Then,

$$\left(\bigwedge_{1\leq i \leq f} \ord_p^{(t_{p,i})}\right)(\psi_{u^\chi}) = (-1)^{d^+\cdot f}\cdot \frac{\det(1-\sigma_p^{-1}\,\big| \, W_p/W_p^0)}{(\#\Delta)^f} \cdot \psi_{\xi^\chi}$$

in $\bigwedge^{d^+}_{\ob{\bQ}_p}  \Hom_G(W_p,\ob{\bQ}_p\otimes U')$.

* Assume Conjectures <ref> and <ref> and <ref>. Fix a topological generator $\gamma$ of $\Gamma$ and put $\varpi_\gamma=\log_p\circ\chi_{\cyc}(\gamma)\in p\Zp$. Then, MRS$_\chi$ is equivalent to the equality

$$\left(\bigwedge_{1\leq i \leq f} \log_p^{(t_{p,i})}\right)(\psi_{u^\chi}) = (-1)^{d^+\cdot f}\cdot \left(-\frac{\varpi_\gamma}{\#\Delta}\right)^f \cdot \psi_{\kappa_\gamma}$$

in ${\bigwedge}^{d^+}_{\cO_p} \Hom_G(T_p,U')$.

Let us prove (1). First of all, a direct computation gives
$$\det(1-\sigma_p^{-1}\,\big| \, W_p/W_p^0)=\prod_{v\in S^{00}_p(k)} (1-\chi^{-1}(v)) \cdot\prod_{1\leq i \leq f} [k_{v_{p,i}}:\Qp],$$

where $S^{00}_p(k)=S_p(k)-\{v_{p,1},\ldots,v_{p,f}\}$ and where we see $\chi$ as a Hecke character over $k$. Consider the operator $\Phi_{V',V}$ of <cit.> (extended by linearity to a $\bC[\Delta]$-linear map), where $V$ and $V'$ are as in <ref>. Its $\chi$-part is given by

$$e_\chi\cdot \Phi_{V',V} = (-1)^{d^+\cdot f} \cdot{(\#\Delta)^f} \cdot \bigwedge_{1 \leq i \leq f}\ord_{w_{p,i}}= (-1)^{d^+\cdot f} \frac{(\#\Delta)^f}{\prod_{1\leq i \leq f} [k_{v_{p,i}}:\Qp]} \cdot \bigwedge_{1 \leq i \leq f}\ord_p^{(t_{p,i})},$$

the last equality being a consequence of Lemma <ref> (1). On the other hand, by Propositions 3.5 and 3.6 of loc. cit., we know that

$$\left(e_\chi \cdot \Phi_{V',V}\right)(\psi_{u^\chi}) = \prod_{v\in S_p^{00}(k)}(1-\chi^{-1}(v))\cdot \psi_{\xi^\chi}.$$

The claim (1) then follows from the above three equations. As for the second claim, it follows immediately from the formula for $\log_p^{(t_{p,i})}$ of Lemma <ref>.

We are now in a position to state and prove a theorem comparing MRS$_\chi$ and EZC$_{\rho,\rho^+}$.

Assume Conjectures <ref> and <ref>.

* MRS$_\chi$ implies EZC$_{\rho,{\rho}^+}$ for all admissible $p$-stabilizations $(\rho^+,W_p^+)$ of $W_p$.

* Conversely, choose any integer $0\leq e \leq d^-$ and any eigenbasis $\{\widetilde{t}_1,\ldots,\widetilde{t}_d\}$ of $W_p$ for $\sigma_p$ such that $\{\widetilde{t}_1,\ldots,\widetilde{t}_{d^++e}\}$ contains a basis of $\HH^0(\Qp,W_p)$. Let $\cB$ be the (finite) set of increasing sequences $\beta=(1\leq i_1<\ldots<i_{d^+}\leq d^++e)$ of integers between $1$ and $d^++e$ and of length $d^+$, and set $W_{p,\beta}^+=<\widetilde{t}_{i_1},\ldots,\widetilde{t}_{i_{d^+}}>_{E_p}$ for any such $\beta$. If $W_{p,\beta}^+$ is admissible and if EZC$_{\rho,{\rho_\beta^+}}$ holds for all $\beta\in \cB$, and if there exists at least one $\beta\in\cB$ such that $\cL(\rho,\rho^+_\beta)\neq 0$, then MRS$_\chi$ holds true.

We know by the first part of Theorem <ref> that EX$_{\rho,\rho^+}$ holds for any $p$-stabilization $(\rho^+,W_p^+)$ of $W_p$. By its second part, we may also assume without loss of generality that Conjecture <ref> (=wEZC$_\chi$) holds and that $\theta_{\rho,\rho^+}$ vanishes at $\mathds{1}$ with multiplicity $\geq \dim \HH^0(\Qp,W_p^-)$ for all $\rho^+$. By Proposition <ref>, MRS$_\chi$ is equivalent to

\begin{equation}\label{eq:MRS_equivalence}
\left(\bigwedge_{1\leq i \leq f} \log_p^{(t_{p,i})}\right)(\psi_{u^\chi}) = (-1)^{d^+\cdot f}\cdot \left(-\frac{\varpi_\gamma}{\#\Delta}\right)^f \cdot \psi_{\kappa_\gamma},
\end{equation}

where $\gamma$ is a fixed generator of $\Gamma$, $\varpi_\gamma=\log_p\circ\chi_{\cyc}(\gamma)\in p\Zp$ and $\kappa_\gamma$ is the bottom layer of the element $\kappa_{\infty,\gamma}\in{\bigwedge}^{d^+}(U_\infty')^\chi$ satisfying $\varepsilon_\infty^\chi=(\gamma-1)^f\cdot\kappa_{\infty,\gamma}$.

Given any $p$-stabilization $(\rho^+,W_p^+)$ of $W_p$, we introduce the following notation: choose any $T_p$-optimal eigenbasis $\omega_p^+=t_1\wedge\ldots\wedge t_{d^+}$ of $\det_{E_p}W_p^+$ for $\sigma_p$ such that $\{t_1,\ldots,t_{f-e}\}$ generates $\HH^0(\Qp,W_p^+)$, where $e=\dim \HH^0(\Qp,W_p^-)$. Let $I^{+,0}=\{1,\ldots,f-e\}$, $I^{+,00}=\{f-e+1,\ldots,d^+\}$ and $I^{-,0}=\{d^++1,\ldots,d^++e\}$ for simplicity, and let

$$\nu_{\rho^+}=\bigwedge_{i\in I^{+,0}}\ord_p^{(t_i)}\wedge \bigwedge_{i\in I^{+,00}} \log_p^{(t_{i})} \in {\bigwedge^{d^+}}_{E_p}\Hom_G(W_p,\cU')^*,$$

where $\cU'=E_p \otimes U'$ as in <ref> and  $X^*=\Hom_{E_p}(X,E_p)$ for any $E_p$-vector space $X$. Note that $\nu_{\rho^+}$ depends on the choice of $\omega_p^+$ and on the ordering of the $t_i$'s, as the construction of $\theta'_{\rho,\rho^+}$ given in Theorem <ref> does. Moreover, it makes sense to apply $\nu_{\rho^+}$ to both sides of the equality (<ref>), and we call for convenience $\textrm{LHS}_{\rho,\rho^+}$ and $\textrm{RHS}_{\rho,\rho^+}$ the left-hand side and the right-hand side of the resulting equality in $E_p$. We claim that, if $\rho^+$ is admissible, then this equality is equivalent to $\textbf{EZC}_{\rho,\rho^+}$. In other words:

If $\rho^+$ is admissible, then $\textrm{LHS}_{\rho,\rho^+}=\textrm{RHS}_{\rho,\rho^+}$ if and only if the $p$-adic measure $\theta_{\rho,\rho^+}'=\sC^\str_{\omega_p^+}(\psi_{\varepsilon_\infty^\chi})$ satisfies the formula stated in Proposition <ref> (2).

