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\hypersetup{pdftitle={Modularity of elliptic curves over imaginary quadratic fields},pdfauthor={OpenAI}}

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\title{Modularity of elliptic curves over imaginary quadratic fields}
\author{OpenAI}
\date{2026}
\begin{document}
\maketitle

\begin{abstract}
We prove that every elliptic curve over an imaginary quadratic field is
modular: it is realized by an automorphic representation of $\GL_2$ of the
field with matching local parameters at every place and matching
$L$-function. The method is a geometric form of Wiles's prime switch. A
single dihedral family of cyclic covers of a quadratic twist of the curve
produces four two-dimensional Galois representations, pairwise congruent
modulo the primes $3$ and a large prime $p$, with an elliptic curve at one
end. Automorphy enters at that end through known results and is carried to
the original curve by three modularity-lifting steps, then descended from a
cyclotomic extension to the field itself. The complex-multiplication case is
treated separately by automorphic induction.
\end{abstract}

\tableofcontents

\section{Introduction}\label{sec:intro}

\subsection{The theorem}

Let $K$ be an imaginary quadratic number field and $E/K$ an elliptic curve.
The modularity conjecture for $E$ asks for an automorphic representation of
$\GL_2(\A_K)$ whose attached Galois representations are the cohomology of
$E$. Concretely, at a place of good reduction this means that the Hecke
polynomial recovers the point count of the reduction; at a bad place it
means that the full local parameter, including inertia and monodromy, is
recovered; at the single complex place it means a fixed parameter dictated by
the Hodge decomposition. We prove that such a representation always exists.

Throughout, $H^1(X)=H^1_{\et}(X_{\overline L},\Q_l)$ for a variety over a
number field $L$, with coefficients made explicit when they matter, and
$G_L=\Gal(\overline L/L)$. The cyclotomic character is $\eps_l$, normalized
to have Hodge--Tate weight $-1$, so an elliptic curve has labeled
Hodge--Tate weights $0,1$ on its $H^1$ and determinant $\eps_l^{-1}$. For a
regular algebraic cuspidal representation $\pi$ of $\GL_2(\A_L)$ and an
isomorphism $\iota:\overline\Q_l\to\C$, we write $r_\iota(\pi)$ for the
associated semisimple $l$-adic representation. The normalization of local
Langlands used on the cohomological side is
$\rec^T_{L_v}(\pi_v)=\rec_{L_v}(\pi_v\otimes|\det|_v^{-1/2})$; see
Section~\ref{sec:conv}.

\begin{theorem}\label{thm:main}
Let $K$ be an imaginary quadratic field and $E/K$ an elliptic curve. There is
an isobaric automorphic representation $\pi_E$ of $\GL_2(\A_K)$ with trivial
central character and weight zero such that for every prime $l$ and every
isomorphism $\iota:\overline\Q_l\to\C$,
\[
 r_\iota(\pi_E)\simeq H^1_{\et}(E_{\overline K},\overline\Q_l).
\]
At every finite place $v\nmid l$ the Frobenius-semisimplified
Weil--Deligne parameter of the right side equals
$\rec_{K_v}(\pi_{E,v}\otimes|\det|_v^{-1/2})$, and the unitary parameter at
the complex place is $z\mapsto\diag(z/|z|,\overline z/|z|)$. In particular
every local factor agrees, so $L(E/K,s)=L(\pi_E,s-\tfrac12)$. If $E$ has no
complex multiplication then $\pi_E$ is cuspidal; if $E$ has complex
multiplication it is an automorphic induction of a Hecke character, or an
isobaric sum when the CM field equals $K$.
\end{theorem}

The phrase ``with matching local parameters at every place'' together with
the equality of $L$-functions is exactly what modularity means: the Hasse
--Weil $L$-function of the curve, an arithmetic object counting points over
all finite fields, is the $L$-function of an automorphic form, an analytic
object with a functional equation and a spectral meaning.

\subsection{Why it is hard}

Over $\Q$, the modularity of every elliptic curve was completed by
Wiles--Taylor and Breuil--Conrad--Diamond--Taylor. A central tool is
\emph{modularity lifting}. One knows automorphy of a residual Galois
representation $\overline r$, and one proves that every lift $r$ of
$\overline r$ with the right local behavior is also automorphic; applied to
the Galois representation of $E$ this gives modularity of $E$. Lifting
theorems require residual hypotheses, and Wiles's $3$--$5$ switch handles
curves for which the residual representation of $E$ itself is unsuitable, by
passing through a resolvent auxiliary curve whose own modularity is already
known.

Over an imaginary quadratic field three new difficulties appear.
\begin{enumerate}[label=(\arabic*)]
\item The relevant cohomology of Bianchi modular forms sits in more than one
degree, so constructing Galois representations and running a patching or
deformation argument both require extra work. Harris--Soudry--Taylor,
Taylor, Harris--Lan--Taylor--Thorne, Scholze, and Calegari--Geraghty supply
these constructions, and potential modularity over CM fields is available
(Allen--Calegari--Caraiani--Gee--Helm--Le Hung--Newton--Scholze--Taylor
--Thorne). But potential modularity may enlarge the field, whereas we must
stay over $K$.
\item The known lifting theorem in the small characteristics available for
starters --- Caraiani--Newton --- imposes residual conditions and local
conditions that $E$ itself need not satisfy. Their theorem already gives
modularity for a positive proportion of curves, and for all curves when
$X_0(15)(K)$ has rank zero, but not for all curves.
\item There is no obvious resolvent auxiliary curve directly attached to
$E$ in the way Wiles's auxiliary curve is attached over $\Q$.
\end{enumerate}

Our contribution is to remove the restriction on the curve and on the
field. We do not build a single auxiliary curve; we build a whole family of
covers, so that the four representations participating in the modularity
chain are \emph{simultaneous eigenspaces of one geometric object}.

\subsection{The geometric prime switch}

Fix a non-CM elliptic curve $E/K$ and write it as $y^2=h(x)$ with origin the
point at infinity $O$. Fix an odd integer $n=3p$, where $p>100$ is a prime
chosen (via Serre's open image theorem) so that the image of $G_K$ on $E[p]$
contains $\SL_2(\F_p)$. Put $F=K(\mu_n)$. For $t$ in a nonempty Zariski-open
subset $U\subset\A^1_K$ we build a genus-$n$ cyclic cover
\[
 C_n(t)\longrightarrow B_t
\]
of the quadratic twist $B_t:h(t)y^2=h(x)$ of $E$, branched at two points,
carrying an involution $w$ that inverts the deck rotations. For each
character $\theta$ of the rotation group $\mu_n$, the $\theta$-eigenspace
\[
 V_\theta=H^1(C_n(t))_\theta
\]
in integral cohomology is a rank-two compatible system with determinant
$\eps_l^{-1}$ and labeled Hodge--Tate weights $0,1$. Choose characters
$\eta$ of order $3$ and $\psi$ of order $p$, and put $\chi=\eta\psi$ of
order $3p$. The point of the construction is the chain of congruences
\begin{equation}\label{intro:chain}
 \underbrace{V_\eta}_{H^1(A)}
 \ \equiv \pmod{p}\ V_\chi
 \ \equiv \pmod{3}\ V_\psi
 \ \equiv \pmod{p}\ \underbrace{V_1}_{H^1(B_t)},
\end{equation}
each congruence being an equality of semisimplified residual Galois
representations. The order-$p$ part of a character becomes trivial modulo
$p$, and the order-$3$ part becomes trivial modulo $3$, so the primary
factors $\eta,\psi,\chi$ collapse onto their neighbours. The order-three
system $V_\eta$ is $H^1$ of an elliptic curve $A/F$, while $V_1$ is $H^1$ of
the twist $B_t$.

Automorphy enters at the left. The prime $5$ is used only to certify that
the auxiliary elliptic curve $A$ is modular: $A$ is chosen (through the same
one-parameter family) with a large mod-$5$ image and non-CM, so the
elliptic-curve starting theorem of Caraiani--Newton applies. Automorphy then
propagates rightwards along \eqref{intro:chain} by three applications of the
Caraiani--Newton lifting theorem, at primes $p,3,p$ respectively. At the end
we know $H^1(B_t)$ is automorphic; because $B_t$ is a quadratic twist of
$E$, twisting by the corresponding Hecke character gives automorphy of
$H^1(E)$. Finally we descend the automorphic form from the cyclotomic field
$F$ to the original field $K$, and handle the CM case separately.

Two geometric specializations of the cover family make this work.
\begin{itemize}
\item At one degeneration the Jacobian of the cover has compact-type limit
$E^n$. Because $E^n$ is a product of copies of $E$, its cohomology --- and
hence the cohomology of every eigenspace --- has its reduction type (good
ordinary, good supersingular, or totally multiplicative) read off from $E$.
A local-constancy argument over the local field of coefficients transfers
this to all nearby fibers at once. This gives simultaneous local conditions
on all four systems for one choice of $t$.
\item At the other degeneration explicit vanishing cycles cut out opposite
unipotent elements on each nontrivial eigenspace. These unipotents give
absolute irreducibility of the residual representations and produce
suitable Frobenius elements, without any assumption that $E$ and its
conjugate are independent. They are also what forces the auxiliary curve to
be non-CM.
\end{itemize}
A Hilbert-irreducibility argument over the restriction of scalars
$\Res_{K/\Q}U$ then finds a single rational point $t$ satisfying prescribed
local conditions together with the residual hypotheses for both conjugates
of $K$ at once. This gives decomposed genericity at every place above a
split rational prime.

We emphasize what is genuinely new here relative to Caraiani--Newton: the
character pieces appear in one integral cohomology lattice, an elliptic curve
occurs at each end of the chain, and the local-constancy and
two-conjugate-specialization arguments are what make each lifting step
applicable to a prescribed curve over a prescribed imaginary quadratic
field.

\subsection{Related work}

The arithmetic groups of imaginary quadratic fields and their action on
hyperbolic three-space go back to Bianchi, with presentations and cohomology
developed by Swan and Harder. The link with elliptic curves is part of the
Langlands program; Langlands's problem list discusses automorphy of local
representations of elliptic curves over global fields. Computations
relating elliptic curves and automorphic forms over imaginary quadratic
fields are due to Grunewald--Helling--Mennicke, Cremona, and
Cremona--Whitley.

