\section{A compatible system of minimal ramification}\label{sec:lift}

Throughout Part~II, $p\ge17$ is a prime with $p\equiv1\pmod3$, $(a,b,c)$ is a solution of $x^3+y^3=z^p$ in coprime nonzero integers, $u$, $w$, $s_0$ are as in \cref{sec:frey}, and $\rho$, $\rhob$ are the Frey representation and its reduction (\cref{def:frey}).  We replace $\rho$, which is ramified at the primes of $c$, by a lift of $\rhob$ that is unramified outside $3p$ and belongs to a compatible system \cite{CEG22}.  In Part~II, representations over finite fields and over $\ell$-adic fields are considered over an algebraic closure where convenient, and \emph{irreducible} means absolutely irreducible.

\subsection{Properties of the Frey representation}\label{sec:F}
Put
\[
\kappa\eqdef\kappa_u^{-\ninf},
\]
\looseness=-1 a character of $G_K$ with values in $\mu_3\subset\OO^\times$.  We shall use the following six properties of~$\rho$.
\begin{enumerate}[(F1)]
\item $\rhob$ is unramified outside $\{\p,v,\bar v\}$ (\cref{prop:local}~(b)).
\item $\rho|_{G_{K_v}}$ and $\rho|_{G_{K_{\bar v}}}$ are crystalline with Hodge--Tate weights $\{0,1,2\}$ (\cref{prop:local}~(f)).
\item $\rho|_{I_\p}$ is $\kappa|_{I_\p}$ times a unipotent representation (\cref{prop:local}~(d)).  Moreover $\kappa^\cc=\kappa^{-1}$ (\cref{sec:conventions}), and $\kappa|_{G_{K_\p}}$ is trivial if $u\equiv\pm1\pmod 9$ and has conductor $\p^2$ otherwise (\cref{lem:cubic-local}).
\item $\rho^\cc\isom\rho^\vee\otimes\varepsilon_p^{-2}$ (\cref{prop:local}~(c)).
\item $\rho$ is unramified at $\lp$ and $\lpb$, and for $\q\in\{\lp,\lpb\}$ the eigenvalues of $\rho(\Frob_\q)$ are the products of $\xi_\q=\kappa(\Frob_\q)\in\mu_3$ with the roots of $P_\q$, where $(P_\lp,P_\lpb)$ is $(P^+,P^-)$ or $(P^-,P^+)$ (\cref{prop:local}~(e), \cref{prop:frob7}).
\item $\rhob$ is absolutely irreducible (\cref{thm:irreducible}).
\end{enumerate}
We say that we are in the \emph{unramified case} if $\kappa$ is unramified at $\p$ (that is, $u\equiv\pm1\pmod 9$), and in the \emph{ramified case} otherwise.  By \cref{rem:normalisation} this is independent of the choice of $u$.

The image of $\rho$ is compact, so $\rho$ is defined over a finite extension of $\Q_p$ and stabilizes a lattice.  By (F6) the lattice is unique up to homothety and its reduction is $\rhob$.  We therefore regard $\rho$ as a homomorphism $G_K\to\GL_3(\overline\Z_p)$, where $\overline\Z_p$ is the ring of integers of $\overline\Q_p$, with reduction $\rhob\colon G_K\to\GL_3(\Fbar_p)$.  Since $\rho|_{I_\p}\otimes\kappa^{-1}$ is unipotent, every constituent of $(\rhob\otimes\bar\kappa^{-1})|_{I_\p}$ is trivial, and therefore
\begin{equation}\label{eq:F3bar}
\rhob|_{I_\p}\ \text{is}\ \bar\kappa|_{I_\p}\ \text{times a unipotent representation.}
\end{equation}

\subsection{Polarizations}\label{sec:polar}
Let $\dK\colon G_\Q\to\{\pm1\}$ be the quadratic character of $K/\Q$.  Let $\ell$ be a prime and let $A$ be $\overline\Q_\ell$, $\Fbar_\ell$, or the ring of integers of a finite extension of $\Q_\ell$.  A continuous action of $G_K$ on $A^3$ is called \emph{polarized} (with multiplier $\varepsilon_\ell^{-2}\dK$) if there is a perfect symmetric $A$-bilinear pairing $\Psi$ on $A^3$ such that
\begin{equation}\label{eq:pairing}
\Psi\big(\sigma x,(\tilde\cc\sigma\tilde\cc)y\big)=\varepsilon_\ell(\sigma)^{-2}\,\Psi(x,y)\qquad(\sigma\in G_K,\ x,y\in A^3).
\end{equation}
This is the notion of a polarized and odd pair of \cite[\S1.5.1]{CEG22} and \cite[\S2.1]{BLGGT14}, for the field $K$, the representation of $G_K$ on $A^3$ and the character $\varepsilon_\ell^{-2}\dK$ of $G_\Q$: taking $\tilde\cc$ as the element of $G_\Q\smallsetminus G_K$ fixed in \cite[\S1.5.1]{CEG22} (denoted $\gamma_0$ there), the two conditions there are \eqref{eq:pairing} and $\Psi(y,x)=-(\varepsilon_\ell^{-2}\dK)(\tilde\cc)\Psi(x,y)=\Psi(x,y)$, and oddness is $(\varepsilon_\ell^{-2}\dK)(\tilde\cc)=-1$.

Let $\Gthree$ be the group scheme of Clozel, Harris and Taylor \cite[\S2.1]{CHT08}: the semidirect product of $\Gthree^0=\GL_3\times\GL_1$ by a group $\{1,\jmath\}$ of order two, with $\jmath(g,t)\jmath^{-1}=(t\cdot{}^tg^{-1},t)$, and let $\mult\colon\Gthree\to\GL_1$ be the character with $\mult(g,t)=t$ and $\mult(\jmath)=-1$.  By \cite[Lemma~2.1.1]{CHT08}, a polarized action of $G_K$ on $A^3$ together with a pairing $\Psi$ is the same as a continuous homomorphism $G_\Q\to\Gthree(A)$ that maps $G_K$ to $\Gthree^0(A)$ by $\sigma\mapsto(\sigma|_{A^3},\varepsilon_\ell(\sigma)^{-2})$, induces an isomorphism of $G_\Q/G_K$ with $\Gthree/\Gthree^0$, and has \emph{multiplier} $\varepsilon_\ell^{-2}\dK$, where the multiplier of a homomorphism to $\Gthree$ is its composite with $\mult$.  It is called a \emph{prolongation}.  Changing $\Psi$ by a scalar conjugates it by an element of $\{1\}\times\GL_1(A)$ \cite[\S1.5.1]{CEG22}.