Assume that $\rho^+$ is admissible. We first compute $\textrm{LHS}_{\rho,\rho^+}$ as follows. Let $t_{d^++1},\ldots,t_{d^++e}$ be any elements of $\HH^0(\Qp,T_p)$ such that $\bigwedge_{i\in I^{0}} t_i=\bigwedge_{1\leq j\leq f}t_{p,j}$, where $I^0=I^{+,0}\bigcup I^{-,0}$.
We have, by definition of $\cL(\rho,{\rho}^+)$, by Proposition <ref> (1), by Lemma <ref> and taking into account the sign rule:

\begin{align*}
\textrm{LHS}_{\rho,\rho^+}&= \nu_{\rho^+}\wedge \left(\bigwedge_{i\in I^0} \log_p^{(t_{i})}\right)(\psi_{u^\chi})\\
&=(-1)^{f-e}\cdot \cL(\rho,{\rho}^+)\cdot \left(\bigwedge_{i\in I^0} \ord_p^{(t_{i})}\right)\wedge \left(\bigwedge_{i\in I^+} \log_p^{(t_i)}\right)(\psi_{u^\chi}) \\
&=(-1)^{f-e}\cdot \cL(\rho,{\rho}^+)\cdot \left(\bigwedge_{i\in I^+} \log_p^{(t_i)}\right)\left((-1)^{d^+\cdot f}\cdot  \frac{\det(1-\sigma_p^{-1}\,\big| \, W_p/W_p^0)}{(\#\Delta)^f}\cdot \psi_{\xi^\chi}\right) \\
&=(-1)^{f-e+d^+\cdot f}\cdot \cL(\rho,{\rho}^+)\cdot \frac{\det(1-\sigma_p^{-1}\,\big| \, W_p/W_p^0)}{(\#\Delta)^f} \cdot \Reg_{{\omega}^+_{p}}(\rho) \cdot j\left(\frac{L^*(\rho^*,0)}{\Reg_{\omega_{\infty}^+}(\rho)}\right).
\end{align*}

Let us explicit the relation between $\textrm{RHS}_{\rho,\rho^+}$ and the $e$-th derivative of $L_p(s):=\kappa^s(\theta_{\rho,\rho^+}')$ at $s=0$. We may write $L_p(s)$ as $(\kappa(\gamma)^s-1)^e\cdot \kappa^s(\theta_\gamma)$, where $\theta_\gamma = \sC^\str_{{\omega}_p^+}((\gamma-1)^{f-e}\cdot \kappa_{\infty,\gamma})\in\Lambda$. Therefore, $\frac{1}{e!}L_p^{(e)}(0)=\varpi_\gamma^e \cdot \mathds{1}(\theta_\gamma)$. On the other hand, if we write $(\gamma-1)^{f-e}\cdot \kappa_{\infty,\gamma}$ as a wedge product of the form $(\gamma-1)\cdot v_1 \wedge \ldots (\gamma-1)\cdot v_{f-e} \wedge v_{f-e+1} \wedge \ldots \wedge v_{d^+}$, then Lemmas <ref> and <ref> together show that

$$\mathds{1}(\theta_\gamma)=\varpi_\gamma^{f-e} \cdot(1-p^{-1})^{f-e}\cdot \prod_{i\in I^{+,00}}\frac{1-p^{-1}\beta_i}{1-\beta_i^{-1}} \cdot \nu_{\rho^+}(\kappa_\gamma),$$

where $\beta_i$ is the eigenvalue of $\sigma_p$ acting on $t_i$. Since $\cE(\rho,{\rho}^+) = \prod_{i\in I^+}(1-p^{-1}\beta_i) \cdot \prod_{i\in I^{-,00}}(1-\beta_i^{-1})$, we have

\begin{align*}
\frac{1}{e!} L_p^{(e)}(0) &= \varpi_\gamma^{f} \cdot(1-p^{-1})^{f-e}\cdot \prod_{i\in I^{+,00}}\frac{1-p^{-1}\beta_i}{1-\beta_i^{-1}} \cdot (-1)^{d^+\cdot f}\cdot \left(-\frac{\#\Delta}{\varpi_\gamma}\right)^f \cdot \textrm{RHS}_{\rho,\rho^+} \\
&=(-1)^{f+d^+\cdot f}\cdot \frac{\cE(\rho,{\rho}^+)}{\det(1-\sigma_p^{-1}\, \big|\, W_p/W_p^0)} \cdot (\#\Delta)^f \cdot \textrm{RHS}_{\rho,\rho^+}.
\end{align*}

Hence, we may infer that

$$ \textrm{LHS}_{\rho,\rho^+}=\textrm{RHS}_{\rho,\rho^+} \Longleftrightarrow \frac{1}{e!} L_p^{(e)}(0)= (-1)^e \cdot \cL(\rho,{\rho}^+)\cdot \cE(\rho,{\rho}^+)\cdot  \frac{\Reg_{{\omega}^+_{p}}(\rho)}{\Reg_{\omega_{\infty}^+}(\rho)}\cdot L^*(\rho^*,0),$$

as claimed.
We end the proof of Theorem <ref>. If MRS$_\chi$ holds, then $\textrm{LHS}_{\rho,\rho^+}=\textrm{RHS}_{\rho,\rho^+}$ obviously holds for any admissible $p$-stabilization $\rho^+$ of $\rho$, so EZC$_{\rho,\rho^+}$ also holds by Lemma <ref>. This shows the first assertion.

Let us prove (2). We may assume without loss of generality that the given family $\{\widetilde{t}_1,\ldots,\widetilde{t}_d\}$ is a basis of $T_p$ and that $\{\widetilde{t}_i\ |\ i\in I^0\}$ is a basis of $\HH^0(\Qp,T_p)$. By hypothesis, we know that $\{\widetilde{t}_i\ |\ i\in I^+\}$ generates an admissible $p$-stabilization of $W_p$, so the space of linear forms on $\Hom_G(W_p,\cU')$ is generated by $\{\ord_p^{(\widetilde{t}_i)}, \ |\ i\in I^0\}\bigcup\{\log_p^{(\widetilde{t}_i)}\ |\ i\in I^+\}$. Therefore, the space $\bigwedge^{d^+}_{E_p}\Hom_G(W_p,\cU')^*$ is generated by the set of all elements of the form
$$\nu_{J^0,J^+}=\bigwedge_{j\in J^0}\ord_p^{(\widetilde{t}_j)}\wedge \bigwedge_{j\in J^+} \log_p^{(\widetilde{t}_{j})},$$

where $J^0$ (resp. $J^+$) runs over all subsets of $I^0$ (resp. of $I^+$) such that $(\#J^0)+(\#J^+)=d^+$. In order to show MRS$_\chi$, it is enough to prove that the equality (<ref>) holds after applying $\nu_{J^0,J^+}$ for all such couples of sets $(J^0,J^+)$.

Let $J^0,J^+$ as above. Assume first that $J^+\cap I^0\neq\emptyset$ and let $i\in J^+ \cap I^0$. We claim that applying $\nu_{J^0,J^+}$ to (<ref>) gives $0=0$, so it is true. The left-hand side is indeed zero because $\log_p^{(\widetilde{t}_i)}$ is a linear combination of elements in $\{\log_p^{(t_{p,j})}\ |\ 1\leq j \leq f \}$. To see that the right-hand side also gives zero, it is enough to prove that the map which sends any norm-compatible sequence $(v_n)_{n\geq 1} \varprojlim_n \widehat{\bQ}^\times_{p,n-1}$ to $\log_p(v_1)$ is identically zero. This follows directly from the exactness of the sequence (<ref>).

To complete the proof of (2), we have to prove that the equality obtained after having applied $\nu_{J^0,J^+}$ to (<ref>) holds true, where $J^0$ (resp. $J^+$) is a subset of $I^0$ (resp. of $I^+$) such that $(\#J^0)+(\#J^+)=d^+$ and that $J^+\cap I^0=\emptyset$. Note that $J^0$ and $J^+$ are disjoint, so their union defines an element $\beta\in\cB$. The key observation here is that $\nu_{J^0,J^+}$ coincides in this case with $\nu_{\rho_\beta^+}$ up to multiplication by a unit. Since we have assumed that $\rho^+_\beta$ is admissible and that EZC$_{\rho,\rho^+_\beta}$ holds, it follows immediately from Lemma <ref> that (<ref>) holds true after applying $\nu_{J^0,J^+}$. This completes the proof of (2) and the proof of the theorem.

Let $W_p^+$ be any admissible $p$-stabilization of $W_p$ containing $\HH^0(\Qp,W_p)$. If EZC$_{\rho,\rho^+}$ holds, then MRS$_\chi$ holds as well.

The assumption $W_p^+ \supseteq \HH^0(\Qp,W_p)$ forces the inequality $f\leq d^+$, so we can take $e=0$ in Theorem <ref> (2) and assume that $\{\widetilde{t}_1,\ldots,\widetilde{t}_{d^+}\}$ is a basis of $W_p^+$. Moreover, $\cL(\rho,\rho^+)=1$ because $\HH^0(\Qp,W_p^-)=0$. The conclusion of the corollary then follows from Theorem <ref> (2).

§ EXAMPLES AND APPLICATIONS

§.§ Deligne-critical motives

The complex $L$-functions of motives and the $p$-adic interpolation of their special values are better understood for motives that admit critical points in the sense of Deligne. With the notations of the introduction, the Artin motives which have this property are precisely those which satisfy $d^+=d$ (the even ones) or $d^+=0$ (the odd ones). In particular, all Dirichlet motives fall in this category. There exists nowadays an extensive literature on the construction and on the properties of $p$-adic $L$-functions in this context, and we will simply recall what is needed.