Over $\Q$, Wiles and Taylor--Wiles proved modularity of semistable curves and
Breuil--Conrad--Diamond--Taylor of all curves. Wiles's $3$--$5$ switch
introduced the idea that a residual representation failing the initial
hypotheses can be replaced by one coming from an auxiliary elliptic curve
with prescribed torsion and local behavior. For imaginary quadratic fields,
Harris--Soudry--Taylor and Taylor constructed Galois representations
attached to Bianchi forms in important cases; Berger--Harcos obtained
Frobenius compatibility for cyclotomic central character away from finitely
many primes; Harris--Lan--Taylor--Thorne and Scholze constructed the general
representations; Calegari--Geraghty developed patching beyond the
Taylor--Wiles method. Potential automorphy over CM fields is due to
Allen--Calegari--Caraiani--Gee--Helm--Le Hung--Newton--Scholze--Taylor--Thorne
and, for genus-one curves over quadratic extensions of totally real fields,
to Boxer--Calegari--Gee--Pilloni. Allen--Khare--Thorne gave residual
modularity results in small characteristic and deduced modularity for a
positive proportion of curves over each CM field not containing $\zeta_5$.
Caraiani--Newton gave a lifting theorem allowing small coefficient primes and
ramified CM fields, and deduced modularity of all curves over imaginary
quadratic $K$ with $X_0(15)(K)$ of rank zero, of $100\%$ of curves ordered by
height over each such $K$ not containing $\zeta_5$, and under residual-image
hypotheses in characteristic $3$ or $5$. Our Theorem~\ref{thm:main} removes
the last restrictions; the Caraiani--Newton lifting theorem and modularity
criterion remain essential inputs.

The geometric ingredients have antecedents: cyclic covers of elliptic
curves with two branch points and dihedral symmetry occur in real
multiplication constructions of Tautz--Top--Verberkmoes, in Ellenberg's work
on endomorphism algebras of Jacobians, and in Mestre's constructions. Darmon
studied congruences removing prime-power parts of monodromy and a conditional
modularity argument by successive lifts for rigid local systems. The
difference operator $2-\zeta-\zeta^{-1}$ in the monodromy computation is the
same one appearing in rank-two monodromy studied by Darmon--Mestre. Taylor's
use of auxiliary abelian varieties with real multiplication is a
higher-dimensional form of the prime switch.

\subsection{Organization}

Section~\ref{sec:conv} fixes conventions. Sections~\ref{sec:geom} and
\ref{sec:mono} build the cover family and compute its integral homology and
monodromy. Section~\ref{sec:sys} constructs the four rank-two systems and
their congruences. Section~\ref{sec:inputs} states the automorphy inputs in
the precise form used and reduces the remaining work to conditions on $t$.
Section~\ref{sec:local} produces the local reduction conditions. Section
\ref{sec:res} produces a single $t$ satisfying all residual and local
conditions at once. Section~\ref{sec:auto} runs the three lifts, descends to
$K$, and treats the CM case.

\section{Conventions}\label{sec:conv}

We write $\eps_l$ for the cyclotomic character with Hodge--Tate weight $-1$.
A bar over a characteristic-zero representation denotes the semisimplification
of the reduction of a chosen stable lattice; this semisimplification is
independent of the lattice. An elliptic curve thus has labeled Hodge--Tate
weights $0,1$ on $H^1$ and determinant $\eps_l^{-1}$.

For a regular algebraic cuspidal representation $\pi$ of $\GL_2(\A_L)$ over a
CM field and $\iota:\overline\Q_l\to\C$, the representation $r_\iota(\pi)$ is
semisimple. We use the Tate-normalized local comparison
\begin{equation}\label{eq:tate}
 \rec^T_{L_v}(\pi_v):=\rec_{L_v}\bigl(\pi_v\otimes|\det|_v^{-1/2}\bigr),
\end{equation}
with geometric-Frobenius reciprocity. The algebraic weight coming from
trivial coefficients is called weight zero. At a complex place the generic
weight-zero parameter, in unitary normalization, is
\begin{equation}\label{eq:inf}
 z\longmapsto\diag(z/|z|,\overline z/|z|),\qquad |z|=(z\overline z)^{1/2}.
\end{equation}
Its Tate-normalized parameter has Hodge types $(1,0)$ and $(0,1)$. If the
finite parameters realize an elliptic curve $E$, then
\begin{equation}\label{eq:Lshift}
 L(E/L,s)=L(\pi,s-\tfrac12).
\end{equation}

For a local field, a Galois representation is \emph{potentially crystalline}
if it becomes crystalline after a finite extension, and is \emph{potentially
ordinary of weight zero} if after a finite extension it has an invariant line
on which inertia acts by an unramified character and a quotient on which
inertia acts by $\eps_l^{-1}$ times an unramified character; this is the
Hodge--Tate ordering $(0,1)$. Local fields used to verify these properties may
be enlarged independently at different places.

A continuous representation $\overline r:G_L\to\GL_2(\overline\F_l)$ over a
number field $L$ is \emph{decomposed generic} if there is a rational prime
$q\ne l$ splitting completely in $L$ such that for every $v\mid q$ it is
unramified at $v$ and the Frobenius eigenvalues $\alpha_v,\beta_v$ satisfy
$\alpha_v/\beta_v\ne q$ and $\beta_v/\alpha_v\ne q$ in $\overline\F_l$. Our
construction will take $q\equiv1\pmod l$, so it suffices to have
$\alpha_v\ne\beta_v$ at every $v\mid q$.

\section{A cyclic-cover family and its two degenerations}\label{sec:geom}

Fix an imaginary quadratic field $K$, a non-CM elliptic curve $E/K$ written
$E:y^2=h(x)$ with $h$ a separable cubic, and the origin $O=[0:1:0]$ at
infinity. Fix an odd integer $n\ge3$ for the moment. We build a family of
cyclic covers of quadratic twists of $E$ and identify two boundary fibers.
At the first the Jacobian becomes a product of copies of $E$; at the second
the cover acquires nodes whose vanishing cycles give explicit monodromy. All
degenerations here are over a characteristic-zero parameter curve.

\subsection{The family, its dihedral symmetry, and geometric connectedness}

For $t$ with $h(t)\ne0$ put
\begin{equation}\label{geom:twist}
 B_t:\quad h(t)y^2=h(x),\qquad R_t=(t,1),\qquad S_t=[n]R_t,
\end{equation}
the $y$-coordinate normalized so that $R_t$ is the point $(t,1)$ on the
twist. Let $U\subset\A^1_k$ over $k\supset K(\mu_n)$ be the open set where
$O,R_t,-R_t,S_t,-S_t$ are pairwise distinct. This is nonempty: over the double
cover $t=x(R)$ parametrized by $R\in E$, the map
$B_{x(R)}\to E$, $(x,y)\mapsto(x,y(R)y)$, identifies $R_t$ with the moving
point $R$, and each excluded equality forces $R$ to be torsion, hence excludes
only finitely many points. We may delete further finitely many points of $U$
whenever a statement requires it.

\begin{lemma}\label{geom:family}
For each divisor $m>1$ of $n$ there is a smooth proper family of
geometrically connected genus-$m$ curves $\cC_m\to U$. It is the
normalization of the smooth proper elliptic family $B\to U$ in the
function-field extension described fiberwise by the Kummer equation
\begin{equation}\label{geom:kummer}
 Y^m=f_t(Z),\qquad
 \operatorname{div}(f_t)=(S_t)-(-S_t)-n(R_t)+n(-R_t),\qquad f_t(O)=1.
\end{equation}
The family carries an involution
\begin{equation}\label{geom:reflection}
 w:(Z,Y)\longmapsto(-Z,Y^{-1})
\end{equation}
and, over a field containing $\mu_m$, rotations
$u_\zeta:(Z,Y)\mapsto(Z,\zeta Y)$ with $wu_\zeta w=u_{\zeta^{-1}}$. For
$d\mid m$ the map $Y\mapsto Y^{m/d}$ exhibits $C_{m,t}\to C_{d,t}$; for $d=1$
the quotient is $B_t$.
\end{lemma}

\begin{proof}
On an elliptic curve a degree-zero divisor is principal if and only if its
sum in the group law is $O$. The divisor in \eqref{geom:kummer} has sum
$S_t-(-S_t)-nR_t+n(-R_t)=2S_t-2nR_t=O$, so it cuts out a rational function,
pinned down by $f_t(O)=1$. Working over $k(U)$ and spreading out gives such a
function on the family. Its divisor changes sign under $Z\mapsto-Z$, so
normalizing at $O$ yields
\begin{equation}\label{geom:inverse}
 f_t(-Z)=f_t(Z)^{-1}.
\end{equation}
This also shows $f_t$ has no vertical divisor: every fiber component is
preserved by inversion, so its valuation is its own negative. Hence the
prescribed disjoint sections are the whole divisor.

The valuation of $f_t$ at $S_t$ is $1$, so the Kummer extension has degree
$m$ even geometrically: it is totally ramified at $S_t$ and $-S_t$ and
unramified elsewhere, because the remaining nonzero valuations $\pm n$ are
divisible by $m$. Riemann--Hurwitz gives $2g(C_{m,t})-2=2(m-1)$, so $g=m$.
The same local computation shows the normalization is smooth over $U$: a
zero or pole of order divisible by $m$ drops out of the Kummer equation, and
at a section of order $\pm1$ one takes a root of an \'etale unit and obtains
the $m$th-root cover of a transverse parameter. Normalization is finite here,
so the resulting family is proper. Equation~\eqref{geom:inverse} gives $w$,
and the quotient statements follow from the Kummer equations.
\end{proof}

We pull this family back to the open set of $E$ where $R$ moves and
$O,\pm R,\pm[n]R$ are distinct; the base elliptic curve there is the constant
curve $E$. We write $C_{m,R}$ for the fiber and $f_R$ for its normalized
function. The normalization at $O$ makes the pullback identification exact:
it introduces no scalar into $Y^m=f_R$.