\needspace{4\baselineskip}\begin{lemma}\label{lem:polarized}
There is a perfect symmetric $\overline\Z_p$-bilinear pairing $\Psi_\rho$ on $\overline\Z_p^3$ satisfying \eqref{eq:pairing} for $\rho$ (with $\ell=p$); it is unique up to a unit.  In particular $\rho$ and $\rhob$ are polarized, and the prolongation of $\rho$ defined by $\Psi_\rho$ lifts the prolongation of $\rhob$ defined by the reduction of $\Psi_\rho$.
\end{lemma}

\begin{proof}
By (F4) there is a nondegenerate $\overline\Q_p$-bilinear pairing $\Psi_\rho$ with \eqref{eq:pairing}: an isomorphism $\rho^\cc\to\rho^\vee\otimes\varepsilon_p^{-2}$ sends $y$ to the linear form $\Psi_\rho(\,\cdot\,,y)$.  This isomorphism maps the lattice $\overline\Z_p^3$ onto a stable lattice of the dual, which by (F6) is a multiple of the dual lattice, and after scaling $\Psi_\rho$ is perfect on $\overline\Z_p^3$.  By Schur's lemma it is unique up to a scalar.  The pairing $(x,y)\mapsto\Psi_\rho(y,x)$ also satisfies \eqref{eq:pairing}: replacing $\sigma$ by $\tilde\cc\sigma\tilde\cc$ in \eqref{eq:pairing} and using $\tilde\cc^2=1$ and $\varepsilon_p(\tilde\cc\sigma\tilde\cc)=\varepsilon_p(\sigma)$ gives $\Psi_\rho((\tilde\cc\sigma\tilde\cc)y,\sigma x)=\varepsilon_p(\sigma)^{-2}\Psi_\rho(y,x)$.  So $\Psi_\rho(y,x)=\pm\Psi_\rho(x,y)$, and the sign is $+$ because a nondegenerate alternating form does not exist in odd dimension.  (This is the argument of the proof of \cite[Lemma~1.5.3]{CEG22}.)
\end{proof}

\subsection{The residual representation is reasonable}
Calegari, Emerton and Gee call a representation $G_K\to\GL_3(\Fbar_p)$ \emph{reasonable} if it is polarizable and odd, its restriction to $G_{K(\zeta_p)}$ is irreducible, $\zeta_p\notin K$, and $p>2(3+1)=8$ \cite[Definition~2.1.6]{CEG22}.

\begin{proposition}\label{prop:reasonable}
The representation $\rhob$ is reasonable.
\end{proposition}

\begin{proof}
It is polarized by \cref{lem:polarized}; $p\ge17$; and $\zeta_p\notin K$.  It remains to prove that $\rhob|_{G_{K(\zeta_p)}}$ is irreducible.  The extension $K(\zeta_p)/K$ is cyclic of degree $p-1$, since $K\cap\Q(\zeta_p)=\Q$.  Suppose that $\rhob|_{G_{K(\zeta_p)}}$ is reducible.  By Clifford's theorem it is a direct sum of irreducible representations that are conjugate under $G_K$; each occurs with multiplicity one, because the quotient $G_K/G_{K(\zeta_p)}$ is cyclic (a constituent extends to its stabilizer).  As the dimension is $3$, there are three distinct conjugate characters, and $\rhob$ is induced from a character of the subgroup of index $3$ containing $G_{K(\zeta_p)}$, that is, of $G_{K'}$ with $K'$ the cubic subfield of $K(\zeta_p)/K$.  Hence $\rhob\isom\rhob\otimes\eta$ for a character $\eta$ of order $3$ of $\Gal(K'/K)$.  The extension $K'/K$ is totally ramified at $v$, so $\eta|_{I_v}=\epsb^{\pm(p-1)/3}|_{I_v}$.

For $d\ge1$ let $\bar\varepsilon_{p,d}\colon I_v\to\Fbar_p^\times$ be a fundamental character of level $d$; so $\bar\varepsilon_{p,1}=\epsb|_{I_v}$, and $\bar\varepsilon_{p,d}^{(p^d-1)/(p-1)}=\bar\varepsilon_{p,1}$.  By (F2) and Fontaine--Laffaille theory \cite[Th\'eor\`eme~5.3]{FontaineLaffaille} ($K_v=\Q_p$, weights $\{0,1,2\}$, $p>3$), the semisimplification of $\rhob|_{I_v}$ is, up to replacing every character by its inverse, one of
\begin{enumerate}[(i)]
\item $1\oplus\bar\varepsilon_{p,1}\oplus\bar\varepsilon_{p,1}^2$;
\item $\bar\varepsilon_{p,1}^{k_1}\oplus\bar\varepsilon_{p,2}^{k_2+pk_3}\oplus\bar\varepsilon_{p,2}^{k_3+pk_2}$ with $\{k_1,k_2,k_3\}=\{0,1,2\}$;
\item $\bar\varepsilon_{p,3}^{x}\oplus\bar\varepsilon_{p,3}^{px}\oplus\bar\varepsilon_{p,3}^{p^2x}$ with $x=k_1+k_2p+k_3p^2$ and $\{k_1,k_2,k_3\}=\{0,1,2\}$.
\end{enumerate}
This multiset of characters is stable under multiplication by $\eta|_{I_v}$, which has order $3$.

In case (i), $\eta\cdot1\in\{1,\bar\varepsilon_{p,1},\bar\varepsilon_{p,1}^2\}$ forces $(p-1)/3\equiv0,\pm1$ or $\pm2\pmod{p-1}$, which is false for $p\ge13$.  In case (ii), $\bar\varepsilon_{p,2}^{k_2+pk_3}$ is not a power of $\bar\varepsilon_{p,1}$, because $k_2+pk_3\equiv k_2-k_3\not\equiv0\pmod{p+1}$; so $\bar\varepsilon_{p,1}^{k_1}$ is the only character of level $1$ in the multiset, and $\eta\bar\varepsilon_{p,1}^{k_1}=\bar\varepsilon_{p,1}^{k_1}$, which is false.  In case (iii), $\eta=\bar\varepsilon_{p,3}^{\pm(p^3-1)/3}$ on $I_v$ permutes the three characters cyclically, so $(p-1)x$ or $(p^2-1)x$ is congruent to $\pm(p^3-1)/3$ modulo $p^3-1$.  Put $D=p^2+p+1$.  The first congruence says $x\equiv\pm D/3\pmod D$.  The second says $(p+1)x\equiv\pm D/3\pmod D$; since $(p+1)(-p)\equiv1\pmod D$ and $p\cdot D/3\equiv D/3\pmod D$ (as $3\mid p-1$), it also gives $x\equiv\pm D/3\pmod D$.  But $x\equiv(k_1-k_3)+(k_2-k_3)p\pmod D$, a nonzero integer of absolute value at most $2p+2<D/3$.  This is a contradiction.
\end{proof}