Keep the notations of the introduction and assume that $\rho$ is even, that $\HH^0(\bQ,\rho)=0$ and that $\rho$ is of type $S$ at $p> 2$, which means that the extension $H/\bQ$ cut out by $\rho$ is linearly disjoint to the $\Zp$-cyclotomic extension. Note this last condition is weaker than being unramified at $p$, and if we denote by $\omega$ the Teichmüller character, then the odd representation $\rho\otimes\omega^{-1}$ is still of type $S$. There exists a $p$-adic measure $\theta_\rho^\DR\in\Lambda$ satisfying the following interpolation property:

\begin{equation}\label{eq:interpolation_deligne_ribet}
\forall\ n\geq2,\quad\forall\ \eta\in\widehat{\Gamma}, \qquad \eta\cdot\kappa^n(\theta^\DR_\rho) = L_{\{p\}}(\rho\otimes\eta\omega^{-n},1-n),
\end{equation}

where the subscript $\{p\}$ means that we removed the Euler factor at $p$. The interpolation property is first proven by the work of Deligne-Ribet [27] for monomial representations and includes $n=1$, and it is proven in general by a simple application of Brauer's induction theorem (see [41]). The fact that $\theta^\DR_\rho\in\Lambda$ and not only in $\Frac(\Lambda)$, the so-called “$p$-adic Artin conjecture” follows from <cit.> as a consequence of the classical Iwasawa Main conjecture. The $p$-adic $L$-function attached to $\rho$ is the $p$-adic analytic function given by
$$L_p(\rho,s)=\kappa^{1-s}(\theta^\DR_\rho) \qquad\qquad (s\in\Zp).$$

Note that since $\rho$ is of type S, the above-mentioned Euler factor at $p$ is in fact equal to $1$ at $n=1$, provided that $\eta\neq \mathds{1}$, and the corresponding $L$-value is non-zero. Therefore, the formula (<ref>) still holds when $n=1$ and $\eta\neq \mathds{1}$.

§.§ Even Artin motives

We expand here upon the relation between Conjecture <ref>, the  classical Iwasawa Main Conjecture over totally real fields and “$p$-adic Stark conjectures at $s=1$”, when $d^+=d$. The only choice of $p$-stabilization is then $W_p^+=W_p$, and there is no extra zeros, so $e=0$ and $\cL(\rho,\rho^+)=1$. On the algebraic side, the corresponding Selmer groups $X_\infty(\rho,\rho^+)$ coincides with the one studied by Greenberg in [41]. It is shown in loc. cit. that $X_\infty(\rho,\rho^+)$ is of $\Lambda$-torsion by using the validity of the weak Leopoldt conjecture, and that its characteristic ideal does not depend on the choice of $T_p$. It is also shown that $\char_\Lambda X_\infty(\rho,\rho^+)$ is generated by $\theta^\DR_\rho$ by invoking Wiles' theorem <cit.>. To make the connection with Conjecture <ref>, one needs to compare $\theta^\DR_\rho$ with $\theta_{\rho,\rho^+}$, that is, to understand the values $\eta(\theta^\DR_\rho)$ for $\eta\in\widehat{\Gamma}$. It is precisely the content of “$p$-adic Stark conjectures at $s=1$” attributed to Serre by Tate in <cit.> and we refer the reader to <cit.> for more detail. With the notations of Conjecture 4.9 of loc. cit., it asserts for the representation $\rho\otimes\eta$ that

\begin{equation}\label{eq:conj_stark_p_adique_s=1}
\eta(\theta^\DR_\rho) \overset{?}{=} \Omega_j(\rho\otimes\eta)\cdot j\left(L_{\{p\}}(\rho\otimes\eta,1)\right)=\left\{\begin{array}{lr}\Omega_j(\rho\otimes\eta)\cdot j\left(L(\rho\otimes\eta,1)\right) & \mbox{if }\eta\neq\mathds{1}\\ \Omega_j(\rho)\cdot\cE(\rho,\rho^+)\cdot j\left(L(\rho,1)\right) & \mbox{if }\eta=\mathds{1}. \end{array}\right.
\end{equation}

Moreover, <cit.> implies that the quantity $(-1)^d\cdot \Omega_j(\rho\otimes\eta)$ coincides with the quotient of regulators (<ref>) for any choice of bases $\omega_\infty^+$ and $\omega_p^+$ of $\det\HH^0(\bR,W_p)=\det W_p^+=\det W_p$ such that $\omega_\infty^+=\omega_p^+$. Therefore, under the validity of (<ref>) for all $\eta\in\widehat{\Gamma}$, the $p$-adic measure $\theta_{\rho,\rho^+}$ exists, it satisfies

and the full Conjecture <ref> holds for any choice of $T_p$. As an example, when $\rho$ is a one-dimensional even character whose associated Dirichlet character $\chi$ has prime-to-$p$ conductor $d\neq1$, for all characters $\eta\in\widehat{\Gamma}$ of conductor $p^n$ we have

$$\Omega_j(\rho\otimes\eta) = \frac{\log_p(\varepsilon^{\chi\otimes\eta}_{\cyc})}{-j\left(\log_\infty(\varepsilon^{\chi\otimes\eta}_{\cyc})\right)},$$

where we have put

$$\varepsilon^{\chi\otimes\eta}_{\cyc} = \prod_{a \mod dp^n} \left(e^{\frac{2i\pi a}{dp^n}}-1\right)^{(\chi\otimes\eta)^{-1}(a)} \in \left(\bZ[\mu_{dp^n}]^\times \otimes_\bZ \bQ(\chi\otimes\eta)\right)^{\chi\otimes\eta}. $$

The conjectural equality (<ref>) follows in this case from the well-known formula $L(\chi\otimes\eta,1)=\g(\chi^{-1}\otimes\eta^{-1})^{-1}\cdot \log_\infty(\varepsilon^{\chi\otimes\eta}_{\cyc})$ and its $p$-adic analogue (Leopoldt's formula), see for instance <cit.>.

§.§ Odd Artin motives

We focus here on the connection between Conjecture <ref> and the “$p$-adic Stark conjecture at $s=0$” (that is, the Gross-Stark conjecture) when $d^+=0$. The only choice of $p$-stabilization is then $W_p^+=0$, and both regulators are equal to $1$. The representation $\widetilde{\rho}=\rho^*\otimes\omega$ is even and of type S, and we may consider $\theta=\textrm{Tw}_{-1}(\theta_{\widetilde{\rho}}^{\DR,\iota})$, where $\iota$ is the involution of $\Lambda$ induced by $\gamma\mapsto \gamma^{-1}$ and  Tw$_{-1}$ is the twist by $\kappa^{-1}$. We thus have

for all $s\in\Zp$ and moreover,

$$\eta(\theta)= L((\rho\otimes\eta)^*,0)$$

holds for all non-trivial characters $\eta\in\widehat{\Gamma}$ by the discussion of <ref>. Therefore, one already can conclude to the existence of the $p$-adic measure $\theta_{\rho,\rho^+}$ and to the equality $\theta=\theta'_{\rho,\rho^+}$, where $\theta'_{\rho,\rho^+}$ is the renormalization of $\theta_{\rho,\rho^+}$ appearing in Proposition <ref>. On the algebraic side, a duality theorem for Selmer groups <cit.> and Wiles' theorem for $\widetilde{\rho}$ proves that $\theta$ is a generator of the characteristic ideal of the Selmer group $X_\infty(\rho,\rho^+)$; hence the first part of Conjecture <ref> is valid. Consider now the extra zeros conjecture for $(\rho,\rho^+)$. The number $e=\dim\HH^0(\Qp,W^-)=\dim\HH^0(\Qp,W)$ is nothing but the order of vanishing of the $\{p\}$-truncated Artin $L$-function $L_{\{p\}}(\rho^*,s)$ at $s=0$. The “Weak $p$-adic Gross-Stark Conjecture” for $\rho^*$ as formulated by Gross in <cit.> states that

$$\frac{1}{e!}L_p^{(e)}(\rho^*\otimes\omega,0)\overset{?}{=}R_p(W_p^*)\cdot j\left( \frac{L^*_{\{p\}}(\rho^*,0)}{(-\log(p))^e}\right)=(-1)^e\cdot R_p(W_p^*)\cdot\cE(\rho,\rho^+)\cdot j\left(L(\rho^*,0)\right),$$

where $R_p(W_p^*)$ is Gross's $p$-adic regulator defined in loc. cit., (2.10) and computed with respect to the set of places $\{p,\infty\}$ of $\bQ$. By <cit.>, we have $R_p(W_p^*)=\cL(\rho,\rho^+)$, so EZC$_{\rho,\rho^+}$ is here equivalent to the Weak $p$-adic Gross-Stark Conjecture. It has already been proven true by Dasgupta, Kakde and Ventullo in [23] for monomial representations. For a general $\rho$, it is proven in <cit.> that Gross-Stark's conjecture holds true under Gross' “Order of Vanishing Conjecture” <cit.> for all monomial representations cutting out the same field extension as $\rho$. This last conjecture states that $\ord_{s=0}L_p(\rho^*\otimes\omega,s)=e$, and it is known to be equivalent to the non-vanishing of $R_p(W_p^*)$ by <cit.>.

§.§ Restriction to the cyclotomic line of Katz's $p$-adic $L$-function

It is shown in [14] that the Iwasawa-theoretic properties of Rubin-Stark elements imply $p$-adic Beilinson type formulae for Katz $p$-adic $L$-functions. We follow the notations of loc. cit..