\subsection{Separating colliding sections}

The two boundary parameters we use are
\begin{equation}\label{geom:boundary}
 R=O,
 \qquad\text{and}\qquad
 R=R_*,\quad [n]R_*\in E[2]\setminus\{O\},
 \quad O,\pm R_*,[n]R_*\ \text{distinct}.
\end{equation}
The second exists after extending scalars and will be used over $\C$.

\begin{lemma}\label{geom:semistable}
In the constant-$E$ family, blow up the surface at the collision point over
either boundary parameter in \eqref{geom:boundary} and normalize the result in
$Y^m=f_R$. This is a regular semistable model near that parameter. Its
special fiber has $m$ geometric nodes over the node of the blown-up elliptic
surface, each with smoothing parameter of order one in a base parameter. The
component over the exceptional projective line is a geometrically connected
rational curve.
\end{lemma}

\begin{proof}
At $R=O$ use parameters $r$ on the parameter curve and $z$ on $E$, both with
the same tangent coordinate at $O$. The sections $R,-R,[n]R,-[n]R$ have
first-order equations $z=r,-r,nr,-nr$; their strict transforms meet the
exceptional line at $1,-1,n,-n$, and the node joining that line to the main
component is at $\infty$. At $R_*$ only $[n]R$ and $-[n]R$ collide, with
distinct tangent slopes, so one blowup separates them from each other and from
the node.

The involution $Z\mapsto-Z$ preserves both components of each blown-up
fiber, so by \eqref{geom:inverse} the valuation of $f_R$ on each component is
zero; no horizontal divisor section passes through the node. Since the
surface is regular, $f_R$ is a unit there, and an $m$th root exists \'etale.
The node downstairs has local equation $xy=b$, so its $m$ lifts have the same
equation and smoothing parameter $b$ rather than a higher power. At the
separated sections the normalization is smooth as in Lemma~\ref{geom:family}.
On the exceptional line the cover is ramified exactly at the two sections of
multiplicities $\pm1$, is connected, and has genus zero by Riemann--Hurwitz.
\end{proof}

\begin{proposition}[First degeneration]\label{geom:compact}
Suppose $k$ contains $\mu_m$. At $R=O$ the special fiber of
Lemma~\ref{geom:semistable} is a rational curve meeting $m$ copies of $E$,
each at one point, all defined over $k$. Rotations cyclically permute the
elliptic components. The Jacobian family of $C_{m,R}$ extends to an abelian
scheme on a Zariski neighborhood of $O$ in the parameter curve, with fiber
$E^m$ at $O$.
\end{proposition}

\begin{proof}
Let $v=z/r$ on the exceptional line. Its normalized function is
\begin{equation}\label{geom:exc}
 f_{\mathrm{exc}}(v)=\frac{v-n}{v+n}\Bigl(\frac{v+1}{v-1}\Bigr)^{n}.
\end{equation}
The divisor gives this up to a scalar; the section $Z=O$ meets the line at
$v=0$ where the value is $1$, and since $n$ is odd \eqref{geom:exc} is also
$1$ at $0$, so the scalar is $1$.

On the main elliptic component $f_R$ has no divisor and is constant, equal to
$f_{\mathrm{exc}}(\infty)=1$ at the node. Thus the cover of the main
component is the disjoint union of the $m$ copies of $E$ indexed by
$Y\in\mu_m$. On the exceptional line the substitution
$Y'=Y\bigl(\tfrac{v-1}{v+1}\bigr)^{n/m}$ gives $Y'^m=(v-n)/(v+n)$; its $m$
points over $v=\infty$ are defined over $k$ and attach one to each elliptic
component.

The dual graph is a tree, so the fiber is of compact type: the
identity-component Picard variety is the product $E^m$ of the component
Jacobians. The marked point $(O,1)$ extends to a section through the smooth
locus, and the relative Picard theorem for nodal curves of compact type makes
the Jacobian family an abelian scheme across this fiber (see
\cite[Section~9.4, Theorem~1, and Section~9.2, Proposition~10]{BLR}). Shrinking
the parameter neighborhood removes the other singular fibers.
\end{proof}

\begin{remark}\label{geom:char0}
Proposition~\ref{geom:compact} concerns an abelian scheme over an open subset
of the characteristic-zero parameter curve $E_k$, and after completing $k$ it
is a family over that local field. No integral model of the Kummer cover in
residue characteristic dividing $m$ is needed; the arithmetic reduction types
of nearby Jacobians are established separately.
\end{remark}

\subsection{The second degeneration and its nodes}

Choose an oriented period basis $a,b$ of $E(\C)$ and take $R_*=a/(2n)$, so
$[n]R_*=a/2$ is a nonzero $2$-torsion point. The first-order equation below
records why a loop about $R_*$ contributes one twist at each node.

\begin{lemma}\label{geom:collision}
For $R_*=a/(2n)$ the special fiber of Lemma~\ref{geom:semistable} consists
of a connected unramified degree-$m$ cover $E'\to E$ and a rational
component, joined at $m$ distinct nodes. In analytic coordinates near each
node the family has equation $XU=-2nt$, where $t=R-R_*$ is an additive local
parameter.
\end{lemma}

\begin{proof}
Write $Q=a/2=[n]R_*$ and use additive coordinates $t=R-R_*$ and $z=Z-Q$. The
colliding sections are $z=nt$ and $z=-nt$, the other divisor sections avoid
this neighborhood, and
\[
 f_R=g(z,t)\,\frac{z-nt}{z+nt}
\]
with $g$ a unit. Taking a local $m$th root of $g$ and $V=Y/g^{1/m}$, the
Kummer equation becomes
\begin{equation}\label{geom:node}
 (z+nt)(V^m-1)=-2nt.
\end{equation}
For each $V=\zeta\in\mu_m$, $U=V^m-1$ is a local coordinate; setting
$X=z+nt$ gives the asserted node equation. This is the local chart of the
normalization of the blowup of Lemma~\ref{geom:semistable}, with $X=0$ the
exceptional component and $U=0$ the main component.

On the main component the limiting divisor is $n(-R_*)-n(R_*)$, whose
unramified Kummer cover is governed by the $m$-torsion class
\[
 \frac nm\bigl((-R_*)-(R_*)\bigr)
 \quad\longleftrightarrow\quad
 -\frac{2n}{m}R_*=-\frac{a}{m}\quad\text{in }E(\C).
\]
This class has exact order $m$, so the cover is connected; being unramified of
degree $m$ over an elliptic curve, it is itself an elliptic curve $E'$. The
exceptional cover is rational by Lemma~\ref{geom:semistable}, and its $m$
attachments are exactly these nodes.
\end{proof}

The first degeneration gives a split product fiber $E^m$ together with a
neighborhood over which nearby smooth fibers can be compared to it; the
second gives $m$ nodes of smoothing order one whose vanishing cycles we count
next in a single integral basis.

\section{Integral homology and explicit monodromy}\label{sec:mono}

Fix the complex elliptic curve $E$ and the constant-$E$ family of
Section~\ref{sec:geom}. Let $n\ge3$ be odd and $m>1$ divide $n$. Write
$\Gamma_m=\mu_m$ and let $u:(Z,Y)\mapsto(Z,e^{2\pi i/m}Y)$ generate the deck
group; throughout $\Z[\Gamma_m]$ is the group ring, not a cyclotomic integer
ring. We build an integral basis permuted by $u$ and compute two monodromy
operators in that one basis. Their reduction on character spaces gives the
residual irreducibility used later.

\subsection{A basis from cutting the torus}

Choose an oriented period basis $a,b$ and identify the underlying oriented
torus with $(\R/\Z)^2$, with $a$ horizontal and $b$ vertical. Consider the
path of parameters
\begin{equation}\label{mon:path}
 R_s=\frac{s}{n}a,\qquad 0<s<\tfrac12,
\end{equation}
whose branch points are $\mp sa$. Let $I_s$ be the horizontal arc from $-sa$
to $sa$ through $O$.

\begin{lemma}\label{mon:slit}
For parameters \eqref{mon:path} the cover $C_{m,R_s}\to E$ is trivial over
$E\setminus I_s$; it is obtained by cutting $m$ copies of $E$ along $I_s$ and
joining adjacent banks cyclically. Crossing the slit changes the sheet index by
$\pm1$ according to the crossing orientation and the initial indexing.
\end{lemma}

\begin{proof}
For small $s$ enclose the colliding sections in a small disk about $O$. By
Proposition~\ref{geom:compact} the model at $R=O$ restricts outside that disk
to $m$ disjoint copies of its complement. The inclusion
$E\setminus\operatorname{int}(D)\into E\setminus I_s$ is a homotopy
equivalence, so the unramified cover of $E\setminus I_s$ is trivial. As $s$
varies in $(0,1/2)$ the arcs $I_s$ move by isotopy and the cover's
topological type is constant. The monodromy around either branch point is a
generator of $\Gamma_m$, since its valuation in $f_R$ is $\pm1$. This gives
the stated gluing.
\end{proof}

On each sheet let $a_i$ be the horizontal loop at height $1/2$ and $b_i$ the
vertical loop at position $1/2$, $i\in\Z/m\Z$. They avoid the slit, meet once
on their common sheet, and are oriented so that
\begin{equation}\label{mon:intersection}
 \langle a_i,b_j\rangle=\delta_{ij},\qquad
 \langle a_i,a_j\rangle=\langle b_i,b_j\rangle=0,
\end{equation}
and $u$ sends $a_i,b_i$ to $a_{i+1},b_{i+1}$.

\begin{proposition}\label{mon:integral}
The cycles $a_i,b_i$ form an integral symplectic basis of
$H_1(C_{m,R_s},\Z)$. Consequently, for every smooth fiber,
\begin{equation}\label{mon:free}
 H_1(C_m,\Z)\simeq\Z[\Gamma_m]^2,
 \qquad H^1(C_m,\Z)\simeq\Z[\Gamma_m]^2.
\end{equation}
After comparison, $H^1_{\et}(C_m,\Z_l)$ is free of rank two over
$\Z_l[\Gamma_m]$ for every prime $l$, including $l\mid m$.
\end{proposition}

\begin{proof}
The genus is $m$, so $H_1$ has rank $2m$. By \eqref{mon:intersection} the
$2m$ displayed classes are independent and span a sublattice with unimodular
pairing; if its index in $H_1$ were $d$, the intersection determinant would
differ by $d^2$, but both are $1$, so $d=1$. The deck action gives the first
isomorphism.