\subsection{Compatible systems}\label{sec:systems}
We recall the terminology of \cite[\S1.5.4]{CEG22} and \cite[\S5.1]{BLGGT14} for the field $K$ and rank $3$.  A \emph{compatible system} $\Rsys$ of three-dimensional representations of $G_K$ with coefficients in a number field $M$ consists of a finite set of primes of $K$ (the \emph{bad} primes), a monic cubic polynomial $Q_\q(X)\in M[X]$ for each prime $\q$ of $K$ that is not bad, a continuous semisimple representation
\[
R_\lambda\colon G_K\to\GL_3(\overline M_\lambda)
\]
for each prime $\lambda$ of $M$, and a multiset $H_\tau$ of three integers for each embedding $\tau\colon K\to\overline M$, such that, if $\ell$ is the residue characteristic of $\lambda$: for every $\q\nmid\ell$ that is not bad, $R_\lambda$ is unramified at $\q$ and $R_\lambda(\Frob_\q)$ has characteristic polynomial $Q_\q$; for $\q\mid\ell$, $R_\lambda|_{G_{K_\q}}$ is de Rham, and crystalline if $\q$ is not bad, and the Hodge--Tate weights of $R_\lambda$ with respect to $\tau$ are $H_\tau$.  The system is \emph{regular} if each $H_\tau$ consists of three distinct integers, and \emph{polarized} (and odd, with multiplier $\varepsilon_\ell^{-2}\dK$) if every $R_\lambda$ is polarized in the sense of \cref{sec:polar}.  It is \emph{weakly irreducible} if there is a set of rational primes $\ell$ of positive Dirichlet density such that $R_\lambda$ is irreducible for all $\lambda\mid\ell$ \cite[Definition~1.5.10]{CEG22}.  We regard $M$ as a subfield of $\Qbar$, with $\overline M=\Qbar$, in such a way that the \emph{$p$-adic member} $R_p$ of the system is the member at the prime of $M$ induced by $\iota_p$, viewed over $\overline\Q_p$ through $\iota_p$.

Let $\rhob_{\mathcal G}\colon G_\Q\to\Gthree(\Fbar_p)$ be the prolongation of $\rhob$ given by the reduction of $\Psi_\rho$.  A \emph{polarized component at $3$} is an irreducible component of the generic fiber of the universal ring of framed liftings of $\rhob_{\mathcal G}|_{G_{\Q_3}}$ to $\Gthree$ with multiplier $\varepsilon_p^{-2}\dK$ and a fixed inertial type.  The prime $3$ is ramified in $K$, and this is the case for which \cite[\S1.5.19]{CEG22} works with $\Gthree$-valued liftings.  A \emph{polarized component at $p$} is an irreducible component of the generic fiber of the universal ring of framed potentially semistable liftings of $\rhob|_{G_{K_v}}$ of a fixed inertial type and fixed Hodge--Tate weights.  As $p$ splits in $K$, this is the same as a component for the $\Gthree$-valued liftings at $p$ \cite[Lemma~1.5.20]{CEG22}.  A polarized component $\mathcal C$ at $p$ with distinct Hodge--Tate weights is \emph{globally realizable} \cite[Definition~2.1.9]{CEG22} if there are a CM field $F$, a regular, polarized and odd, weakly irreducible compatible system of representations of $G_F$ whose residual representation at $p$ is reasonable, and a place of the maximal totally real subfield of $F$ with completion $\Q_p$ at which the prolongation of the $p$-adic member of the system has residual representation isomorphic to $\rhob_{\mathcal G}|_{G_{\Q_p}}$ and defines a point of $\mathcal C$.

\begin{theorem}[Calegari--Emerton--Gee]\label{thm:CEG}
Let $\Cthree$ be a polarized component at $3$, and let $\Cp$ be a polarized component at $p$ with Hodge--Tate weights $\{0,1,2\}$ which is globally realizable.  Then there is a regular, polarized and odd, weakly irreducible compatible system $\Rsys=\{R_\lambda\}$ of three-dimensional representations of $G_K$ such that:
\begin{enumerate}[(1)]
\item the $p$-adic member $R_p$ is a lift of $\rhob$, and it is unramified outside $\{\p,v,\bar v\}$;
\item the prolongation of $R_p$, restricted to $G_{\Q_3}$, defines a point of $\Cthree$, and $R_p|_{G_{K_v}}$ defines a point of $\Cp$.
\end{enumerate}
\end{theorem}

\begin{proof}
This is \cite[Theorem~4.2.11]{CEG22} (for components at places of the maximal totally real subfield that are ramified in the CM field, see \cite[\S1.5.19 and Convention~1.5.22]{CEG22}) for the CM field $K$, with maximal totally real subfield $\Q$, in dimension $n=3$ (the theorem requires $n$ odd or $n=2$), for the residual representation $\rhob$ with the character $\epsb^{-2}\bar\delta_{K/\Q}$, the set of places $\{3,p\}$ of $\Q$, and the lift $\varepsilon_p^{-2}\dK$ of that character.  Its hypotheses hold: $\rhob$ is polarized and reasonable (\cref{lem:polarized}, \cref{prop:reasonable}); the set $\{3,p\}$ contains $p$ and the primes at which $\rhob$ or $\epsb^{-2}\bar\delta_{K/\Q}$ is ramified, by (F1); the character $\varepsilon_p^{-2}\dK$ is de Rham and unramified outside $\{3,p\}$; a polarized component is given at each of the two places; and the component at the place $p$ is globally realizable, by hypothesis (in all uses below this is \cref{lem:realizable}, with the field of \cite[Definition~2.1.9]{CEG22} equal to $K$).

\emph{The place $3$.}  At a place not above $p$ of the set $S$ of \cite[Theorem~4.2.11]{CEG22}, here $\{3,p\}$, that theorem asks only for a polarized component in the sense of \cite[\S1.5.19]{CEG22}, where such components are defined, following Bellovin and Gee, at every finite place of the maximal totally real subfield, including the places that are inert or ramified in the CM field; global realizability is asked only at the places above $p$.  So $\Cthree$ may be any polarized component at $3$ (\cref{sec:systems}).

In the proof of \cite[Theorem~4.2.11]{CEG22}, a component at a place of $S$ not above $p$ is imposed on auxiliary compatible systems by \cite[Lemmas~4.2.3 and~4.2.4]{CEG22}, and kept through the component swapping and descent of \cite[Lemmas~4.1.8, 4.2.1 and~4.2.9]{CEG22}.  For such a component these steps use two results of Bellovin and Gee, which allow at a place prime to $p$ any component of a $\Gthree$-valued lifting ring of fixed inertial type, also when the place is inert or ramified in the CM field \cite[\S5.1]{BellovinGee19}: global deformation rings with given components at all places of $S$ have Krull dimension at least one \cite[Corollary~5.1.1]{BellovinGee19}, which is \cite[Proposition~1.5.24]{CEG22}; and, given a suitable potentially diagonalizably automorphic lift over a CM extension, there is a lift on the given components \cite[Theorem~5.2.1]{BellovinGee19}, which gives the lift in \cite[Lemma~4.2.3]{CEG22}.  The latter result asks for potential diagonalizability, a notion defined for $\GL_n$-valued liftings \cite[\S1.4]{BLGGT14}, only at the places above $p$, which split in the CM field in \cite[Lemma~4.2.3]{CEG22}; and the finiteness of the deformation rings rests on \cite[Lemma~1.5.27]{CEG22}, which passes to a base change in which every place of $S$ splits \cite[Lemma~1.2.3]{BLGGT14}.