Let $k$ be a CM field of degree $2g$ and let $k^+$ be its maximal totally real subfield. Let $p$ be an odd prime which splits totally in $k^+$. Assume as in <ref> that $\rho$ is induced from a non-trivial character $\chi : \Gal(H/k) \longrightarrow E^\times$ of prime-to-$p$ order and that $T_p=\cO_p[G] \otimes_{\cO_p[\Gal(H/k)]}\cO_p(\chi)$. Hence, $d=2g$ and $d^+=g$. Under the assumption that every prime of $k^+$ above $p$ splits in $k$ (known as Katz's $p$-ordinarity condition), one may pick a $p$-adic CM type of $k$. This amounts to choosing a subset $\Sigma\subseteq S_p(k)$ such that $\Sigma$ and its complex conjugate form a partition of $S_p(k)$. To any $p$-adic CM type $\Sigma$ one can attach a (motivic) $p$-stabilization $(\rho^+_\Sigma,W_{p,\Sigma}^+)$ of $W_p$ by taking the linear span of $\{g\otimes 1\ \big|\ g\in G,\ g^{-1}(v_p)\in \Sigma\}$, where $v_p$ is the place of $k$ defined by $\iota_p$. By Shapiro's lemma, one can show that $X_\infty(\rho,\rho^+)$ is pseudo-isomorphic over $\Lambda$ to the $\chi$-isotypic component of the Galois group $\Gal(M_{\Sigma}/H\cdot \bQ_\infty)$ of the maximal pro-$p$ abelian extension of $H\cdot\bQ_\infty$ which is $\Sigma$-ramified (as in <cit.>). In particular, it is of $\Lambda$-torsion by <cit.>.

Consider now the Galois group $\Gamma_\infty$ of the compositum of all $\Zp$-extensions $k_\infty$ of $k$. The Iwasawa main conjecture for CM fields [49] asserts that

\begin{equation}\label{eq:IMC_CM_fields}
\char_{\hat{\cO}_p[[\Gamma_\infty]]} (\hat{\cO}_p \otimes\Gal(M_{\infty,\Sigma}/H\cdot k_\infty))^\chi\overset{?}{=}(L^{\iota}_{p,\chi,\Sigma}),
\end{equation}

where $\hat{\cO}_p$ is the completion of the maximal unramified extension of $\cO_p$, $M_{\infty,\Sigma}$ is the maximal pro-$p$ abelian extension of $H\cdot k_\infty$ which is $\Sigma$-ramified, and $ L_{p,\chi,\Sigma}$ is Katz's $p$-adic $L$-function for $\chi$ and $\Sigma$ [55, 48]. This latter is a $p$-adic measure which $p$-adically interpolates the algebraic part of critical Hecke $L$-values for $\chi^{-1}$ twisted by certain characters of $\Gamma_\infty$. The $L$-values $L((\chi\otimes\eta)^{-1},0)$ do not belong to its range of interpolation, because they are non-critical. Assuming the equality (<ref>), one can deduce by a descent argument that

$$\char_\Lambda X_\infty(\rho,\rho_\Sigma^+)\overset{?}{=}(L^{\cyc,\iota}_{p,\chi,\Sigma}),$$

where $L^{\cyc}_{p,\chi,\Sigma}=L_{p,\chi,\Sigma}\big|_\Gamma$ is the cyclotomic restriction of $L_{p,\chi,\Sigma}$. More precisely, we apply <cit.> and we claim that $X_\infty(\rho,\rho_\Sigma^+)$ has the same characteristic ideal as the extended Selmer group $\widetilde{\HH}^2_\ff(G_{k,S},\bT_\Gamma,\Delta_\Sigma)$ of loc. cit. as follows. By Shapiro's lemma, the exact sequence (26) of loc. cit. can be written (with the notations of <ref>) as

0 \rar & \coker(\Loc_+^\str) \rar & \widetilde{\HH}^2_\ff(G_{k,S},\bT_\Gamma,\Delta_\Sigma) \rar & \ker\left[\HH^2_{\Iw,\ff,p}(\bQ,\widecheck{T}_p) \longrightarrow \bigoplus_{v\in\Sigma}\HH^2_\Iw(k_v,\widecheck{T}_p)\right] \rar & 0.
\end{tikzcd}
Moreover, the characteristic ideal of the last term of the sequence can be computed as $\cA^e \cdot \char_\Lambda \Sha^2_\Iw(\widecheck{T}_p)$, where $\cA\subset \Lambda$ is the augmentation ideal and  $e=\#\{v|p,\ v\notin\Sigma, \ |\ \chi(v)=1\}=\dim(\Qp,W_{p,\Sigma}^-)$. Our claim then easily follows from (<ref>) and Lemma <ref>.

The (cyclotomic part of the) explicit reciprocity conjecture for Rubin-Stark elements states that

\begin{equation}\label{eq:explicit_rec_conj_BS}
\sC_{\omega_p^+}^\str(\varepsilon_\infty^\chi)\overset{?}{=}L^{\cyc,\iota}_{p,\Sigma},
\end{equation}

where $\sC_{\omega_p^+}^\str$ is the operator introduced in <ref> (see <cit.> and <cit.>).

Assume as above that $p$ is an odd prime which totally splits in a CM field $k$ and that $\rho\simeq \Ind_k^\bQ \chi$ for some non-trivial character $\chi$ of prime-to-$p$ order and conductor which satisfies Conjectures <ref> and <ref>. If (<ref>) and (<ref>) both hold for some $p$-adic CM type $\Sigma$, then IMC$_{\chi}$ and IMC$_{\rho,\rho^+}$ hold for any choice of $p$-stabilization $\rho^+$ of $\rho$.

In particular, when $k$ is an imaginary quadratic field whose class number is prime to $p$, IMC$_\chi$ and IMC$_{\rho,\rho^+}$ are valid for any $p$-stabilization $\rho^+$ of $\rho$.

We know that $\sC_{\omega_p^+}^\str(\varepsilon_\infty^\chi)=\theta_{\rho,\rho_\Sigma^+}'=U\cdot \theta_{\rho,\rho_\Sigma^+}$ for some unit $U\in\Lambda^\times$ by Theorem <ref> and Proposition <ref>. Therefore, IMC$_{\rho,\rho_\Sigma^+}$ is a consequence of (<ref>) and (<ref>). As $X_\infty(\rho,\rho_\Sigma^+)$ is of $\Lambda$-torsion, we deduce from Theorem <ref> (2) that IMC$_{\chi}$ and IMC$_{\rho,\rho^+}$ hold for any choice of $\rho^+$.

When $k$ is an imaginary quadratic field whose class number is prime to $p$, both (<ref>) and (<ref>) hold for any choice of $\Sigma$ by the work of Rubin [72] and Yager [82]. Furthermore, Conjectures <ref> and <ref> hold for $\chi$, as explained in the proof of Proposition <ref> below. So the last claim follows from the first one.

The admissibility of $W_{p,\Sigma}^+$ is equivalent to the $\Sigma$-Leopoldt conjecture which appear for instance in Hida-Tilouine's work. Moreover,  $\cL(\rho,\rho^+_{\Sigma})$ is equal to the cyclotomic $\cL$-invariant defined in <cit.>; accordingly, the implication MRS$_\chi \Longrightarrow$ EZC$_{\rho,\rho^+_\Sigma}$  of Theorem <ref> can be seen as a reformulation of <cit.> in the cyclotomic direction. A similar result was obtained by Betina and Dimitrov for anticyclotomic Katz $p$-adic $L$-functions [3]. Finally, it is worth pointing out that, when $[k:\bQ]=2g>2$, there are $2^g$ distinct $p$-adic CM types, whereas ${\bigwedge}^{d^+}W_p$ has dimension $(2g)!(g!)^{-2}>2^g$, so Conjecture <ref> predicts in particular the existence of more $p$-adic $L$-functions for $\chi$ than the ones constructed by Katz.

§.§ Two-dimensional odd monomial representations

Fix an imaginary quadratic field $k$ and an odd prime $p$ which is unramified in $k$. The prime $p$ is allowed to be split or inert in the following proposition.

Let $\chi:G_k\longrightarrow E^\times$ be a non-trivial character of prime-to-$p$ order and conductor, and consider $\rho=\Ind_k^\bQ \chi$. Then EX$_{\rho,\rho^+}$ and EZC$_{\rho,\rho^+}$ are valid for all $p$-stabilizations $\rho^+$ of $\rho$.

We first recall why Conjectures <ref> and <ref> hold for $\chi$. The Rubin-Stark conjecture over $\bZ$ for abelian extensions of $k$ in the rank $1$ scenario is a consequence of Stark's work (see, for instance, <cit.>). In particular, the statements concerning $\varepsilon_n^\chi$ for $n\geq 1$ in Conjectures <ref> and <ref> hold. One can draw the same conclusion for $\xi^\chi$ defined in (<ref>) but a priori not for $u^\chi$, as we may have $d^++f=1+f>1$.

Let $L/k$ be the abelian extension cut out by $\chi$ and let $g=[L:k]$. By <cit.>, Rubin-Stark's conjecture over $\bZ[\frac{1}{g}]$ for $u^\chi$ follows from that for $\xi^\chi$ and from a Gras-type conjecture, which is proven in <cit.> (the missing case $(k,p)=(\bQ(\sqrt{-3}),3)$ where $p|\#\cO_k^\times$ is ruled out by our assumptions). Since we assumed $p\nmid g$, we may infer that the statements for $u^\chi$ in Conjectures <ref> and <ref> hold as well.