The dual of a regular module is regular: the coefficient-of-identity pairing
$\Z[\Gamma_m]\times\Z[\Gamma_m]\to\Z$, $(x,y)\mapsto[1](xy)$, is perfect and,
with the pullback action on the dual, identifies
$\Hom_{\Z}(\Z[\Gamma_m],\Z)$ with $\Z[\Gamma_m]$. This proves the
cohomological assertion without dividing by $m$. Topological local
triviality transports the result through the smooth parameter open, and
comparison gives the \'etale statement.
\end{proof}

\begin{remark}
The compact-type degeneration gives a second view: the Jacobian extends to an
abelian scheme with special fiber $E^m$ on which $\Gamma_m$ permutes the
factors regularly, and the integral homology of that fiber is the regular
module twice. The slit construction makes this concrete by identifying cycles
that continue toward the other degeneration.
\end{remark}

\subsection{Vanishing cycles and a common monodromy basis}

Let $s\to1/2$ in \eqref{mon:path}. The branch points meet at $a/2$ while the
slit fills all but a shrinking gap in the horizontal circle. Take a small disk
about $a/2$ containing both branch points; its boundary has $m$ closed lifts
$\delta_i$, the neck circles at the $m$ nodes of Lemma~\ref{geom:collision},
hence the vanishing cycles.

\begin{lemma}\label{mon:vanishing}
After a cyclic indexing of the lifts,
\begin{equation}\label{mon:delta}
 [\delta_i]=\pm(a_i-a_{i+1}),\qquad i\in\Z/m\Z,
\end{equation}
or the same formula with $i-1$ in place of $i+1$.
\end{lemma}

\begin{proof}
Arrange the boundary circle to cross the slit transversely twice. A lifted
boundary circle has its upper arc on one sheet and its lower arc on an adjacent
sheet; index the upper sheet by $i$, so the lower is $i+1$ or $i-1$ by
Lemma~\ref{mon:slit}. The circle is disjoint from every $a_j$, and crosses the
two corresponding vertical loops once each with opposite signs and no other
$b_j$. These intersections determine its class in the integral symplectic basis
of Proposition~\ref{mon:integral}, giving \eqref{mon:delta}.
\end{proof}

\begin{theorem}[Monodromy transvections]\label{mon:transvections}
In one integral basis $a_i,b_i$ as above, the geometric monodromy of the
constant-$E$ family contains operators $T_a,T_b$ with
\begin{align}
 T_a(a_j)&=a_j,&
 T_a(b_j)&=b_j+\sigma_a(2a_j-a_{j-1}-a_{j+1}),\label{mon:Ta}\\
 T_b(b_j)&=b_j,&
 T_b(a_j)&=a_j+\sigma_b(2b_j-b_{j-1}-b_{j+1}),\label{mon:Tb}
\end{align}
where $\sigma_a,\sigma_b\in\{1,-1\}$ are independent of $j$. Equivalently,
their off-diagonal entries in the free $\Z[\Gamma_m]$-module are
$\sigma_a(2-u-u^{-1})$ and $\sigma_b(2-u-u^{-1})$. The geometric monodromy of the
family over $U$ contains these actions under the pullback identification.
\end{theorem}

\begin{proof}
The Picard--Lefschetz formula for a positively oriented loop about a
semistable fiber is the product of the Dehn twists at its nodes (see
\cite[Expos\'e~XV]{SGA7II}). Every node has smoothing order one by
\eqref{geom:node}, so every twist has exponent one. The vanishing cycles are
disjoint, so their mutual intersection pairings vanish, and a class $\gamma$
maps to
\begin{equation}\label{mon:PL}
 \gamma\longmapsto\gamma+\sigma\sum_i\langle\gamma,\delta_i\rangle[\delta_i]
\end{equation}
with one sign $\sigma$ fixed by the intersection convention. Substituting
\eqref{mon:delta} gives zero on the $a_j$ and, up to the sign,
\begin{equation*}
 (a_j-a_{j+1})-(a_{j-1}-a_j)=2a_j-a_{j-1}-a_{j+1}
\end{equation*}
on $b_j$, proving \eqref{mon:Ta}. Replacing the horizontal path by
$R_s=(s/n)b$ gives \eqref{mon:Tb}, initially in the basis from the vertical
slit.

It remains to compare the two bases, since separate triangular matrices in
unrelated bases would not suffice. Choose both starting parameters close to
$O$. The horizontal loop at $y=1/2$ and the vertical loop at $x=1/2$ avoid one
fixed small disk about $O$ and both short slits. Outside this disk the covering
extends across $R=O$ as $m$ disjoint sheets; trivialize it over a small simply
connected parameter disk containing $O$ and use this trivialization to label the
lifts of the two fixed loops. Parallel transport between the starting
parameters then identifies both sets of $a_i,b_i$ with the same basis. There is
no monodromy ambiguity from looping $O$, because the sheet trivialization
extends over $O$. A different initial labeling only imposes a common cyclic
shift, leaving \eqref{mon:Ta}--\eqref{mon:Tb} unchanged.

Both loops therefore give these operators in a single geometric monodromy
group. Pullback along the open parameter curve of $E\to U$ maps its fundamental
group into that of $U$, so the operators occur in the latter image too, and
deleting further finitely many points does not remove them.
\end{proof}

The operator $2-u-u^{-1}$ is the cyclic difference operator coming from the
adjacent-sheet vanishing classes. The next section uses the integral module of
Proposition~\ref{mon:integral} to specialize rotation characters, and the two
matrices above to keep the resulting residual representations irreducible.

\section{Rank-two systems and the congruent chain}\label{sec:sys}

The cyclic covers now supply the Galois representations that transfer
automorphy. The involution pairs opposite rotation characters, making each
character space two-dimensional with the determinant and Hodge--Tate weights of
an elliptic curve. The integral regular-module computation then yields
congruences even at primes dividing the covering degree.

\subsection{Standing setup}

Return to the imaginary quadratic field $K$ and a non-CM elliptic curve $E/K$.
By Serre's open image theorem \cite[\S4.4, Th\'eor\`eme~3]{SerreOpenImage}
choose a prime $p>100$ such that the image of $G_K$ on $E[p]$ contains
$\SL_2(\F_p)$, and set
\begin{equation}\label{sys:fields}
 n=3p,\qquad F=K(\mu_n).
\end{equation}
Both $K/\Q$ and $\Q(\mu_n)/\Q$ are abelian, so $F/\Q$ is abelian; $F$ is a CM
field, since complex conjugation is central with totally real fixed field and
$F$ has no real embedding (it contains $K$). Moreover
\begin{equation}\label{sys:nozeta}
 \zeta_5\notin F.
\end{equation}
Indeed $\Q(\mu_n)$ is unramified at $5$, so the ramification index of $F$ at
$5$ is at most $[K:\Q]=2$, whereas a field containing $\Q(\zeta_5)$ has
ramification index divisible by $4$ at $5$.

Apply Lemma~\ref{geom:family} with this $n$: let $U\subset\A^1_K$ be its
parameter open and $B_t$ the quadratic twist of $E$. We shrink $U$ whenever a
smooth proper model of one of the finitely many families is needed. Until
Section~\ref{sec:res} the point $t\in U(K)$ is arbitrary.

\subsection{Character spaces}

Fix an embedding $\overline\Q\into\C$ and put $M=\Q(\mu_n)$. For each prime
$l$ choose $j_l:\overline\Q\into\overline\Q_l$ and an isomorphism
$\iota_l:\overline\Q_l\to\C$ whose composite with $j_l$ is the fixed complex
embedding. Every rotation character is a homomorphism
$\theta:\mu_n(F)\to M^\times$. For the fiber $C_n=C_n(t)$ define
\begin{equation}\label{sys:def}
 V_{\theta,l}=\bigl\{x\in H^1_{\et}(C_{n,\overline F},\overline\Q_l):
 u^*x=j_l(\theta(u))x\ \text{for all }u\in\mu_n(F)\bigr\},
\end{equation}
where $u$ denotes the deck transformation with multiplier $u$. All deck
transformations are defined over $F$, so these are $G_F$-representations; write
$V_\theta$ for the system of realizations. In any congruence at $l$ all
characters use the same $j_l$, which is essential when comparing reductions.

\begin{proposition}\label{sys:properties}
For every $t\in U(K)$ and every rotation character $\theta$, the system
$V_{\theta,l}$ satisfies:
\begin{enumerate}[label=(\arabic*)]
\item it is semisimple of dimension two with determinant $\eps_l^{-1}$;
\item at every place of $F$ above $l$ it is potentially semistable with
labeled Hodge--Tate weights $\{0,1\}$;
\item there is a finite set of places $S$ of $F$, independent of $l$, such
that $V_{\theta,l}$ is unramified outside $S$ at places of residue
characteristic $\ne l$, with Frobenius characteristic polynomial the image under
$j_l$ of a polynomial in $M[T]$ independent of $l$.
\end{enumerate}
\end{proposition}

\begin{proof}
The rank-two regular module of Proposition~\ref{mon:integral}, followed by
comparison between singular and $l$-adic cohomology, gives $\dim V_{\theta,l}=2$.

Let $\langle\,,\rangle$ be the cup-product pairing on $H^1(C_n)$ into
$\overline\Q_l(-1)$. It pairs the $\theta$-space perfectly with the
$\theta^{-1}$-space, and the involution $w$, defined over $F$ and preserving cup
product, exchanges them. Thus
\[
 [x,y]_\theta=\langle x,w^*y\rangle,\qquad x,y\in V_{\theta,l},
\]
is a perfect $G_F$-equivariant pairing into $\overline\Q_l(-1)$, alternating
because $\langle y,w^*x\rangle=\langle w^*y,x\rangle=-\langle x,w^*y\rangle$. A
perfect alternating pairing on a two-dimensional space trivializes the
determinant against the target line, so $\det V_{\theta,l}=\eps_l^{-1}$.

For the Hodge numbers, fix an embedding of $F$ into $\C$ and let $a_\theta$ be
the multiplicity of $\theta$ in the holomorphic differentials of the resulting
curve. The holomorphic involution gives $a_\theta=a_{\theta^{-1}}$, and the
Hodge decomposition together with complex conjugation identifies the dimension
of the full $\theta$-space with $a_\theta+a_{\theta^{-1}}$. As this dimension is
two, $a_\theta=1$ for every embedding and character; the comparison theorems
then give labeled weights $0,1$. Potential semistability comes from semistable
reduction and the semistable comparison theorem for smooth proper varieties
\cite{Tsuji}, and passes to character summands.

Outside a finite set $C_n$ has good reduction and the rotations extend to its
smooth proper model; smooth proper base change gives unramifiedness. The
projector onto the $\theta$-space is
$e_\theta=\tfrac1n\sum_{u\in\mu_n(F)}j_l(\theta(u))^{-1}u^*$, used only in
characteristic zero. The Lefschetz trace formula for a rotation composed with a
positive power of Frobenius shows the trace on $H^1(C_n)$ is independent of
$l$; applying the projector gives traces in $M$ compatible under the $j_l$.
Together with the known determinant this gives the common characteristic
polynomial.