The theorem gives a regular, polarized and odd, weakly irreducible compatible system together with a compatible system of characters of $G_\Q$ whose $p$-adic member is $\varepsilon_p^{-2}\dK$.  A compatible system of characters is determined by one member, so its member at $\lambda\mid\ell$ is $\varepsilon_\ell^{-2}\dK$, and every $R_\lambda$ is polarized in the sense of \cref{sec:polar}.  Conclusions (1) and (2) are those of \cite[Theorem~4.2.11]{CEG22}, read with \cite[Convention~1.5.22 and Lemma~1.5.20]{CEG22}: the $p$-adic member lifts $\rhob$, is unramified outside the places above $3$ and $p$, that is, outside $\{\p,v,\bar v\}$, and lies on the given components.

\emph{A second proof.}  For the component $\Cp$ of \cref{sec:components}, the only one used below, the theorem also holds without the component swapping of \cite[\S4.2]{CEG22}.  As $p$ splits in $K$ and the points of $\Cp$ are potentially diagonalizable \cite[Lemma~1.4.3~(2)]{BLGGT14}, it follows from the lifting argument of Bellovin and Gee \cite[Theorem~5.2.1]{BellovinGee19}, written out in \cref{lem:3zp-lift} for $p\equiv2\pmod 3$, together with the proof of \cref{lem:members-polarized}, as \cref{rem:3zp-CEG} shows.
\end{proof}

\subsection{The two components}\label{sec:components}
\emph{At $p$.}  By (F2), $\rho|_{G_{K_v}}$ is a point of the generic fiber of the universal ring of framed crystalline liftings of $\rhob|_{G_{K_v}}$ with Hodge--Tate weights $\{0,1,2\}$.  This generic fiber is regular, and it is a union of irreducible components of the generic fiber of the ring of semistable liftings with the same weights, namely those on which the monodromy operator vanishes identically \cite[Theorem~3.3.7 and the paragraph before it]{BellovinGee19} (see also \cite{Kisin08}).  Let $\Cp$ be its irreducible component through $\rho|_{G_{K_v}}$.  It is a polarized component at $p$, with trivial inertial type, and every point of $\Cp$ is crystalline.  To show that $\Cp$ is globally realizable (\cref{lem:realizable}), we first put $\rho$ itself in a compatible system.

\begin{lemma}\label{lem:members-polarized}
The representation $\rho$ is the $p$-adic member of a compatible system $\{\rho_\lambda\}$ of three-dimensional representations of $G_K$, with coefficient field $M_\rho\subset\Qbar$, which is regular, weakly irreducible, and polarized and odd with the multipliers $\varepsilon_\ell^{-2}\dK$: every $\rho_\lambda$ is polarized in the sense of \cref{sec:polar}.
\end{lemma}

\begin{proof}
(1) \emph{The system.}  We apply \cite[Theorem~5.5.1]{BLGGT14} to $(\rho,\varepsilon_p^{-2}\dK)$.  Its hypotheses hold.  The field $K$ is an imaginary CM field, $\zeta_p\notin K$, and $p\ge17\ge2(3+1)$.  The pair $(\rho,\varepsilon_p^{-2}\dK)$ is a totally odd, regular algebraic, polarized $p$-adic representation in the sense of \cite[\S2.1]{BLGGT14}: the pairing $\Psi_\rho$ of \cref{lem:polarized} is symmetric and satisfies \eqref{eq:pairing}, and $(\varepsilon_p^{-2}\dK)(\tilde\cc)=-1$; $\rho$ is unramified outside the primes dividing $3pc$, and de Rham with distinct Hodge--Tate weights at $v$ and $\bar v$ by (F2).  It is potentially diagonalizable at $v$ and $\bar v$ by \cite[Lemma~1.4.3~(2)]{BLGGT14}, because $K_v=K_{\bar v}=\Q_p$, $\rho$ is crystalline there and its weights lie in $[0,p-2]$.  And $\rhob|_{G_{K(\zeta_p)}}$ is irreducible (\cref{prop:reasonable}).  The theorem puts $\rho$ in a compatible system $\{\rho_\lambda\}$ of representations of $G_K$, with coefficient field $M_\rho\subset\Qbar$, which we embed so that $\rho$ is the member at the prime of $M_\rho$ induced by $\iota_p$: each $\rho_\lambda\colon G_K\to\GL_3(\overline M_\lambda)$ is semisimple, and there are a finite set $S$ of primes of $K$, which we take stable under $\cc$ and containing the primes above $3p$, and polynomials $Q_\q\in M_\rho[X]$ for $\q\notin S$, such that $\rho_\lambda$ is unramified at every $\q\notin S$ prime to $\lambda$ and $\rho_\lambda(\Frob_\q)$ has characteristic polynomial $Q_\q$.  Its Hodge--Tate weights are those of $\rho$, $\{0,1,2\}$ at both embeddings, so it is regular.

(2) \emph{Duality.}  For $\q\notin S$ let $Q_\q^*(X)\in M_\rho[X]$ be the monic polynomial whose roots are the $\Nm\q^2/\alpha$, $\alpha$ running over the roots of $Q_\q$.  Since $\tilde\cc\Frob_\q\tilde\cc$ is a Frobenius element at $\bar\q$ and $\varepsilon_\ell(\Frob_\q)=\Nm\q^{-1}$, the characteristic polynomials of $\rho_\lambda^\cc(\Frob_\q)$ and of $(\rho_\lambda^\vee\otimes\varepsilon_\ell^{-2})(\Frob_\q)$ are $Q_{\bar\q}$ and $Q_\q^*$.  For $\lambda$ the prime of $\rho$ they are equal, because $\rho^\cc\isom\rho^\vee\otimes\varepsilon_p^{-2}$ (F4); so $Q_{\bar\q}=Q_\q^*$ in $M_\rho[X]$, and by the Chebotarev density theorem and the Brauer--Nesbitt theorem $\rho_\lambda^\cc\isom\rho_\lambda^\vee\otimes\varepsilon_\ell^{-2}$ for every $\lambda$, $\ell$ being the residue characteristic of $\lambda$.

(3) \emph{Constituents.}  Fix $\lambda$ and write $\rho_\lambda=\bigoplus_{\alpha\in\mathcal A}V_\alpha$ with $V_\alpha$ irreducible.  The $V_\alpha$ are pairwise non-isomorphic, since the Hodge--Tate weights of $\rho_\lambda$ at each embedding are distinct.  By (2) there is an involution $\alpha\mapsto\alpha^*$ of $\mathcal A$ with $V_\alpha^\cc\isom V_{\alpha^*}^\vee\otimes\varepsilon_\ell^{-2}$.