We can now apply Theorem <ref> and deduce that EX$_{\rho,\rho^+}$ holds for every $\rho^+$. EZC$_{\rho,\rho^+}$ follows likewise from MRS$_\chi$, which is proven in <cit.> (and whose proof is built on earlier works [12, 7]).

We now give a concrete description of the $p$-stabilizations, $p$-adic regulators and $\cL$-invariants attached to $\rho$ when $\rho\simeq \Ind_k^\bQ \chi$ for some non-trivial character $\chi$ which we fix once and for all. We don't necessarily assume that $\chi$ has order prime to $p$. Write $p\cO_k$ as $\gp$ when $p$ remains inert in $k$, and as $\gp\ob{\gp}$ (where $\gp$ is the prime determined by the embedding $\iota_p$) when $p$ splits in $k$. Note that $\sigma_p\in G_k$ if and only if $p$ splits in $k$. We denote by $\tau$ both the complex conjugation and the non-trivial element of $\Gal(k/\bQ)$, and we let $\chi^\tau$ be the character $\chi(\tau \cdot \tau)$ defined over $k$. We assume for simplicity that $\chi\neq\chi^\tau$, so that $\rho$ is irreducible. This assumption is not restrictive, as the case $\rho=\chi^+\oplus\chi^-$ (for some Dirichlet characters $\chi^\pm$) can be derived from <ref>-<ref> and the Artin formalism (Remark <ref>).

There exists a unique basis $\{w_1,w_2\}$ of $W$ (up to scaling) in which $\rho$ is of the form:

$$\rho(g)\sim \begin{pmatrix}\chi(g) & 0 \\ 0 & \chi^\tau(g)\end{pmatrix},\qquad \rho(\tau)\sim  \begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix},$$

where $g\in G_k$. The eigenvalues of $\sigma_p$ are $\chi(\gp)$ and $\chi(\ob{\gp})$ in the split case, and $\pm \sqrt{\chi(\gp)}$ in the inert case respectively. A natural eigenbasis for $\sigma_p$ is $\{w_1,w_2\}$ when $p$ splits in $k$, and $\{-aw_1+ w_2,aw_1+w_2\}$ (where $a=\pm \sqrt{\chi(\gp)}\chi(\sigma_p^{-1}\tau)$) when $p$ is inert in $k$. Therefore, there are exactly two (motivic) choices of $p$-stabilization $\rho^+$ of $\rho$, except in the split case when $\chi(\gp)=\chi(\ob{\gp})$, where any line is a $p$-stabilization.

Extending some of the ideas present in <cit.>, we now express $p$-adic regulators and $\cL$-invariants in terms of Minkowski units. Given a character $\eta\in\widehat{\Gamma}$, let $H_\eta/\bQ$ be the Galois extension cut out by $\rho\otimes\eta$, let $E_\eta$ be its coefficient field and put $C_\eta=\Gal(H_\eta/k)$. We drop the subscript in $H_\eta,E_\eta,C_\eta$ when $\eta=\mathds{1}$. By Minkowski's theorem, there exists a unit $u\in \cO_{H_\eta}^\times$ such that $\{g(u)\ |\ 1\neq g\in C_\eta\}$ generates a sublattice of $\cO_{H_\eta}^\times$ of finite index. Moreover, one can impose $u$ to be fixed by $\tau$, so that the only relation between elements of $\{g(u)\ |\ g\in C_\eta\}$ is given by the kernel of the norm map of the extension ${H_\eta/k}$.

Shapiro's lemma provide us with an isomorphism

\begin{equation}\label{eq:isom_shapiro_CM}
\begin{array}{rcl}
\HH^1_\ff((\rho\otimes\eta)^*(1)) & \overset{\simeq}{\longrightarrow} & (E_\eta\otimes \cO_{H_\eta}^\times)^{\chi\eta} \\
\psi & \mapsto & \psi(w_1),
\end{array}
\end{equation}

where  $({E_\eta}\otimes \cO_{H_\eta}^\times)^{\chi\eta}$ is the $\chi\eta$-isotypic component of the group of units of $H_\eta$. This latter space is generated by the element (written in additive notation)

\begin{equation}\label{eq:u_chi}
u_{\chi\eta}:= \sum_{g\in C_\eta} \chi\eta(g) \otimes g^{-1}(u),
\end{equation}

and its complex conjugate $\tau(u_{\chi\eta})$ generates $(E_\eta\otimes \cO_{H_\eta}^\times)^{\chi^\tau\eta}$.

We let $\omega_{\ff,\eta}$ be the basis of $\HH^1_\ff((\rho\otimes\eta)^*(1))$ corresponding to $u_{\chi\eta}$ under (<ref>). Given a $p$-stabilization $\rho^+$ generated by an element $\omega_p^+=w_1+sw_2$ (where $s\in \bP^1(\ob{\bQ}_p)$), we then have

As $\rho$ is assumed to be irreducible, we have $\chi\eta\neq\chi^\tau\eta$, so $\log_p(u_{\chi\eta})$ and $\log_p(\tau(u_{\chi\eta}))$ are $\ob{\bQ}$-linearly independent by the Baker-Brumer theorem. In particular, $\rho^+$ is always $\eta$-admissible, except when $\chi(\gp)=\chi(\ob{\gp})$ and $s$ equals

\[\sS(\chi\eta):=-\frac{\log_p(u_{\chi\eta})}{\log_p(\tau(u_{\chi\eta}))} \in \ob{\bQ}_p\setminus\ob{\bQ}. \]

Let us describe the $\cL$-invariant attached to the admissible $p$-stabilization $\rho^+$ given by the line $\ob{\bQ}_p\cdot w_1$ in the split case, and by $\ob{\bQ}_p\cdot (-aw_1+w_2)$ in the inert case. Extra zeros for $\rho^+$ only arise when $\chi(\ob{\gp})=1$ in the split case (resp. when $\chi({\gp})=1$ and $a=\chi(\sigma_p^{-1}\tau)$ in the inert case), which we assume. Denote by $v_0$ the place of $H$ determined by $\iota_p$, let $y_0$ be any element of $\cO_H[\tfrac{1}{v_0}]^\times$ with $\ord_p(y_0)\neq 0$ and put

\begin{equation}\label{eq:u_p_chi}
u_{\gp,\chi}= \sum_{g\in C} \chi(g) \otimes g^{-1}(y_0),  \quad \textrm{and }\  u_{\ob{\gp},\chi}=\sum_{g\in C} \chi(g) \otimes g^{-1}\cdot\tau(y_0)\quad \textrm{ (only when $p$ splits in $k$).}
\end{equation}

Under our hypotheses, (<ref>) defines a non-trivial $\gp$-unit (resp. a $\ob{\gp}$-unit) in $H$. Moreover, a simple computation based on Lemma <ref> gives

\begin{align*}
\ord_p(\tau(u_{\ob{\gp},\chi}))\cdot\Reg_{{\omega}^+_{p}}(\rho)\cdot\cL(\rho,\rho^+) =\det\begin{pmatrix}\log_p(u_\chi) & \log_p(u_{\ob{\gp},\chi}) \\ \log_p(\tau(u_{\chi})) & \log_p(\tau(u_{\ob{\gp},\chi}))\end{pmatrix} \qquad & \textrm{when $p$ splits in $k$,} \\ \ord_p(u_{\gp,\chi})\cdot\Reg_{{\omega}^+_{p}}(\rho)\cdot\cL(\rho,\rho^+) =-2\cdot \det\begin{pmatrix}\log_p(u_\chi) & \log_p(u_{{\gp},\chi}) \\ \log_p(\tau(u_{\chi})) & \log_p(\tau(u_{{\gp},\chi}))\end{pmatrix} \qquad & \textrm{when $p$ in inert in $k$.}
\end{align*}

Note that replacing $\rho^+$ by $\ob{\bQ}_p\cdot w_2$ in the split case and assuming $\chi(\gp)=1$ would give the same formula as above, with a minus sign and with $u_{\ob{\gp},\chi}$ replaced by $u_{{\gp},\chi}$ in the determinant.

We record in the next proposition the interpolation formulae satisfied by the two $p$-adic $L$-functions attached to $\chi$ when $p$ is inert in $k$.

Let $\chi:G_k\longrightarrow E^\times$ be a character of prime-to-$p$ order and conductor of an imaginary quadratic field $k$ in which $p$ is inert. Assume $\chi\neq \chi^\tau$. Let $\alpha\in E^\times$ be a square root of $\chi(\gp)$ and put $a=\alpha\cdot\chi(\sigma_p^{-1}\tau)$. There exists a measure $\theta^\alpha_\chi\in\Lambda$ with the following interpolation property. For all $\eta\in\widehat{\Gamma}$ of conductor $p^n$, we have

\[\eta(\theta^\alpha_\chi)= \left(1+\alpha\frac{\eta(p)}{p}\right)\left(1-\frac{\eta(p)}{\alpha}\right)\frac{(-\alpha)^n}{\g(\eta^{-1})}(-a\log_p(u_{\chi\eta})+\log_p(\tau(u_{\chi\eta}))\cdot j\left(\frac{L'((\chi\eta)^{-1},0)}{\log_\infty(u_{\chi\eta}\cdot \tau(u_{\chi\eta}))}\right), \]

where $u_{\chi\eta}$ is defined as in (<ref>). Moreover, if $\alpha=1$, then $\mathds{1}(\theta^\alpha_\chi)=0$ and

\[\lim_{s\rightarrow 0}\frac{\kappa^s(\theta^\alpha_\chi)}{s}=\left(1+\frac{1}{p}\right)\frac{2}{\ord_p(u_{\gp,\chi})}\det\begin{pmatrix}\log_p(u_\chi) & \log_p(u_{{\gp},\chi}) \\ \log_p(\tau(u_{\chi})) & \log_p(\tau(u_{{\gp},\chi}))\end{pmatrix}\cdot j\left(\frac{L'(\chi^{-1},0)}{\log_\infty(u_{\chi}\cdot \tau(u_{\chi}))}\right), \]

where $u_{{\gp},\chi}$ is the $\gp$-unit defined in (<ref>).