Finally $H^1(C_n)$ is the first cohomology of its Jacobian, so Faltings'
semisimplicity theorem \cite[\S5, Satz~3]{Faltings} makes it a semisimple
$G_F$-representation. Extension of scalars and passage to a stable summand
preserve semisimplicity.
\end{proof}

\subsection{The two elliptic-curve realizations}

Characters of order one and three connect the systems to elliptic curves. For
the order-three character the involution produces a genus-one quotient rather
than only an abstract rank-two summand.

\begin{proposition}\label{sys:elliptic}
If $\theta$ has order $m>1$ dividing $n$, the system $V_\theta$ is realized by
the corresponding character space in $H^1(C_m)$. For the trivial character,
$V_{1,l}\simeq H^1_{\et}(B_{t,\overline F},\overline\Q_l)$. Let
$D_t=C_3(t)/\langle w\rangle$ and $A_t=\Jac(D_t)$. Then $D_t$ has genus one,
$A_t$ is an elliptic curve over $F$, and for either nontrivial order-three
character $\eta$ there are $G_F$-equivariant isomorphisms
\[
 V_{\eta,l}\simeq H^1_{\et}(A_{t,\overline F},\overline\Q_l)
 \qquad\text{for every }l.
\]
\end{proposition}

\begin{proof}
The quotient $C_n\to C_m$ sends $Y$ to $Y^{n/m}$. Pullback is injective on
rational cohomology since its composite with the trace is multiplication by the
degree, and its image is the fixed space of the kernel of $\mu_n\to\mu_m$; the
rank computation identifies the order-$m$ character summand with
$V_{\theta,l}$. The same argument for $C_n\to B_t$ identifies the trivial
character space.

On $H^1(C_3)$ the trivial rotation space is $H^1(B_t)$, where $w$ acts by
$[-1]^*=-1$, and $w$ exchanges the two nontrivial rotation spaces, each of
dimension two. Their direct sum therefore contains a two-dimensional
$w$-invariant subspace. Rational cohomology of a finite quotient is the
invariant space, so $H^1(D_t)\simeq H^1(C_3)^{w=1}$ has dimension two: $D_t$ has
genus one and its Jacobian $A_t$ is an elliptic curve, with or without an
$F$-rational point. Projection of $x+w^*x$ onto the $\eta$-space is $x$, so this
projection is a $G_F$-equivariant isomorphism from the invariant space to
$V_{\eta,l}$. Finally $H^1(D_t)\simeq H^1(\Jac(D_t))$ depends on no choice of
geometric base point, since translations act trivially on cohomology.
\end{proof}

\subsection{Evaluation of the integral group ring}

A character projector carries the denominator $n$ and cannot by itself give a
congruence at $3$ or $p$. Instead we use the integral freeness of
Proposition~\ref{mon:integral}: evaluating a free group-ring module at a
character yields a lattice before reduction, and two characters with equal
residual values then yield equal residual representations.

\begin{lemma}\label{sys:integral}
Fix $t\in U(K)$ and a prime $l$. Let $L/\Q_l$ be a finite extension containing
$j_l(M)$, with ring of integers $\cO$, maximal ideal $\mathfrak m$, and residue
field $k$. For a rotation character $\theta$ the module
\[
 \mathcal T_\theta
 =H^1_{\et}(C_{n,\overline F},\Z_l)\otimes_{\Z_l[\mu_n],\,j_l\theta}\cO
\]
is a free rank-two $G_F$-stable lattice realizing $V_{\theta,l}$ after
extension to $\overline\Q_l$. If $j_l\theta$ and $j_l\theta'$ have the same
reduction as characters $\mu_n\to k^\times$, then
$\mathcal T_\theta/\mathfrak m\mathcal T_\theta\simeq
\mathcal T_{\theta'}/\mathfrak m\mathcal T_{\theta'}$ as $k[G_F]$-modules.
\end{lemma}

\begin{proof}
By Proposition~\ref{mon:integral} and comparison,
$H^1_{\et}(C_{n,\overline F},\Z_l)\simeq\Z_l[\mu_n]^2$ as group-ring modules,
and the Galois action commutes with the group ring because the rotations are
defined over $F$. Tensoring by evaluation at $j_l\theta$ gives a free rank-two
$\cO$-module with continuous Galois action; in characteristic zero the group
algebra is semisimple and its quotient by character evaluation is the
corresponding summand, so the rational representation is $V_{\theta,l}$. After
reduction the tensor product becomes
$H^1_{\et}(C_{n,\overline F},\Z_l)\otimes_{\Z_l[\mu_n],\bar\theta}k$, which
depends only on the residual evaluation $\bar\theta$. This holds without
assuming $l$ prime to $n$.
\end{proof}

Choose a primitive character $\chi$ of order $n=3p$ with primary decomposition
$\chi=\psi\eta$, $\ord(\psi)=p$, $\ord(\eta)=3$. For a system $V_\theta$ write
$\overline V_{\theta,l}$ for the semisimplified reduction of a stable lattice,
scalars extended to $\overline\F_l$; this is independent of the lattice.

\begin{corollary}\label{sys:congruences}
For every $t\in U(K)$,
\begin{equation}\label{sys:chain}
 \overline V_{\eta,p}\simeq\overline V_{\chi,p},
 \qquad
 \overline V_{\chi,3}\simeq\overline V_{\psi,3},
 \qquad
 \overline V_{\psi,p}\simeq\overline V_{1,p},
\end{equation}
holding with the elliptic-curve realizations of Proposition~\ref{sys:elliptic}
at the two ends.
\end{corollary}

\begin{proof}
A root of unity of $l$-power order reduces to $1$ in characteristic $l$. Thus
$\chi$ and $\eta$ agree modulo $p$, $\chi$ and $\psi$ agree modulo $3$, and
$\psi$ and $1$ agree modulo $p$. Lemma~\ref{sys:integral} gives the three
congruences, and the rational isomorphisms with smaller covers and elliptic
curves in Proposition~\ref{sys:elliptic} preserve semisimplified reduction even
if their natural integral pullbacks have index divisible by $l$.
\end{proof}

The direction is now fixed: automorphy will pass from $V_\eta$ to $V_\chi$ at
$p$, to $V_\psi$ at $3$, and to $V_1$ at $p$. The congruences place no
condition on $t$; the local and residual arguments choose one $t$ for which all
three lifts apply.

\section{Automorphy inputs}\label{sec:inputs}

The congruences of Corollary~\ref{sys:congruences} tie the cohomology of the
auxiliary elliptic curve $A$ to that of the twist $B_t$ of $E$. To turn them
into automorphy we use two results of Caraiani--Newton, stated here in the
exact form needed, and then isolate what the parameter $t$ must still
satisfy. All references to \cite{CN} use the numbering of its third arXiv
version.

The residual hypothesis below uses the notion of decomposed genericity
introduced in Section~\ref{sec:conv}.

We also record the automorphic condition paired with potential ordinarity of
Galois representations in the convention of Section~\ref{sec:conv}. It is
called \emph{$\iota$-ordinarity of weight zero} and is the ordinary Hecke
condition used in \cite[Theorem~5.2]{CN}. It is imposed at a sufficiently deep
level, so it allows ramified finite-order characters. We obtain it only for
quadratic twists of Steinberg, via the local computation of
\cite[Lemma~6.1.3(4)]{CN}.

\begin{theorem}[Caraiani--Newton lifting, specialized form]\label{aut:lift}
Let $L$ be a CM number field and $l$ an odd prime other than $5$. Let
$r:G_L\to\GL_2(\overline\Q_l)$ be continuous, unramified outside finitely many
places, with determinant $\eps_l^{-1}$. Suppose that:
\begin{enumerate}[label=(\roman*)]
\item $r$ is potentially semistable with labeled Hodge--Tate weights $0,1$ at
every $v\mid l$;
\item $\overline r$ is decomposed generic;
\item $\overline r|_{G_{L(\zeta_l)}}$ is absolutely irreducible.
\end{enumerate}
Suppose there are a cuspidal weight-zero regular algebraic representation
$\pi$ of $\PGL_2(\A_L)$ and an isomorphism $\iota:\overline\Q_l\to\C$ with
$\overline{r_\iota(\pi)}\simeq\overline r$. At each $v\mid l$ assume one of:
\begin{enumerate}[label=(\arabic*)]
\item both local Galois representations are potentially crystalline; one is
potentially ordinary of weight zero if and only if the other is; and
$\rec_{L_v}(\pi_v)$ has zero monodromy;
\item neither local Galois representation is potentially crystalline, and
$\pi$ is $\iota$-ordinary of weight zero at $v$.
\end{enumerate}
Then $r\simeq r_\iota(\Pi)$ for a cuspidal weight-zero regular algebraic
representation $\Pi$ of $\PGL_2(\A_L)$.
\end{theorem}

This is the sufficient-hypothesis specialization of \cite[Theorem~5.2]{CN}.
Its inverse-cyclotomic determinant hypothesis permits $l=3$, including residual
image $\SL_2(\F_3)$ \cite[Remark~5.2.2]{CN}. No residual adequacy is assumed.
Our three applications below have $l=3$ or $l=p>100$.

\begin{theorem}[Elliptic starting point]\label{aut:seed}
Let $L$ be a CM number field not containing $\zeta_5$ and let $A/L$ be a non-CM
elliptic curve. If its mod-$5$ representation is decomposed generic and is
absolutely irreducible on $G_{L(\zeta_5)}$, then $H^1(A)$ is realized by a
cuspidal weight-zero regular algebraic representation of $\PGL_2(\A_L)$.
\end{theorem}

This is \cite[Theorem~6.1 and Lemma~6.1.3]{CN}, dualizing the Tate-module
convention there if necessary; absolute irreducibility and decomposed
genericity in dimension two are preserved by duality.