(4) \emph{An automorphic restriction.}  By \cite[Lemma~5.3.1~(1)]{BLGGT14} there is a finite Galois extension $K^1/K$ such that, for every $\lambda$, $\sigma\mapsto\rho_\lambda(\sigma)$ induces an isomorphism of $\Gal(K^1/K)$ with $G_\lambda/G_\lambda^0$, where $G_\lambda$ is the Zariski closure of $\rho_\lambda(G_K)$ and $G_\lambda^0$ its identity component.  Apply \cite[Theorem~4.5.1]{BLGGT14} with $F=F_0=K$, the single representation $(\rho,\varepsilon_p^{-2}\dK)$ (its hypotheses were checked in (1)), an isomorphism $\imath\colon\overline\Q_p\to\C$, and $F^{(\mathrm{avoid})}=K^1$: there are a finite CM extension $F'/K$, Galois and linearly disjoint from $K^1$ over $K$, and a regular algebraic, cuspidal, polarized automorphic representation $(\pi,\chi)$ of $\GL_3(\A_{F'})$ in the sense of \cite[\S2.1]{BLGGT14} with $(r_{p,\imath}(\pi),r_{p,\imath}(\chi)\varepsilon_p^{-2})\isom(\rho|_{G_{F'}},\varepsilon_p^{-2}\dK|_{G_{(F')^+}})$.  For each $\lambda$ fix an isomorphism $j_\lambda\colon\overline M_\lambda\to\C$ inducing $\imath\circ\iota_p$ on $M_\rho$.  Both $\rho_\lambda|_{G_{F'}}$ (as $F'/K$ is Galois) and $r_{\ell,j_\lambda}(\pi)$ are semisimple.  At a prime $w$ of $F'$ at which $\pi$ is unramified, above a prime $\q\notin S$ of $K$ prime to $\ell p$, both $\rho_\lambda(\Frob_w)$ and $r_{\ell,j_\lambda}(\pi)(\Frob_w)$ have as characteristic polynomial the image under $j_\lambda^{-1}$ of that of $\rec(\pi_w\otimes|\det|^{-1})(\Frob_w)$: for $r_{\ell,j_\lambda}(\pi)$ by \cite[Theorem~2.1.1~(2)]{BLGGT14}, and for $\rho_\lambda$ because its characteristic polynomial is given by the same polynomial over $M_\rho$ as that of $\rho(\Frob_w)=r_{p,\imath}(\pi)(\Frob_w)$.  So $\rho_\lambda|_{G_{F'}}\isom r_{\ell,j_\lambda}(\pi)$ by the Chebotarev density and Brauer--Nesbitt theorems.  Likewise $r_{\ell,j_\lambda}(\chi)=\dK|_{G_{(F')^+}}$: this holds for $\ell=p$, so $\chi$ has finite order, and the $r_{\ell,j}(\chi)$ form a compatible system of characters.

The Zariski closure $H_\lambda$ of $\rho_\lambda(G_{F'})$ has finite index in $G_\lambda$, so it contains $G_\lambda^0$; and $G_{F'}$ maps onto $\Gal(K^1/K)=G_\lambda/G_\lambda^0$.  So $H_\lambda=G_\lambda$.  Hence each $V_\alpha$ is irreducible as a representation of $G_{F'}$, and $V_\alpha\not\isom V_\beta$ as representations of $G_{F'}$ for $\alpha\ne\beta$ (a map commuting with $\rho_\lambda(G_{F'})$ commutes with $H_\lambda=G_\lambda$).  As $F'$ is a CM field, $\tilde\cc$ normalizes $G_{F'}$, so the same holds for the $V_\alpha^\cc$ and the $V_\beta^\vee\otimes\varepsilon_\ell^{-2}$.

(5) \emph{The pairing.}  For a closed subgroup $H\subset G_K$ normalized by $\tilde\cc$, let $\mathcal B_H$ be the space of bilinear forms $\Psi$ on the space of $\rho_\lambda$ with $\Psi(\sigma x,(\tilde\cc\sigma\tilde\cc)y)=\varepsilon_\ell(\sigma)^{-2}\Psi(x,y)$ for $\sigma\in H$; sending $\Psi$ to $y\mapsto\Psi(\cdot\,,y)$ identifies $\mathcal B_H$ with $\Hom_H(\rho_\lambda^\cc,\rho_\lambda^\vee\otimes\varepsilon_\ell^{-2})$.  By (3), (4) and Schur's lemma, $\Hom_H(V_\alpha^\cc,V_\beta^\vee\otimes\varepsilon_\ell^{-2})$ has dimension $1$ if $\beta=\alpha^*$ and $0$ otherwise, both for $H=G_K$ and for $H=G_{F'}$.  So $\mathcal B_{G_K}\subset\mathcal B_{G_{F'}}$ have the same dimension $|\mathcal A|$, and they are equal.  By \cite[Theorem~2.1.1~(1)]{BLGGT14} and the definitions of \cite[\S2.1]{BLGGT14}, $(\rho_\lambda|_{G_{F'}},\varepsilon_\ell^{-2}\dK|_{G_{(F')^+}})$ is a totally odd polarized representation: taking for the complex conjugation at an infinite place of $(F')^+$ the element $\tilde\cc$, which lies in $G_{(F')^+}$, there is a nondegenerate symmetric $\Phi\in\mathcal B_{G_{F'}}$.  As $\mathcal B_{G_{F'}}=\mathcal B_{G_K}$, $\Phi$ satisfies \eqref{eq:pairing} for all $\sigma\in G_K$, and $\rho_\lambda$ is polarized.

(6) \emph{Weak irreducibility.}  By \cite[Theorem~1.7]{PatrikisTaylor15} there is a set $L'$ of rational primes of positive Dirichlet density such that $r_{\ell,j}(\pi)$ is irreducible for every $\ell\in L'$ and every $j$.  For $\lambda$ above $\ell\in L'$, $\rho_\lambda|_{G_{F'}}$ is irreducible, hence so is $\rho_\lambda$.
\end{proof}

\needspace{5\baselineskip}\begin{lemma}\label{lem:realizable}
The component $\Cp$ is globally realizable.
\end{lemma}

\begin{proof}
By \cref{lem:members-polarized}, $(\{\rho_\lambda\},\{\varepsilon_\ell^{-2}\dK\})$ is a regular, polarized and odd, weakly irreducible compatible system of representations of $G_K$.  Its residual representation at $p$ is $\rhob$, which is reasonable (\cref{prop:reasonable}); the prolongation of $\rho$ defined by $\Psi_\rho$ (\cref{lem:polarized}) has multiplier $\varepsilon_p^{-2}\dK$, whose restriction to $G_{\Q_p}$ is the multiplier of $\Cp$; and the restriction of this prolongation to $G_{\Q_p}$, at the place $p$ of $\Q$, lies on $\Cp$ by the definition of $\Cp$ (\cite[Lemma~1.5.20]{CEG22}, as $p$ splits in $K$).  These are the conditions of \cite[Definition~2.1.9]{CEG22} with $F=K$.
\end{proof}