We apply Proposition <ref> to $\rho^+$ given by the line generated by $-aw_1+w_2$ and we put $\theta^\alpha_\chi=\theta'_{\rho,\rho^+}$ (cf. Proposition <ref>). Then $\sigma_p$ acts on $W_p^+$ by $-\alpha$ and on $W_p^-$ by $\alpha$. In particular, we have $e=0$ and $\cE(\rho,\rho^+)=(1+\alpha p^{-1})(1-\alpha^{-1})$, unless $\alpha=1$, in which case $e=1$ and $\cE(\rho,\rho^+)=(1+\alpha p^{-1})$. To compute $\Reg_{\omega_{\infty}^+}(\rho\otimes\eta)$ we use the basis $w_1+w_2$ of $\HH^0(\bR,W)$, and the basis $u_{\chi\eta}$ of $(E_\eta\otimes \cO_{H_\eta}^\times)^{\chi\eta}$, which yields $\Reg_{\omega_{\infty}^+}(\rho\otimes\eta)=\log_\infty(u_{\chi\eta}\cdot \tau(u_{\chi\eta}))$. The other terms in the interpolation formula for $\theta^\alpha_\chi$ have already been computed.

Let $\rho=\Ind_k^\bQ \chi$ with $\chi\neq\mathds{1}$. There exists at least one admissible $p$-stabilization $\rho^+$ of $\rho$ such that $\cL(\rho,\rho^+)\neq 0$. Moreover, if $\chi$ is of prime-to-$p$ order, then the weak exceptional zero conjecture wEZC$_\chi$ (Conjecture <ref>) for $\chi$ is true.

The second statement follows from the first one and from Theorem <ref> (2). Assume first that $\chi=\chi^\tau$. Then $\rho$ is the sum of an even and an odd Dirichlet character $\chi^+$ and $\chi^-$, and $\rho^+=\chi^+$ defines an admissible $p$-stabilization, the $p$-adic regulator essentially being $\log_p(\varepsilon_{\cyc}^{\chi^+})$ with the notations of <ref>. It then follows from the $p$-adic Artin formalism (Remark <ref>) and from <ref> that $\cL(\rho,\rho^+)$ is Gross's regulator attached to $\chi^-$, so it does not vanish.

Assume now that $\chi\neq \chi^\tau$. If $f=\dim \HH^0(\Qp,\rho)=\#\{\mathfrak{q}|p\ | \ \chi(\mathfrak{q})=1\}\leq 1$, then there is always a choice of admissible $\rho^+$ for which $e=\dim \HH^0(\Qp,\rho^-)=0$. For such a $\rho^+$, there is no extra zero and $\cL(\rho,\rho^+)=1$. Hence, we may assume that $f=2$, i.e., $p$ splits in $k$ and $\chi(\gp)=\chi(\ob{\gp})=1$. Since $H$ is the Galois closure of the field cut out by $\chi$, $p$ must split completely in $H$. We show that at least one of the $p$-stabilizations $\rho^+=\ob{\bQ}_p\cdot w_1$ or $\rho^+=\ob{\bQ}_p\cdot w_2$ has a non-zero $\cL$-invariant. With the notations introduced above, it is enough to show that the matrix

$$ M=\begin{pmatrix} \log_p(u_\chi) & \log_p(u_{\gp,\chi}) & \log_p(u_{\ob{\gp},\chi}) \\ \log_p(\tau(u_{\chi})) & \log_p(\tau(u_{{\gp},\chi})) & \log_p(\tau(u_{\ob{\gp},\chi})) \end{pmatrix}$$

has rank 2. This follows from a straightforward adaptation of the proof of <cit.> to the cyclotomic setting, which we explain for the sake of completeness.

By Roy's Six Exponentials Theorem <cit.>, we only have to show that $M$ has $\bQ$-linearly independent rows and $\bQ$-linearly independent columns. The two rows are linearly independent because $\sS(\chi)$ is transcendental. Since $\log_p(\textrm{N}_{H/k}(u))=\log_p(\textrm{N}_{H/\bQ}(y_0))=0$, any linear equation $a \cdot \log_p(u_\chi) + b\cdot \log_p(u_{{\gp},\chi}) +c \cdot \log_p(u_{\ob{\gp},\chi})$ with $a,b,c\in \bZ$ can be written as

$$\sum_{1\neq g\in C}(\chi(g)-1)(a\cdot \log_p(g^{-1}(u))+b\cdot \log_p(y_0)) + \sum_{g\in C}(c\cdot\chi(g)-b)\log_p(g^{-1}\tau(y_0))=0.$$

In loc. cit., the same equation appears, with $+b$ instead of the $-b$ in the second sum. We then may repeat verbatim their arguments to show that $a=b=c=0$, as wanted.

Betina and Dimitrov proved in <cit.> an analogue of Proposition <ref> where $\cL(\rho,\rho^+)$ is replaced by an anticyclotomic $\cL$-invariant.

The Gross-Kuz'min conjecture holds for abelian extensions $L$ of $k$ and all odd primes $p$, that is, the module of $\Gamma$-coinvariants of

$$A'_\infty=\varprojlim_{L\subseteq M \subseteq L\cdot \bQ_\infty}A'(M)$$

is finite, where $A'(M)$ is the $p$-Sylow of the $p$-split class group of $M$ and  the transition maps are the norm maps.

Let $L$ be an abelian extension of $k$ of Galois group $C$. By <cit.>, we may replace $L$ by its Galois closure over $\bQ$ without loss of generality. Then $A'_\infty$ is a module over $\Zp[[\Gamma]]$ which carries a linear action of $C$, so it is enough to show that the $\chi$-part $A'_\infty$ has finite $\Gamma$-coinvariants for all multiplicative characters $\chi$ of $C$. When $\chi=\mathds{1}$, this follows from the validity of the Gross-Kuz'min conjecture for $k$ which is implicitly proven in [37]. When $\chi\neq \mathds{1}$ and when $p$ is unramified in $L$, this is precisely the content of Conjecture <ref> for $\rho=\Ind_k^\bQ \chi$, which is valid by Theorem <ref> (2) and by Proposition <ref>.

We now indicate how to proceed when $p$ is ramified in $L$. This case is easier than the previous one because of the absence of trivial zeros. Let $\{1\}\neq G_p \subseteq G$ be the decomposition group at the place above $p$ defined by $\iota_p$. By <cit.>, it is enough to prove the Gross-Kuz'min conjecture for a well-chosen subfield $L'$ of $L$ such that all the $p$-adic primes of $L'$ are non-split in $L$. In particular, we may reduce the proof to the case where $G_p$ is abelian. Indeed, if there is a unique $p$-adic place in $k$, then it suffices to consider $L'=L^{C\cap G_p}$. Take a faithful ($2$-dimensional) $p$-adic representation $(\rho,W_p)$ of $G$ and fix $T_p\subseteq W_p$ a $G$-stable lattice. Since $G_p\neq \{1\}$ is abelian, we can choose a $G_p$-stable line $W_p^+$ of $W_p$ such that $G_p$ acts non-trivially on the quotient $W_p^-=W_p/W_p^+$ and we can form a Selmer group $\Sel^\str_n(\rho,\rho^+)$ for any $n\geq 0$ as in Definition <ref>. Since $\HH^0(\Qp,W_p^-)=0$, the module $\HH^1(\Gamma,\HH^0(\bQ_{p,\infty},D_p^-))$ is finite, and it is not hard to see that the restriction map $\Sel^\str_0(\rho,\rho^+) \longrightarrow \Sel^\str_\infty(\rho,\rho^+)^\Gamma$ has a finite kernel and cokernel (see for instance the argument in <cit.>). On the other hand, $\Sel^\str_0(\rho,\rho^+)$ is finite by <cit.>, so the same conclusion holds for $\Sel^\str_\infty(\rho,\rho^+)^\Gamma$. Thus $\Hom_G(T_p,A_\infty')_\Gamma$ is finite by Lemma <ref> and Diagram (<ref>).