For the comparison after modularity of an elliptic curve is known, we use:

\begin{theorem}[Compatibility for modular elliptic curves]\label{aut:compat}
Let $D/L$ be a non-CM elliptic curve over a CM field $L$. Suppose a regular
algebraic cuspidal representation $\pi$ realizes $H^1(D)$ at one coefficient
prime. Then $\pi$ has weight zero and trivial central character, realizes
$H^1(D)$ at every coefficient prime and every coefficient isomorphism, and
\[
 \iota\WD\bigl(H^1(D,\overline\Q_l)|_{G_{L_v}}\bigr)^{\Fss}
 \simeq\rec^T_{L_v}(\pi_v)\qquad(v\nmid l).
\]
If $D$ has potentially multiplicative reduction at $v\mid l$, then $\pi$ is
$\iota$-ordinary of weight zero there.
\end{theorem}

This is \cite[Lemma~6.1.3]{CN}. For the intermediate systems, which need not
come from elliptic curves, we use the general existence and good-place
compatibility of $r_\iota(\pi)$ from \cite{HLTT,Scholze}, together with Varma's
equality of the underlying semisimplified Weil representations at $v\nmid l$
\cite[Theorem~1]{Varma}. The local analysis of Section~\ref{sec:local} supplies
the extra facts needed in Theorem~\ref{aut:lift}.

We can now state what remains for a single parameter $t\in U(K)$. By
Proposition~\ref{sys:properties}, the determinant and Hodge--Tate conditions
hold for every $t$. At places above $3p$ we will make the Jacobian of
$C_n(t)$ potentially good ordinary, potentially good with all Newton slopes
$1/2$, or potentially totally toric, according to the reduction type of $E$.
This makes all four systems respectively potentially crystalline and
potentially ordinary, potentially crystalline but not potentially ordinary, or
not potentially crystalline. Section~\ref{sec:local} obtains these from
nonempty local open conditions on $t$. Section~\ref{sec:res} then picks a
parameter in those opens for which the residual hypotheses hold. The pairs
$(m,l)=(3,5),(3,p),(p,3)$ give the elliptic starting point and the first two
lifts; the last lift uses the large mod-$p$ image of $E$. The two conjugates of
$K$ must be treated together to get decomposed genericity. Once these
properties hold, Section~\ref{sec:auto} carries automorphy along the three
congruences and descends to $K$.

\section{Prescribing the local reduction type}\label{sec:local}

The degeneration at the origin lets us impose the same local behavior on every
character summand of the covering Jacobian. We first explain why the reduction
type of an abelian variety is controlled, locally in a characteristic-zero
family, by a sufficiently large finite torsion module, and then apply this to
the Jacobian family and translate the reduction into conditions on the
two-dimensional systems.

\subsection{Finite torsion controls the reduction}

Throughout, a local field is a finite extension of $\Q_\ell$; here $\ell$ is
the residue characteristic of the base field, and $d$ is an auxiliary prime
different from $\ell$. For an abelian variety $A$ write
$V_d(A)=T_d(A)\otimes_{\Z_d}\Q_d$.

\begin{lemma}[Local constancy of finite torsion]\label{loc:torsion}
Let $L$ be a local field, $X$ a smooth $L$-variety, $\mathcal A\to X$ an
abelian scheme. For $x_0\in X(L)$ and $N\ge1$ there is an analytic
neighborhood $\mathcal N$ of $x_0$ in $X(L)$ such that
$\mathcal A_x[N](\overline L)\simeq\mathcal A_{x_0}[N](\overline L)$ as
$G_L$-modules for all $x\in\mathcal N$.
\end{lemma}

\begin{proof}
Since $X$ has characteristic zero, $\mathcal A[N]$ is finite \'etale over
$X$; equivalently, full level-$N$ structures form a finite \'etale torsor
under the constant group $\GL_{2g}(\Z/N\Z)$, with $g$ the relative dimension.
Take a finite Galois extension $L'/L$ splitting the fiber at $x_0$. The inverse
function theorem for an \'etale morphism gives, through each point of the fiber,
a local section over a neighborhood of $x_0$ in $X(L')$; shrinking, these
sections enumerate every point of every nearby fiber. Intersecting finitely
many Galois conjugates and shrinking again keeps the neighborhood Galois-stable
and preserves the Galois permutations; the level-group action is preserved the
same way. Restricting to $X(L)$ identifies the fibers as Galois sets with
level-group action, hence the torsion modules.
\end{proof}

Finite torsion will first force semistability, then identify a Frobenius
polynomial. For the second step we use the \emph{full} Tate module, which
keeps both toric and lattice parts of the semistable representation.

\begin{lemma}[Finitely many semistable Frobenius polynomials]\label{loc:frob}
Fix a local field $L$ with residue field of cardinality $Q$, an integer
$g\ge1$, and a prime $d$ different from the residue characteristic. For a
semistable abelian variety $A/L$ of dimension $g$ let
$P_A(T)=\det(T-\Frob_L\mid V_d(A))$ for any lift of arithmetic Frobenius.
Then $P_A(T)\in\Z[T]$, is independent of the lift and of $d$, and ranges over
a finite set depending only on $g,Q$. It determines the toric rank of $A$; if
that rank is zero it also determines the Newton polygon of the good reduction.
\end{lemma}

\begin{proof}
Let $r$ be the toric rank. The semistable filtration identifies the graded
pieces of $V_d(A)$ with the rational Tate module of a rank-$r$ torus, the
rational Tate module of a $(g-r)$-dimensional abelian variety with good
reduction, and a rank-$r$ lattice \cite[Expos\'e IX]{SGA7I}. The character
lattice of the torus and the lattice quotient act through finite unramified
characters, so the arithmetic Frobenius eigenvalues on those pieces are $Q$
times roots of unity and roots of unity respectively; the middle piece has the
$2(g-r)$ eigenvalues of an abelian variety over $\F_Q$, all of complex absolute
value $Q^{1/2}$.

Each graded piece has an integral characteristic polynomial independent of
$d$: for the toric and lattice pieces this follows from the finite-order
integral actions and the toric Tate twist, for the abelian piece it is the
usual finite-field Frobenius polynomial. Inertia acts trivially on the graded
pieces, so the product is independent of the Frobenius lift. Every root has
absolute value at most $Q$, so the coefficient of $T^{2g-j}$ has absolute value
at most $\binom{2g}{j}Q^j$; integral polynomials of fixed degree with these
bounds are finite in number.

Exactly $r$ roots have absolute value $1$, exactly $r$ have absolute value
$Q$, and the rest have $Q^{1/2}$; as $Q>1$ these three values distinguish $r$.
If $r=0$, semistability is good reduction and the normalized valuations of the
roots of $P_A$ are the Newton slopes.
\end{proof}

\begin{proposition}[Local constancy of the semistable type]\label{loc:const}
Let $L_0$ be a local field, $X$ a smooth $L_0$-variety, $\mathcal A\to X$ an
abelian scheme of relative dimension $g$. For $x_0\in X(L_0)$ there exist a
finite extension $L/L_0$ and an analytic neighborhood $\mathcal N\subset
X(L_0)$ of $x_0$ such that for every $x\in\mathcal N$, $\mathcal A_x$ is
semistable over $L$ with the same toric rank as $\mathcal A_{x_0}$ over $L$,
and if $\mathcal A_{x_0}$ has good reduction over $L$ then $\mathcal A_x$ has
the same Newton polygon over $L$.

In particular, if $\mathcal A_{x_0}\simeq E_0^g$, every nearby fiber has, after
the same finite extension, the same one of: good ordinary reduction; good
reduction with all Newton slopes $1/2$; semistable reduction of toric rank $g$.
\end{proposition}

\begin{proof}
Choose an odd prime $d$ different from the residue characteristic and a finite
$L/L_0$ over which $\mathcal A_{x_0}$ is semistable with rational $d$-torsion.
Lemma~\ref{loc:torsion} after base change to $L$ shows nearby fibers also have
rational $d$-torsion over $L$.

Nearby fibers are semistable over $L$. Indeed, an inertia element acts on their
$d$-adic Tate module as $1+dB$ with $B$ integral, so each eigenvalue $\lambda$
satisfies $v_d(\lambda-1)\ge1$, $v_d(d)=1$. By the local monodromy theorem
$\lambda$ is a root of unity; for odd $d$ the only such root of unity with
$v_d(\lambda-1)\ge1$ is $1$ (a root with nontrivial prime-to-$d$ part has
$v_d(\lambda-1)=0$, and a primitive $d^a$th root has
$v_d(\lambda-1)=1/(d^{a-1}(d-1))<1$). Thus inertia acts unipotently and
Grothendieck's criterion gives semistability \cite[Expos\'e IX]{SGA7I}.

Let $\mathcal P$ be the finite set of Lemma~\ref{loc:frob} for dimension $g$
and residue cardinality of $L$; choose $j$ with reduction mod $d^j$ injective
on $\mathcal P$. A second application of Lemma~\ref{loc:torsion} gives, on a
smaller neighborhood, $\mathcal A_x[d^j](\overline L)\simeq
\mathcal A_{x_0}[d^j](\overline L)$ as $G_L$-modules, so the full Frobenius
polynomials are congruent mod $d^j$ and hence equal. Lemma~\ref{loc:frob} now
gives the assertions on toric rank and Newton polygon; pull the neighborhood
back to $X(L_0)$.

For the final assertion, choose $L$ so $E_0$ has good or split multiplicative
reduction. In the good case $E_0$ is ordinary or supersingular, the latter with
slopes $1/2$; in the multiplicative case $E_0^g$ has toric rank $g$. Enlarging
$L$ to rationalize $d$-torsion preserves these, so the argument applies.
\end{proof}

\subsection{Monodromy of a totally toric abelian variety}

The third possibility of Proposition~\ref{loc:const} must force nonzero
monodromy on \emph{each} character summand, including when the coefficient
prime is the residue characteristic. The needed rank statement is a
consequence of the valuation pairing in toric uniformization, and we treat the
residue characteristic as well, through Kummer extensions.

\begin{lemma}\label{loc:toric}
Let $L/\Q_\ell$ be finite and $A/L$ an abelian variety of dimension $g$ that
acquires semistable reduction of toric rank $g$ after a finite extension. For
every prime $d$ the monodromy operator on the Weil--Deligne representation of
$H^1_{\et}(A_{\overline L},\Q_d)$ has rank $g$; for $d=\ell$ the
Weil--Deligne representation is the one attached to the potentially semistable
representation by $\ell$-adic Hodge theory.
\end{lemma}

\begin{proof}
Monodromy rank is unchanged by finite extension, so assume $A$ has split toric
uniformization $A^{\mathrm{an}}\simeq(\Gm^g)^{\mathrm{an}}/\Lambda$ with
$\Lambda\simeq\Z^g$ rational over $L$. In bases the periods form a matrix
$(q_{ij})\in(L^\times)^{g\times g}$, and the valuation pairing is nondegenerate:
\begin{equation}\label{loc:val}
 \bigl(v_L(q_{ij})\bigr)_{1\le i,j\le g}\ \text{is nonsingular.}
\end{equation}
This is the toric case of the monodromy pairing \cite[Expos\'e
IX]{SGA7I}; its de Rham realization is in \cite{ColemanIovita}.