\emph{At $3$.}  The prolongation of $\rho$ defined by $\Psi_\rho$ (\cref{lem:polarized}), restricted to $G_{\Q_3}$, is a lifting of $\rhob_{\mathcal G}|_{G_{\Q_3}}$ with multiplier $\varepsilon_p^{-2}\dK$.  Let $\Cthree$ be any irreducible component through it of the generic fiber of the lifting ring of its inertial type.

\begin{lemma}\label{lem:lifts-at-3}
Let $\rho'\colon G_{K_\p}\to\GL_3(\overline\Z_p)$ be a continuous representation whose reduction, restricted to $I_\p$, is $\bar\kappa|_{I_\p}$ times a unipotent representation.  Then there is a nilpotent matrix $N$ such that $\rho'(\sigma)=\kappa(\sigma)\exp(t_p(\sigma)N)$ for all $\sigma\in I_\p$.  In other words, $\WD(\rho')=(\phi,N)$ with $\phi|_{I_\p}=\kappa|_{I_\p}$, a scalar.  (The proof uses only that $p\notin\{2,3,13\}$.)
\end{lemma}

\begin{proof}
Let $I_\p^{\mathrm w}\subset I_\p$ be the wild inertia subgroup, a pro-$3$ group.  The kernel of reduction is a pro-$p$ group and $p\ne3$, so $\rho'(I_\p^{\mathrm w})$ maps isomorphically onto its reduction $\bar\kappa(I_\p^{\mathrm w})$, of order $1$ or $3$ (a unipotent representation over $\Fbar_p$ is trivial on a pro-$3$ group).  An element of order dividing $3$ of $\GL_3(\overline\Z_p)$ whose reduction is a scalar is a scalar, since its eigenvalues lie in $\mu_3$, which injects into the residue field.  So $\rho'\otimes\kappa^{-1}$ is trivial on $I_\p^{\mathrm w}$.  Let $g$ be the image under $\rho'\otimes\kappa^{-1}$ of a topological generator of the tame inertia group.  Its reduction is unipotent, so its eigenvalues are congruent to $1$.  The residue field of $K_\p$ has three elements, so a Frobenius element conjugates the tame inertia group by $\sigma\mapsto\sigma^{3^{\pm1}}$; hence $g$ is conjugate to $g^3$, the multiset of eigenvalues of $g$ is stable under $t\mapsto t^3$, and each eigenvalue satisfies $t^{3^j-1}=1$ for some $j\in\{1,2,3\}$: its order divides $2$, $8$ or $26$.  A root of unity congruent to $1$ has $p$-power order, and $p\nmid2\cdot8\cdot26=2^5\cdot13$ (this is where $p\ne2,13$ is used).  So $g$ is unipotent, the closure of the group generated by $g$ is $\{\exp(xN):x\in\Z_p\}$ with $N=\log g$, and the tame inertia group acts through its quotient $\Z_p(1)$.
\end{proof}

\subsection{The system and its \texorpdfstring{$p$}{p}-adic member}\label{sec:S}
From now on, $\Rsys=\{R_\lambda\}$ denotes a compatible system given by \cref{thm:CEG} for the component $\Cp$ of \cref{sec:components} (\cref{lem:realizable}) and some polarized component at $3$.  Unless said otherwise the component at $3$ is $\Cthree$.  Let $M\subset\Qbar$ be its field of coefficients.  Its $p$-adic member $R_p$ has the following properties.
\begin{enumerate}[(P1)]
\item $R_p$ is a lift of $\rhob$ and is unramified outside $\{\p,v,\bar v\}$.
\item $R_p$ is crystalline at $v$ and $\bar v$ with Hodge--Tate weights $\{0,1,2\}$.  Hence $H_\tau=\{0,1,2\}$ for both embeddings $\tau$ of $K$.
\item $R_p|_{I_\p}\isom\kappa\otimes\exp(t_pN)$ for a nilpotent matrix $N$.
\end{enumerate}
Indeed (P1) is \cref{thm:CEG}~(1).  For (P2): $R_p|_{G_{K_v}}$ defines a point of $\Cp$, all of whose points are crystalline with weights $\{0,1,2\}$; and $R_p^\cc\isom R_p^\vee\otimes\varepsilon_p^{-2}$, which turns the weights $h$ at $v$ into the weights $2-h$ at $\bar v$.  For (P3): apply \cref{lem:lifts-at-3} to $R_p|_{G_{K_\p}}$, using \eqref{eq:F3bar}.  (P3) holds for \emph{every} lift of $\rhob|_{G_{K_\p}}$; the choice of the component at $3$ only decides whether $N$ can be zero (\cref{prop:notscalar}).

\subsection{What the other members inherit}\label{sec:members}

\begin{lemma}\label{lem:lgc-members}
Let $\Rsys=\{R_\lambda\}$ be a regular, polarized and odd, weakly irreducible compatible system of three-dimensional representations of $G_K$, with coefficient field $M$ (for instance a system given by \cref{thm:CEG}), and let $\q$ be a prime of $K$.  There are an embedding $\imath_\q\colon M\to\C$ and a Weil--Deligne representation $W_\q$ of $W_{K_\q}$ over $\C$ such that
\[
j\,\WD\big(R_\lambda|_{G_{K_\q}}\big)^{\Fss}\isom W_\q
\]
for every prime $\lambda$ of $M$, including those above the residue characteristic of $\q$, and every isomorphism $j\colon\overline M_\lambda\to\C$ that restricts to $\imath_\q$ on $M$.  (For $\lambda$ above the residue characteristic of $\q$, $\WD$ is Fontaine's Weil--Deligne representation of the de Rham representation $R_\lambda|_{G_{K_\q}}$.)  In particular:
\begin{enumerate}[(a)]
\item if $R_{\lambda_0}$ is unramified at $\q$ for some prime $\lambda_0$ of $M$ not above the residue characteristic of $\q$, or crystalline at $\q$ for some $\lambda_0$ above it, then $R_\lambda$ is unramified at $\q$ for every $\lambda$ not above the residue characteristic of $\q$, and crystalline at $\q$ for every $\lambda$ above it;
\item the restrictions to $I_\q$ of the Weil--Deligne representations of the $R_\lambda|_{G_{K_\q}}$ correspond to one another under isomorphisms of their coefficient fields that are the identity on $M$, and their monodromy operators have the same Jordan type.
\end{enumerate}
\end{lemma}