§.§ Weight one modular forms

We consider in this paragraph $E$-valued Artin representations $(\rho,W)$ that are two-dimensional and irreducible with $d^+=1$. The proof by Khare and Wintenberger of Serre's modularity conjecture implies that $\bC \otimes_E W$ is the Deligne-Serre representation of a cuspidal newform $f(q)=\sum_{n\geq 1}a_nq^n\in E[[q]]$ of weight 1 <cit.>. It has level equal to the Artin conductor $N$ of $\rho$, and for all primes $\ell \nmid N$, we have

\[ \textrm{tr}(\rho(\sigma_\ell))=a_\ell, \]

where $\sigma_\ell$ is the Frobenius substitution at $\ell$ (see [28]). Some aspects of Iwasawa theory for $f$ and an odd prime $p$ were studied in [62] and [44] under the hypothesis that $p\nmid N$ (i.e., $\rho$ is unramified at $p$) and that $f$ is regular at $p$. The last assumption means that

\[\alpha\neq \beta,\]

where $\alpha$ and $\beta$ are the eigenvalues of $\sigma_p$. We propose in this paragraph a new conjectural description of the $p$-adic $L$-function of $f$ that is considered in loc. cit.. We therefore assume that $p\nmid N$ and $\alpha\neq \beta$.

As $\sigma_p$ has distinct eigenvalues, there exist exactly two choices of $p$-stabilizations of $\rho$, namely the two lines $W_\alpha=W_p^{\sigma_p=\alpha}$ and $W_\beta=W_p^{\sigma_p=\beta}$. Both $p$-stabilizations are motivic, hence $\eta$-admissible for all $\eta\in\widehat{\Gamma}$ by Lemma <ref>.

Let us consider $W_p^+=W_\beta$ (so $W_p^-\simeq W_\alpha$), that we let correspond to the $p$-stabilization $f_\alpha(q)=f(q)-\beta f(q^p)$ of $f$. Then $f_\alpha$ is a normalized eigen cuspform of level $Np$ sharing the same eigenvalues with $f$ under the action of Hecke operators of prime-to-$p$ level, and moreover, $U_p f_\alpha = \alpha f_\alpha$. A theorem of Hida and Wiles asserts that $f_\alpha$ belongs to a primitive Hida family $\cF$ of tame level $N$. The uniqueness of $\cF$ up to Galois conjugacy is proved by Bellaïche and Dimitrov in [2]. The existence of a local parameterization of $\cF$ around $f_\alpha$ proven in <cit.> yields, in particular, the following result.

There exists a finite flat extension $\cO$ of $\Zp$ containing the Fourier coefficients of $f_\alpha$ and, for any integer $m>>0$, a $p$-ordinary newform $g_m(q)=\sum_{n\geq 1}a_n(g_m)\in\cO[[q]]$ of weight $(p-1)p^m+1$ and level $N$, such that

\[\lim_{m\to +\infty} g_{m,\alpha_m}(q) = f_\alpha(q),\]

where $g_{m,\alpha_m}(q)$ is the unique $p$-ordinary $p$-stabilization of $g_m$. In particular, for all integers $n\geq 1$ coprime with $p$, we have  $\lim_{m\to +\infty} a_n(g_m) = a_n$ in $\cO$.

Assume henceforth that $p$ does not divide the order of the finite group $\rho(G_\bQ)$. The residual representation $\ob{\rho}$ is then irreducible and $p$-distiguished, and $W_p$ has a unique Galois-stable $\cO$-lattice $T_p$ (up to homothety). One can attach to $\cF$ a two-variable $p$-adic $L$-function à la Mazur-Kitagawa, and the choice of Shimura's periods for each member of the family $\cF$ can be made in a canonical way by the results of Emerton, Pollack and Weston (see <cit.>). We obtain, for all integers $m>>0$, a complex period $\Omega_m^+\in\bC^\times$ and a $\cO$-valued measure $\theta_m$ on $\Gamma$ such that, for all characters $\eta \in\widehat{\Gamma}$ of conductor $p^n$,

\begin{equation}\label{eq:formule_theta_m}
\eta(\theta_m)=\left(\frac{p}{\alpha_m}\right)^{n} \left(1-\frac{\eta(p)}{\alpha_m}\right)\left(1-\frac{\eta(p)\beta_m}{p}\right)\cdot j\left(\frac{L(g_m\otimes \eta,1)}{\g(\eta)(-2i\pi)\Omega_m^+}\right),
\end{equation}

where $\alpha_m$ and $\beta_m$ are the roots of the $p$-th Hecke polynomial of $g_m$ with $\alpha_m\in\cO^\times$ (see <cit.>).

The construction of Emerton-Pollack-Weston also provides us with a $p$-adic measure $\theta_{f,\alpha}\in\Lambda=\cO[[\Gamma]]$ which is defined as the specialization at $f_\alpha$ of the $p$-adic measure attached to $\cF$. Changing the choice of the canonical periods in $\cF$ results in multiplying $\theta_{f,\alpha}$ by a unit in $\cO$. By <cit.>, we have

\begin{equation}\label{eq:lim_theta_m}
\lim_{m\to +\infty}\theta_m=\theta_{f,\alpha}
\end{equation}

in $\Lambda$ (endowed with its topology of local ring).

Let $\omega_p^+$ (resp. $\omega_{\infty}^+$) be a $T_p$-optimal basis of $W_p^+$ (resp. of $W^{\sigma_\infty=1}$) and assume that EX\(_{\rho,\rho^+}\) and EZC\(_{\rho,\rho^+}\) hold for the $p$-stabilization $(\rho^+,W_p^+)$ given by $W_p^+=W_\beta$ as above. For all characters $\eta \in\widehat{\Gamma}$ of conductor $p^n$, we have

\begin{equation}\label{eq:formule_theta_f_alpha}
\eta(\theta_{\rho,\rho^+})=\left(\frac{p}{\alpha}\right)^{n} \left(1-\frac{\eta(p)}{\alpha}\right)\left(1-\frac{\eta(p)\beta}{p}\right) \Reg_{\omega_p^+}(\rho\otimes\eta)\cdot j\left(\frac{L(f\otimes \eta,1)}{\g(\eta)(-2i\pi)\Reg_{\omega_\infty^+}(\rho\otimes\eta)}\right).
\end{equation}

Notice first that $d^+=d^-=1$ and $\sigma_p$ acts by multiplication by $\alpha$ on $W_p^-$, so $\det(\rho^-)(\sigma_p)=\alpha$. Let $\eta\in\widehat{\Gamma}$. As $\eta$ is even, we have $\tau(\eta)=\g(\eta^{-1})=p^n\g(\eta)^{-1}$ by Lemma <ref> (1). The equality (<ref>) then follows from Lemma <ref> if $\eta\neq \mathds{1}$, as $\eta(p)=0$. In the case where $\eta=\mathds{1}$, (<ref>) follows from the same lemma, after noticing that $e=\dim \HH^0(\Qp,W_p^-)=0$ if and only if $\alpha\neq 1$, in which case we obtain $\cL(\rho,\rho^+)=1$ and $\cE(\rho,\rho^+)=\left(1-\frac{1}{\alpha}\right)\left(1-\frac{\beta}{p}\right)$ from the definition.

The interpolation formulae (<ref>) and (<ref>) are very similar, except that there is an extra $p$-adic regulator in the expression of $\eta(\theta_{\rho,\rho^+})$. This discrepancy comes from the fact that the Selmer groups $\HH^1_\ff(g_m\otimes \eta)$ and  $\HH^1_\ff((g_m\otimes \eta)^*(1))$ of the motive attached to $g_m\otimes \eta$ and its dual vanish by the work of Kato (<cit.>), unlike with $\HH^1_\ff((\rho\otimes \eta)^*(1))$.

As $\lim_{m\to +\infty} \eta(\theta_m)=\eta(\theta_{f,\alpha})$ by (<ref>), it is tempting to raise the following question.

Under the running assumptions $p\nmid N\cdot(\sharp \rho(G_{\bQ}))$ and $\alpha\neq \beta$, is it true that

\begin{equation}\label{eq:question}
\theta_{f,\alpha}=u\cdot \theta_{\rho,\rho^+}
\end{equation}

for some unit $u\in \cO^\times$?

In other terms, the construction of $\theta_{f,\alpha}$ yields a potential definition of $\theta_{\rho,\rho^+}$. Note that Question <ref> makes sense. Indeed, both $\theta_{f,\alpha}$ and $\theta_{\rho,\rho^+}$ are well-defined up to a unit in $\cO^\times$, as they depend on the choice of $p$-normalized periods.

Assume Conjecture <ref> for $(\rho,\rho^+)$. There exists an element $U \in \Lambda \otimes \Qp$ such that

\[\theta_{f,\alpha}=U\cdot \theta_{\rho,\rho^+}.\]

We know from <cit.> that $\theta_{f,\alpha}$ belongs to $\char_{\Lambda\otimes \Qp} X_\infty(\rho,\rho^+) \otimes \Qp$. The latter is generated by $\theta_{\rho,\rho^+}$ over $\Lambda \otimes \Qp$ under IMC$_{\rho,\rho^+}$. Therefore, there exists $U \in \Lambda \otimes \Qp$ such that \(\theta_{f,\alpha}=U\cdot \theta_{\rho,\rho^+}\), as wanted.