For $d\ne\ell$ the same pairing describes the monodromy map from the lattice
quotient to the toric subspace, and nondegeneracy gives rank $g$. We prove rank
$g$ for $d=\ell$. Toric uniformization gives an exact sequence
\begin{equation}\label{loc:tate}
 0\longrightarrow\Q_\ell(1)^g\longrightarrow V_\ell(A)
   \longrightarrow\Q_\ell^g\longrightarrow0,
\end{equation}
whose extension classes are the Kummer classes of the $q_{ij}$; it is
semistable by \cite{ColemanIovita}. Let $L_{\mathrm{ur}}$ be the maximal
unramified subfield and $D_{\mathrm{st}}$ Fontaine's functor. Monodromy is zero
on the two outer terms, so it induces $\nu:D_{\mathrm{st}}(\Q_\ell^g)\to
D_{\mathrm{st}}(\Q_\ell(1)^g)$. Frobenius on the two spaces is $\sigma$ and
$\ell^{-1}\sigma$ with $\sigma$ the Frobenius of $L_{\mathrm{ur}}/\Q_\ell$, so
$N\varphi=\ell\varphi N$ makes $\ker\nu\subset L_{\mathrm{ur}}^g$
$\sigma$-stable.

Suppose $\nu$ has rank $<g$. Galois descent gives nonzero
$a=(a_j)\in\Q_\ell^g\cap\ker\nu$. Pull \eqref{loc:tate} back along the line
spanned by $a$ in its trivial quotient: the result is a semistable extension of
$\Q_\ell$ by $\Q_\ell(1)^g$ with zero monodromy, hence crystalline. By
Kummer theory the Bloch--Kato crystalline subspace $H_f^1(L,\Q_\ell(1))$ is the
kernel of the valuation map \cite[Example~3.9]{BlochKato}, and applying this in
each coordinate gives $\sum_j a_j v_L(q_{ij})=0$ for $1\le i\le g$, contrary to
\eqref{loc:val}. Hence $N$ has rank $g$ on $V_\ell(A)$, and dualizing gives rank
$g$ on $H^1_{\et}(A_{\overline L},\Q_\ell)$.
\end{proof}

\subsection{Open conditions on the original parameter}

Return to $E/K$, the odd integer $n=3p$, and $F=K(\mu_n)$. By
Proposition~\ref{geom:compact} the Jacobian of the constant-$E$ cover $C_n$
extends over a neighborhood of $O$ in the parameter curve $E$, with fiber
$E^n$ at $O$, so Proposition~\ref{loc:const} applies at every finite completion
of $F$. The next statement re-expresses the resulting neighborhoods in the
original parameter $t\in U\subset\A^1_K$, to combine with Hilbert
irreducibility.

\begin{proposition}\label{loc:opens}
For every place $v_0\mid3p$ of $K$ there is a nonempty open
$\Omega_{v_0}\subset U(K_{v_0})$ with the following property. Suppose
$t\in U(K)$ lies in $\Omega_{v_0}$ for all $v_0\mid3p$, and set
$J_t=\Jac(C_{n,t})$. At each place $v\mid3p$ of $F$, $J_t$ has the same one of
the following potential reduction types as $E^n$:
\begin{enumerate}[label=(\arabic*)]
\item good ordinary reduction;
\item good reduction with all Newton slopes $1/2$;
\item semistable reduction of toric rank $n$.
\end{enumerate}
For each $v$ the extensions giving these properties may be chosen in advance,
independently of $t$ in the opens.
\end{proposition}

\begin{proof}
Fix $v_0\mid3p$. For each $v\mid v_0$ apply Proposition~\ref{loc:const} to the
extended Jacobian family over $F_v$ at the point $O$ of its parameter curve,
obtaining a neighborhood of $O$ in $E(F_v)$ and a fixed finite extension of
$F_v$ on which the reduction type holds. The inverse images of these finitely
many neighborhoods in $E(K_{v_0})$ intersect in a neighborhood of $O$.

Choose $R_0\ne O$ in this intersection, close to $O$ and avoiding the finitely
many excluded parameters and the ramification points of $x:E\to\Pbf^1$. The
inverse function theorem gives a nonempty neighborhood
$\Omega_{v_0}\subset U(K_{v_0})$ of $x(R_0)$ and an analytic lift
$R(t)=(t,y(t))\in E(K_{v_0})$, $y(t)^2=h(t)$, in the chosen intersection;
shrink so $y(t)\ne0$ and all excluded parameters are avoided.

The map $B_t\to E$, $(x,y)\mapsto(x,y(t)y)$, is an isomorphism over
$K_{v_0}$ carrying $(t,1)$ to $R(t)$; it sends the defining function of $C_n(t)$
to that of the constant-$E$ cover (their divisors agree and both are $1$ at
$O$), so it identifies the covers and Jacobians at every $v\mid v_0$. The
reduction conclusions follow from the choice of neighborhoods; do this
independently at the finitely many $v_0\mid3p$.
\end{proof}

\subsection{Local representations of the character summands}

For a prime $\ell$ and rotation character $\theta$, let $V_{\theta,\ell}$ be
the $\ell$-adic realization of Proposition~\ref{sys:properties}. Recall from
Section~\ref{sec:conv} that an ordinary filtration of weight zero has an
unramified character on its line and $\eps_\ell^{-1}$ times an unramified
character on its quotient, after a finite extension; in the ordinary case below
the line is preserved already before the extension.

\begin{corollary}\label{loc:chars}
Choose $t$ satisfying Proposition~\ref{loc:opens}. Let $v\mid3p$ be a place of
$F$ of residue characteristic $\ell$. Then simultaneously for every rotation
character $\theta$:
\begin{enumerate}[label=(\arabic*)]
\item if $E$ has potentially good ordinary reduction at $v$, then
$V_{\theta,\ell}|_{G_{F_v}}$ is potentially crystalline and potentially
ordinary of weight $0$;
\item if $E$ has potentially good supersingular reduction at $v$, then
$V_{\theta,\ell}|_{G_{F_v}}$ is potentially crystalline and not potentially
ordinary of weight $0$;
\item if $E$ has potentially multiplicative reduction at $v$, then the monodromy
of $V_{\theta,d}|_{G_{F_v}}$ has rank one for every coefficient prime $d$,
including $d=\ell$; in particular $V_{\theta,\ell}|_{G_{F_v}}$ is not
potentially crystalline.
\end{enumerate}
\end{corollary}

\begin{proof}
Choose a finite $L/F_v$ on which $J_t$ has the reduction of
Proposition~\ref{loc:opens}. In the two good-reduction cases, crystalline
comparison makes $H^1_{\et}(J_{t,\overline L},\Q_\ell)$, and each character
summand, crystalline over $L$.

If the reduction is ordinary, the connected--\'etale sequence of the ordinary
$\ell$-divisible group, dualized to $H^1$, gives a filtration with an
unramified sub and a quotient that is $\eps_\ell^{-1}$ times unramified. This
filtration is functorial, hence stable under the rotations, and restricts to
each $V_{\theta,\ell}$. By Proposition~\ref{sys:properties} this summand has
dimension two and labeled Hodge--Tate weights $0,1$; the unramified part is
weight $0$ and the quotient weight $1$, so both parts are one-dimensional,
giving the required ordinary filtration. After a finite Galois extension, the
weight-zero line is determined by its inertia action, since $\eps_\ell$ has
infinite image on inertia; conjugation by $G_{F_v}$ preserves it, so it descends
to the original decomposition group.

If all slopes are $1/2$, every summand has that slope, and slopes are
unchanged by finite extension. An ordinary filtration of weight $0$ would give
a crystalline rank-one subrepresentation of parallel Hodge--Tate weight zero,
whose slope is forced to be zero by weak admissibility, contradiction. So no
potential ordinary filtration exists.

If $J_t$ has potential toric rank $n$, Lemma~\ref{loc:toric} gives monodromy of
rank $n$ on the $2n$-dimensional $H^1$ at every coefficient prime. Monodromy
commutes with rotations, so preserves the $n$ two-dimensional summands; since
$N^2=0$, each summand has rank at most one, the ranks sum to $n$, and there are
$n$ summands, so every rank is one. Nonzero monodromy survives finite extension
and excludes potential crystallinity.
\end{proof}

\section{A single parameter with all residual properties}\label{sec:res}

We now choose one parameter for the whole congruence chain. The geometric
monodromy gives irreducibility and suitable Frobenius elements; Hilbert
irreducibility lets us keep them after imposing the local conditions of
Section~\ref{sec:local}. To get decomposed genericity we must control every
place above a split rational prime, so we specialize the two conjugate
parameter families together.

Throughout, $K,E,p,n,F,U$ are as in Section~\ref{sec:sys}, so $p>100$, $n=3p$,
$F=K(\mu_n)$. The pairs of covering degree and residual characteristic needing
geometric monodromy are
\begin{equation}\label{res:pairs}
 (m,\ell)=(3,p),\quad(3,5),\quad(p,3).
\end{equation}
In each pair $\ell\nmid m$, so residual cohomology decomposes into
rotation-character spaces after extending its finite coefficient field.

\subsection{Consequences of the two monodromy transformations}

\begin{lemma}\label{res:geom}
For each pair in \eqref{res:pairs} the geometric monodromy of
$H^1(C_m,\overline\F_\ell)$ on every nontrivial rotation-character space is
absolutely irreducible, its image contains opposite nonidentity root
unipotents on that space, and one element of the geometric monodromy acts
regular semisimply on all these nontrivial character spaces simultaneously.
These statements also hold for the conjugate family over $K$.
\end{lemma}

\begin{proof}
Work first with homology; duality transfers the conclusions. On the character
space with generator eigenvalue $\lambda\in\overline\F_\ell^\times$ the
transformations of Theorem~\ref{mon:transvections} are
\[
 T_a=\begin{pmatrix}1&\sigma c_\lambda\\0&1\end{pmatrix},\qquad
 T_b=\begin{pmatrix}1&0\\\tau c_\lambda&1\end{pmatrix},\qquad
 c_\lambda=2-\lambda-\lambda^{-1},
\]
with $\sigma,\tau\in\{1,-1\}$. Since $c_\lambda=-(\lambda-1)^2/\lambda$, it is
nonzero when $\lambda\ne1$. The two matrices have different unique invariant
lines, even over the algebraic closure, so their common representation is
absolutely irreducible.