\begin{proof}
Let $r$ be the residue characteristic of $\q$.  By \cite[Theorem~2.1.16]{CEG22}, applied to $\Rsys$, its compatible system of multipliers $\{\varepsilon_\ell^{-2}\dK\}$ and the set $\{r\}$ of places of $\Q$, there is a finite Galois extension $F'/K$ of CM fields such that $r$ splits completely in its maximal totally real subfield $(F')^+$ and the restriction of $\Rsys$ to $G_{F'}$ is automorphic \cite[\S1.5.4]{CEG22}: there are a regular algebraic, cuspidal, polarized automorphic representation $(\Pi',\chi')$ of $\GL_3(\A_{F'})$ in the sense of \cite[\S2.1]{BLGGT14} and an embedding $\imath_\q\colon M\to\C$ such that, at almost all primes $w$ of $F'$, $\Pi'_w$ is unramified and $\rec(\Pi'_w\otimes|\det|^{-1})(\Frob_w)$ has as characteristic polynomial the image under $\imath_\q$ of that of the $R_\lambda(\Frob_w)$ \cite[\S5.1]{BLGGT14}.  Let $\lambda$ be a prime of $M$, $\ell$ its residue characteristic, and $j\colon\overline M_\lambda\to\C$ an isomorphism restricting to $\imath_\q$ on $M$.  Both $R_\lambda|_{G_{F'}}$ (as $F'/K$ is Galois) and the representation $r_{\ell,j}(\Pi')$ of \cite[Theorem~2.1.1]{BLGGT14} are semisimple, with the same characteristic polynomials of Frobenius at almost all primes, by \cite[Theorem~2.1.1~(2)]{BLGGT14}; so $R_\lambda|_{G_{F'}}\isom r_{\ell,j}(\Pi')$ by the Chebotarev density and Brauer--Nesbitt theorems.  Let $y$ be a place of $F'$ above $\q$, and $y^+$ the place of $(F')^+$ below $y$.  As $r$ splits completely in $(F')^+$, the completion of $(F')^+$ at $y^+$ is $\Q_r$; as $F'=(F')^+K$, the completion $F'_y$ is generated over it by $K_\q$, so $F'_y=K_\q$, for every choice of $y$.  Hence $y^+$ is split, inert or ramified in $F'$ as $\q$ is over $r$; for $\q=\p$ it is ramified.  Put $W_\q=\rec(\Pi'_y\otimes|\det|^{-1})$.

If $\ell\ne r$, then $j\WD(R_\lambda|_{G_{F'_y}})^{\Fss}\isom W_\q$ by \cite[Theorem~2.1.1~(2)]{BLGGT14}, which is stated for a regular algebraic, cuspidal, polarized pair in the sense of \cite[\S2.1]{BLGGT14} over a CM field, and for every place of that field not above $\ell$.  If $\ell=r$, the same isomorphism, monodromy operators included, is the second assertion of \cite[Theorem~1.1]{BLGGTlgc2}, applied to $F'$, $(\Pi',\chi')$, $j$ and the place~$y$.

That theorem concerns an imaginary CM field $F$, a RAECSDC (regular algebraic, essentially conjugate self-dual, cuspidal) pair $(\Pi,\chi)$ of $\GL_m(\A_F)$, an isomorphism $j\colon\Ql\to\C$, and every place $x\mid\ell$ of $F$, with no condition on the behavior of $x$ in $F/F^+$.  For every weight it gives $j\WD(r_{\ell,j}(\Pi)|_{G_{F_x}})^{\Fss}\prec\rec(\Pi_x\otimes|\det|^{(1-m)/2})$: the two sides have isomorphic semisimplifications, but the Jordan type of the monodromy operator of the left side may be smaller, in the dominance order, than that of the right side.  For Shin-regular weight it gives $j\WD(r_{\ell,j}(\Pi)|_{G_{F_x}})^{\Fss}\isom\rec(\Pi_x\otimes|\det|^{(1-m)/2})$, monodromy operators included.  By the definition in \cite[\S1]{BLGGTlgc2}, every weight is Shin-regular when $m$ is odd (when $m$ is even, a condition on the weight is required), so for $m=3$ the isomorphism holds.  For every $m\ge2$, every CM field and every place above $\ell$, the isomorphism with no condition on the weight is \cite[Theorem~1.1]{Caraiani14}, stated for cohomological conjugate self-dual $\Pi$ (one reduces to these by a twist, as in the proof of \cite[Theorem~2.1]{BLGGTlgc2}); that paper proves the case of even $m$ and refers to \cite{BLGGTlgc1,BLGGTlgc2} for odd $m$ \cite[\S1]{Caraiani14}, so for $m=3$ it is not an independent source.  The monodromy operator is needed when (a) is applied at the primes above $p$ (\cref{prop:members,prop:three-unram}), where that of $W_\q$ must vanish, and when (b) is applied at $\p$ to the members above $3$ (\cref{prop:three-wd}), where it must be regular.

A regular algebraic, cuspidal, polarized pair in the sense of \cite[\S2.1]{BLGGT14}, such as $(\Pi',\chi')$, is RAECSDC (the definition of a polarized pair asks in addition that $\chi_x(-1)=(-1)^m$ for $x\mid\infty$), and the two papers attach to it the same representation $r_{\ell,j}(\Pi')$, both being semisimple with the characteristic polynomials of Frobenius given by $\Pi'$ at almost all places.  The proof of \cite[Theorem~1.1]{BLGGTlgc2} treats a place that does not split over the maximal totally real subfield by passing to a solvable CM extension in which the place splits completely, so that its completion does not change, and in which the places above $\ell$ of the maximal totally real subfield split \cite[proof of Theorem~3.1 and Corollary~3.2]{BLGGTlgc2}, \cite[\S7]{BLGGTlgc1}.  As $F'_y=K_\q$, this gives the isomorphism when $\ell=r$, whether $\q$ is split, inert or ramified in $K$; all three cases occur ($\q\in\{v,\bar v\}$, $\q=5\OO$, $\q=\p$).

For (a): a representation of $G_{K_\q}$ over $\overline M_\lambda$, with $\ell\ne r$, is unramified if and only if its Weil--Deligne representation is unramified with zero monodromy, and a de Rham representation of $G_{K_\q}$, with $\ell=r$, is crystalline if and only if its Weil--Deligne representation is unramified with zero monodromy \cite{Fontaine94pst}.  For (b): Frobenius semisimplification changes neither the restriction to inertia nor the monodromy operator.
\end{proof}

\begin{proposition}\label{prop:members}
Let $\lambda$ be a prime of $M$ of residue characteristic $\ell\notin\{3,p\}$.
\begin{enumerate}[(M1)]
\item $R_\lambda$ is unramified outside $\p$ and the primes of $K$ above $\ell$.
\item $R_\lambda$ is crystalline at the primes of $K$ above $\ell$, with Hodge--Tate weights $\{0,1,2\}$ for each embedding of $K$.
\item $R_\lambda|_{I_\p}\isom\kappa_\lambda\otimes\exp(t_\ell N_\lambda)$, where $\kappa_\lambda\colon I_\p\to\mu_3\subset\overline M_\lambda^\times$ is $\kappa|_{I_\p}$ or its inverse (after an identification of the groups $\mu_3$ of the two coefficient fields), and $N_\lambda$ is a nilpotent matrix of the same rank as the matrix $N$ of \textup{(P3)}.
\end{enumerate}
Moreover, for every prime $\q\nmid3p\ell$ of $K$, the characteristic polynomials of $R_\lambda(\Frob_\q)$ and of $R_p(\Frob_\q)$ are the images of the same polynomial $Q_\q\in M[X]$.
\end{proposition}

\begin{proof}
Apply \cref{lem:lgc-members}, comparing $R_\lambda$ with $R_p$.