Assuming that $f$ is a dihedral form, Ferrara made a conjecture in [32] which is equivalent to a weak version of (<ref>) where $u$ is taken in $\Frac(\cO)^\times$. When $f$ has CM by an imaginary quadratic field $k$ in which $p$ splits, he compared $\theta_m$ and $\theta_{f,\alpha}$ with Katz's $p$-adic $L$-function and deduced from a limit argument that his conjecture holds. In this special case, Conjecture <ref> holds by Propositions <ref> and <ref> under the technical assumption $p\nmid h_k$.

§.§ Adjoint of a weight one modular form

We explore the connection between the “variant of Gross-Stark conjecture” formulated by Darmon, Lauder and Rotger in [25] and EZC$_{\rho,\rho^+}$ for $\rho=\rho_g\otimes \rho_h$, where $\rho_g,\rho_h$ are the Deligne-Serre representations attached to weight one newforms $g,h$ (see <ref>). We restrict ourselves to the adjoint case, i.e. $h$ is the dual form of $g$, in which Darmon-Lauder-Rotger's conjecture is proven by Rivero and Rotger in [70].  Letting $\ad \rho_g$ be the traceless adjoint of $\rho_g$, we have in this case

\[\rho_g \otimes \rho_h \simeq \rho_g \otimes \rho_g^* \simeq \ad \rho_g \oplus \mathds{1}\]

as $G_{\bQ}$-representations.

Consider a cuspidal newform $g$ of weight one with coefficients in $\cO_p$ and let $\rho=\ad \rho_g$. Then $\rho$ does not contain the trivial representation and $(d,d^+)=(3,1)$. Assume that $\rho_g$ is unramified at $p$, residually irreducible, and $\alpha\neq \pm\beta$ modulo the maximal ideal of $\cO_p$, where $\alpha$ and $\beta$ are the roots of the $p$-th Hecke polynomial of $g$. Assume also that $g$ does not have real multiplication by a real quadratic field in which $p$ splits.

The Frobenius substitution $\sigma_p$ at $p$ acts on $W$ with distinct eigenvalues $1,\alpha/\beta,\beta/\alpha$, and there are accordingly three choices $W_1,W_{\alpha/\beta},W_{\beta/\alpha}$ of $p$-stabilizations of $W_p$. Note that these three lines are motivic, hence $\eta$-admissible for all $\eta\in\widehat{\Gamma}$ by Lemma <ref>.

The theorem of Rivero-Rotger for $\ad (g_\alpha)$ can be formulated in terms of the $\cL$-invariant introduced in <ref> as follows.

Take $W_p^+$ to be $E_p \otimes_E W_{\beta/\alpha}$. Then $e=\dim \HH^0(\Qp,W_p^-)=1$ and

\begin{equation}\label{eq:rivero_rotger}
I_p(g_\alpha)=\cL(\rho,\rho^+) \quad \mod E^\times,
\end{equation}

where $I_p(g_\alpha)\in E_p$ is the $p$-adic period attached to $g_\alpha$ (see <cit.> for its definition). Moreover, for any $E$-basis $\omega_p^+$ of the line $W_{\beta/\alpha}$, we have

\begin{equation}\label{eq:gross-stark_unit}
\Reg_{{\omega}^+_{p}}(\rho)= \log_p(u_{g_\alpha}) \mod E^\times,
\end{equation}

where $u_{g_\alpha}\in E\otimes \cO_H^\times$ is the Gross-Stark unit appearing in the Elliptic Stark Conjecture and its variants [24, 25].

As $\alpha/\beta$ and $\beta/\alpha$ are $\neq 1$, one has $\HH^0(\Qp,W_p^{-})=E_p\otimes W_1$ which is a line, so \(e=1\). Consider the matrices $A^\pm,B^\pm,O^-$ defined in (<ref>) using motivic bases $\{t_1^+,t_1^-\}$ and $\{\psi_1,\psi'_1\}$. That is, $t_1^+\in W_{\beta/\alpha}\subset W_{p}^+$, $t_1^-\in W_1$ and $\psi_1,\psi'_1\in \Hom_{G_{\bQ}}(W,U'_0)$ where $U'_0=E \otimes \cO_H[\tfrac{1}{p}]^\times$.

<cit.> states that

$$I_p(g_\alpha)=\det \begin{pmatrix}A^+ & B^+ \\ A^- & B^- \end{pmatrix} (\det A^+)^{-1} \mod E^\times.$$

As $\det(O^-)=\ord_p(\psi_1'(t_1^-))\in E^\times$, we conclude from Lemma <ref> that (<ref>) holds.

The $p$-adic Stark regulator $\Reg_{{\omega}^+_{p}}(\rho)$, computed with respect to the bases $\omega_{p}^+=t_1^+$ and $\omega_{\ff}=\psi_1$, is equal to $\det A^+=\log_p(\psi_1(t_1^+))$. Recall that $u_{g_\alpha}$ is, by definition, any generator of the $E$-line of $\psi_1(W)\subset E\otimes \cO_H^\times$ on which $\sigma_p$ acts with eigenvalue $\beta/\alpha$. As $\psi_1(t_1)$ belongs to the same $E$-line and is non-trivial, their $p$-adic logarithms must coincide modulo $E^\times$.

The $p$-adic iterated integral of (<ref>) can be recast as the leading term of a certain $p$-adic $L$-function as we now recall.

Consider as in <ref> the unique primitive Hida family  $\cG$ passing through $g_\alpha$. Bellaiche-Dimitrov's theorem [2] implies that $\cG$ can be parameterized by the weight variable $k$ around $g_\alpha$. Let $L^\HH_p(\ad(\cG),k,s)$ be Hida's two-variable $p$-adic $L$-function attached to $\ad(\cG)$ in [46] and put $L^\HH_p(\ad(g_\alpha),s)=L^\HH_p(\ad(\cG),k,s)_{\big|k=1}$.
By <cit.> and <cit.>, the quantity $I_p(g_\alpha)$ can be interpreted as the value at $s=1$ of the first derivative of $L^\HH_p(\ad(g_\alpha),s)$. Therefore, Proposition <ref> reads:

\begin{equation}\label{eq:derivative_hida}
(L^\HH_p)'(\ad(g_\alpha),1)= \cL(\rho,\rho^+)\mod E^\times.
\end{equation}

EZC$_{\rho,\rho^+}$ gives a similar formula for the $p$-adic $L$-function defined by $L_p(\rho,\rho^+,s)=\kappa^{1-s}(\theta'_{\rho,\rho^+})$, where $\theta'_{\rho,\rho^+}$ is computed with respect to a basis $\omega_p^+$ of the $E$-line $W_{\beta/\alpha}\subseteq W_p^+$, namely:

\begin{equation}\label{eq:derivative_motivic}
L'_p(\rho,\rho^+,1)= \Reg_{{\omega}^+_{p}}(\rho)\cdot\cL(\rho,\rho^+)=\log_p (u_{g_\alpha})\cdot\cL(\rho,\rho^+) \mod E^\times.
\end{equation}

Note that $\log_p (u_{g_\alpha})$ should be transcendental according to the weak $p$-adic Schanuel conjecture (Conjecture <ref>). In particular, Hida's construction does not provide the right candidate for the $p$-adic $L$-function of $(\rho,\rho^+)$. A similar observation for higher weight specializations of $L^\HH_p(\ad(\cG),k,s)$ is already raised by Hida in his monograph [47] (see also <cit.>). Given a weight $k\geq 2$ specialization $g_k$ of $\cG$, $L^\HH_p(\ad(\cG),k,s)$ interpolates the critical $L$-values of $g_k$ divided by periods of automorphic nature which do not match with the motivic periods given for instance by the conjectures of Coates and Perrin-Riou ([21]). Further, Hida related in [45] the quotient of these periods with the size of the congruence module of $g_k$, and it was proposed in [47] and in <cit.> that, up to fudge factors arising from primitivity issues, his $p$-adic $L$-function could be written as

\begin{equation}\label{eq:conj_Hida}
\end{equation}

where $L_p(\ad(\cG),k,s)$ is a two-variable Iwasawa function interpolating $L$-values of the $g_k$'s normalized with respect to motivic periods, and $C(k)$ is a generator of the congruence ideal of $\cG$.

It is plausible that $L_p(\ad(\cG),k,s)_{|k=1}$ equals $L_p(\rho,\rho^+,s)$ up to a fudge factor. In light of (<ref>), (<ref>) and (<ref>), one may then expect to find a simple relation between $C(k)_{|k=1}$ and $\log_p (u_{g_\alpha})$. We shall provide some evidences concerning this latter fact in a sequel, where it will be shown that, under mild assumptions on $g$, there exists a generator $C(k)$ of the congruence ideal of $\cG$ such that

\[C(k)_{\big|k=1}=\left(1-\frac{\beta}{\alpha}p^{-1}\right)\left(1-\frac{\alpha}{\beta}\right)(\#\Sha(T_p))\cdot \log_p (u_{g_\alpha})\in E^\times\cdot \log_p (u_{g_\alpha}),\]

where $T_p$ is a Galois-stable $\cO_p$-lattice of $W_p$, $\Sha(T_p)$ is the Tate-Shafarevitch group of $T_p$ and $u_{g_\alpha}$ is assumed to be $T_p$-normalized.

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