(M1) and the last assertion: at a prime $\q\nmid3p\ell$, $R_p$ is unramified by (P1), so $R_\lambda$ is unramified at $\q$.  For every isomorphism $j\colon\overline M_\lambda\to\C$ restricting to $\imath_\q$ on $M$, $j$ maps the characteristic polynomial of $R_\lambda(\Frob_\q)$ to that of $W_\q(\Frob_\q)$.  Composing $j$ with the automorphisms of $\C$ that fix $\imath_\q(M)$ shows that the coefficients of the latter lie in $\imath_\q(M)$ (the fixed field in $\C$ of the automorphisms fixing a subfield $k$ is $k$); so the characteristic polynomial of $R_\lambda(\Frob_\q)$ is the image of $Q_\q=\imath_\q^{-1}\big(\mathrm{char}\,W_\q(\Frob_\q)\big)\in M[X]$, the same polynomial for every $\lambda$.  So $\q$ may be removed from the bad primes, which gives the last assertion; the same argument applies to $R_p$.  At $\q\in\{v,\bar v\}$, $R_p$ is crystalline by (P2), so $R_\lambda$ is unramified at $\q$.

(M2): at $\q\mid\ell$, $R_p$ is unramified by (P1), so $R_\lambda$ is crystalline at $\q$, with the weights $H_\tau$ of (P2).

(M3): take $\q=\p$.  By (P3), the Weil--Deligne representation of $R_p|_{G_{K_\p}}$ restricted to $I_\p$ is the scalar character $\kappa$, with values in $\mu_3$, and its monodromy operator is conjugate to $N$; so by \cref{lem:lgc-members}~(b) the same holds for $R_\lambda$, with $\kappa$ replaced by its image $\kappa_\lambda$ and $N$ by a nilpotent matrix $N_\lambda$ of the same Jordan type.  For an $\ell$-adic representation the restriction to inertia is recovered from the Weil--Deligne representation $(\phi,N_\lambda)$ as $\sigma\mapsto\phi(\sigma)\exp(t_\ell(\sigma)N_\lambda)$ (\cref{sec:conventions}).
\end{proof}

\subsection{Irreducibility of the members}

Weak irreducibility says nothing about a given member.  The test at the primes above $7$ that gave the irreducibility of $\rhob$ does.

\begin{proposition}\label{prop:member-irreducible}
Let $\lambda$ be a prime of $M$ of residue characteristic $\ell\notin\{3,7,p\}$.  Then $R_\lambda$ is irreducible.
\end{proposition}

\begin{proof}
$R_\lambda$ is semisimple.  If it is reducible, it has a one-dimensional summand $\theta\colon G_K\to\overline M_\lambda^\times$.  By \cref{prop:members}, $\theta$ is unramified outside $\p$ and the primes above $\ell$; $\theta|_{I_\p}=\kappa_\lambda$ (a unipotent element acts trivially on a stable line), which has order dividing $3$ and is trivial on $1+\p^2$ by (F3); and $\theta$ is crystalline at the primes above $\ell$, with Hodge--Tate weights $k,k'\in\{0,1,2\}$ at the two embeddings of $K$.  A crystalline character of $G_K$ is the $\lambda$-adic realization of an algebraic Hecke character \cite[Chapter~III, Appendix and \S2.3, Theorem~2; Chapter~II, \S2]{SerreAbelian}, here of conductor dividing $\p^2$ and of infinity type $(k,k')$: there are an embedding $\tau\colon K\to\overline M_\lambda$ and a character $\eta_0$ of $(\OO/\p^2)^\times$, of order dividing $3$, with
\[
\theta(\Frob_{(x)})=\tau\big(\eta_0(x)\,x^{k}\,\bar x^{k'}\big)\qquad\text{for all }x\in\OO\text{ prime to }3\ell .
\]
(Here $\Frob_{(x)}$ denotes the product of the $\Frob_\q^{v_\q(x)}$ over the primes $\q$ dividing $x$, in the abelianized Galois group, and we regard $\eta_0$ as a character with values in $\mu_3\subset K$.)  The right side must be $1$ for $x\in\mu_6$.  As $\mu_6\to(\OO/\p^2)^\times$ is an isomorphism, this gives $\eta_0(\zeta)=\zeta^{k'-k}$ for $\zeta\in\mu_6$; since $\eta_0(-1)=1$, $k\equiv k'\pmod 2$, so $(k,k')$ lies in the set $\mathcal T$ of the proof of \cref{thm:irreducible}, and $x\mapsto\eta_0(x)x^k\bar x^{k'}$ is one of the five Hecke characters $\psi_{(0,0)},\psi_{(1,1)},\psi_{(2,2)},\psi',\psi''$ of that proof.  Complex conjugation of the values permutes these five characters, and replacing $\tau$ by $\tau\circ\cc$ in the display replaces the character by its complex conjugate.  So we may take for $\tau$ the restriction to $K$ of an embedding $\overline M\to\overline M_\lambda$ that is the identity on $M$, and there is $\psi$ among the five such that, for every prime $\q\nmid3\ell$ of $K$, the element $\psi(\q)\in K\subset\Qbar=\overline M$ is a root of the polynomial $Q_\q$ of $\Rsys$.

Take $\q\in\{\lp,\lpb\}$; this is allowed because $\ell\ne7$.  By \cref{prop:members}, $\iota_p(\psi(\q))$ is an eigenvalue of $R_p(\Frob_\q)$, so its reduction is an eigenvalue of $\rhob(\Frob_\q)$, because $R_p$ lifts $\rhob$ (P1).  By (F5), $P_\q(\xi_\q^{-1}\psi(\q))$ lies in the prime $\mathfrak P$ of $\OO$ induced by $\iota_p$, for both $\q=\lp$ and $\q=\lpb$.  As in Step~4 of the proof of \cref{thm:irreducible}, this gives $(k,k')\in\mathcal T$ and $\zeta_1,\zeta_2\in\mu_3$ with \eqref{eq:test7}, which Step~5 of that proof excludes for $p\ge17$.
\end{proof}

\begin{remark}\label{rem:eisenstein}
Properties \textup{(M1)--(M3)} alone do not exclude a reducible system: in the unramified case with $N=0$ they are satisfied by the system with members $1\oplus\varepsilon_\ell^{-1}\oplus\varepsilon_\ell^{-2}$.  The proof excludes such systems using only that $\psi(\q)$ is an eigenvalue of Frobenius at the two primes above $7$, that is, the congruence with $\rhob$ there.
\end{remark}