The product $T_aT_b^k$ has determinant one and trace
\begin{equation}\label{res:trace}
 2+\sigma\tau\,k\,c_\lambda^2.
\end{equation}
In odd characteristic a determinant-one matrix is regular semisimple exactly
when its trace is neither $2$ nor $-2$. If $m=3$ then
$\lambda+\lambda^{-1}=-1$, so $c_\lambda=3$ for both nontrivial characters; take
$k=2$, so the two possible traces in \eqref{res:trace} are $20$ and $-16$.
Modulo $5$ these are $0$ and $4$, and modulo $p>100$ they are $20$ and $-16$;
in neither case do they equal $\pm2$.

For $m=p$ and $\ell=3$ take $k=1$; failure of regular semisimplicity would
force $c_\lambda^2=4$ or $-4$, since $c_\lambda\ne0$ and the excluded traces are
$\pm2$. Over $\F_3$ the elements $4$ and $-4$ are $\pm1$, so
$c_\lambda^2=\pm1$; equivalently $c_\lambda=\pm1$ or $\pm i$. Multiplying
$c_\lambda=2-\lambda-\lambda^{-1}$ through by $\lambda^2$ gives, in each case, a
degree-four polynomial over $\F_3$ satisfied by $\lambda$. Hence
$\lambda$ has degree $d\le4$ over $\F_3$ and its order divides
$3^d-1\le80$, contradicting the fact that its order is the prime $p>100$.
Thus the same product works in this case too.

The transformations come from loops in the constant-elliptic-curve parameter
space, which by Theorem~\ref{mon:transvections} is a pullback of the family over
$U$; its monodromy image lies in the geometric monodromy over $U$. Duality
preserves absolute irreducibility, nonidentity unipotents, and regular
semisimplicity, giving the cohomological claims. The geometric argument works
for every complex elliptic curve, hence for the conjugate of $E$.
\end{proof}

\subsection{The original elliptic curve and its conjugate}

The trivial character needs a separate argument, because the geometric family
varies only a quadratic twist of $E$; its large residual image was built into
$p$. The count below needs no independence hypothesis between $E$ and its
conjugate.

\begin{lemma}\label{res:triv}
For every $t\in U(K)$ the representation $\overline V_{1,p}$ is absolutely
irreducible on $G_{F(\zeta_p)}$ and decomposed generic over $F$.
\end{lemma}

\begin{proof}
Let $\Gamma$ be the image of $G_K$ on $E[p]$. Since $F/K$ is abelian, the image
of $G_F$ contains $[\Gamma,\Gamma]$. As $\Gamma$ contains $\SL_2(\F_p)$ and this
group is perfect for $p>3$, its commutator subgroup is $\SL_2(\F_p)$ (one
inclusion from perfectness, the other from the determinant). The determinant is
trivial on $G_F$ because $\mu_p\subset F$, so
\begin{equation}\label{res:SL2}
 \rho_{E,p}(G_F)=\SL_2(\F_p).
\end{equation}
The same equality holds for the conjugate $E^c$, $c$ the nontrivial automorphism
of $K/\Q$, because $F/\Q$ is normal.

Consider the joint image $J\subset\SL_2(\F_p)\times\SL_2(\F_p)$ of $G_F$ on
$E[p]$ and $E^c[p]$; both projections are surjective. In $\SL_2(\F_p)$ there
are exactly $p^2$ matrices of trace $2$: writing
$\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)$, trace and
determinant give $d=2-a$ and $bc=-(a-1)^2$; $a=1$ yields $2p-1$ choices of
$(b,c)$, and each of the other $p-1$ values of $a$ yields $p-1$ choices, for a
total $p^2$. Negating gives the same count for trace $-2$.

The order of $\SL_2(\F_p)$ is $p(p^2-1)$, and each projection of $J$ has
equal-sized fibers, so the proportion of $J$ whose image in at least one factor
has trace $\pm2$ is at most
\[
 2\cdot\frac{2p^2}{p(p^2-1)}=\frac{4p}{p^2-1}<1.
\]
Hence some element of $G_F$ acts regular semisimply on both $E[p]$ and
$E^c[p]$.

Take a finite Galois extension of $\Q$ containing $F$ and the fields of these
two residual actions. Conjugation by $G_\Q$ either preserves or exchanges the
two curves and conjugates their residual matrices, so the chosen element and all
its conjugates act regular semisimply on the representation of $E$. Chebotarev
gives rational primes $q$ splitting completely in $F$ with this property at
every $v\mid q$ of $F$. Excluding $p$ and the finitely many bad primes and using
$\mu_p\subset F$, these satisfy $q\equiv1\pmod p$; distinct Frobenius
eigenvalues give the ratio condition of decomposed genericity.

The curve $B_t$ is a quadratic twist of $E$; twisting multiplies residual
matrices by scalar signs, preserving absolute irreducibility and eigenvalue
ratios. Excluding the finitely many primes ramified in this twist leaves the
same Chebotarev argument. Since $\zeta_p\in F$ and the dual of $E[p]$ gives the
residual $H^1$-realization, Proposition~\ref{sys:elliptic} proves both claims for
$\overline V_{1,p}$.
\end{proof}

\subsection{Simultaneous specialization}

We have the required geometric elements and the trivial character is handled.
It remains to keep the geometric elements at one parameter while arranging their
Frobenius properties above a rational prime; restriction of scalars puts the two
conjugate families in one finite cover.

\begin{lemma}\label{res:finite}
Let $W$ be a nonempty open subset of $\A^d_\Q$, $D/\Q$ finite Galois, and $H$ a
finite quotient of $\pi_1(W)$ through which the constant-field map factors as
$\pi_1(W)\surj H\surj\Gal(D/\Q)$. Prescribe nonempty open subsets of $W(\Q_r)$
for finitely many rational primes $r$. Then there is $x\in W(\Q)$ in all these
opens for which the specialization $s_x:G_\Q\to H$ is surjective; for this $x$,
\begin{equation}\label{res:kernel}
 s_x(G_D)=\ker\bigl(H\to\Gal(D/\Q)\bigr).
\end{equation}
\end{lemma}

\begin{proof}
The quotient $H$ defines a connected finite \'etale Galois cover of $W$.
Hilbert irreducibility with weak approximation gives a rational point in the
prescribed opens whose fiber is connected \cite[Proposition~3.3.1 and
Theorem~3.5.3]{SerreHilbert}. Connectedness of the Galois fiber is equivalent to
surjectivity of $s_x$ (geometric connectedness is not required). The composite
of $s_x$ with $H\to\Gal(D/\Q)$ is the restriction map on absolute Galois
groups, so an element of the displayed kernel has a preimage in $G_D$.
\end{proof}

\begin{proposition}\label{res:special}
Let $S_K$ be a finite set of finite places of $K$ containing all places above
$3p$ and, for each $v_0\in S_K$, prescribe a nonempty open
$\Omega_{v_0}\subset U(K_{v_0})$. There is a single $t\in U(K)$ in every
$\Omega_{v_0}$ such that, over $F=K(\mu_{3p})$:
\begin{enumerate}[label=(\arabic*)]
\item for each $(m,\ell)$ in \eqref{res:pairs} and each character $\theta$ of
order $m$, the representation $\overline V_{\theta,\ell}$ is absolutely
irreducible on $G_{F(\zeta_\ell)}$ and decomposed generic;
\item the representation $\overline V_{1,p}$ has the same two properties;
\item the elliptic curve $A_t=\Jac(C_3(t)/\langle w\rangle)$ is non-CM.
\end{enumerate}
In particular, with the opens above $3p$ taken from Proposition~\ref{loc:opens},
this $t$ also satisfies all conclusions of Corollary~\ref{loc:chars}.
\end{proposition}

\begin{proof}
Put $W=\Res_{K/\Q}U$, an open subset of $\A^2_\Q$ since $U\subset\A^1_K$. Choose
finite Galois $D/\Q$ containing $F$ and $\zeta_\ell$ for every residual
characteristic in \eqref{res:pairs}. Let $c$ be the nontrivial automorphism of
$K/\Q$ and write $U^c$, $\cC_m^c$ for the conjugate parameter open and family.
Then $W_D\simeq U_D\times(U^c)_D$. For all three pairs pull back the finite
local system $H^1(C_m,\F_\ell)$ from the first factor and $H^1(C_m^c,\F_\ell)$
from the second.

Their joint monodromy has finite image on $\pi_1(W_D)$. Take, in $\pi_1(W)$, the
normal core of the common kernel; it is open normal, contained in $\pi_1(W_D)$,
and the resulting finite quotient $H$ records every listed action and has
quotient $\Gal(D/\Q)$.

Over $\C$ a loop in one parameter factor acts trivially on local systems pulled
back from the other, so the geometric monodromy of Lemma~\ref{res:geom} is
available independently in both factors. For each pair $(m,\ell)$ choose an
element $h_{m,\ell}$ of the geometric monodromy image that is regular semisimple
on every nontrivial character space of both factors (product of the two factor
loops from the lemma); it maps trivially to $\Gal(D/\Q)$. The quotient also
contains the individual opposite unipotents on every character space.

The local conditions are compatible with restriction of scalars: for a prime
$r$, $W(\Q_r)=\prod_{v_0\mid r}U(K_{v_0})$, so the $\Omega_{v_0}$ give
nonempty open subsets of $W(\Q_r)$ for finitely many $r$ (at unprescribed
factors impose no condition). Apply Lemma~\ref{res:finite}: it gives
$x\in W(\Q)$, i.e.\ $t\in U(K)$, satisfying all local conditions and inducing a
surjection $G_\Q\surj H$. Over $D$ this point has coordinates $(t,t^c)$; smooth
proper base change identifies the specialized local systems with the residual
cohomology of $C_m(t)$ and $C_m^c(t^c)$.

By \eqref{res:kernel} every geometric monodromy element used above is the image
of an element of $G_D$. The opposite unipotents therefore survive on each
specialized character space over $G_D$, proving its absolute irreducibility; as
$G_D\subset G_{F(\zeta_\ell)}$, this also holds on the latter group. Because
$\ell\nmid m$, taking rotation eigenspaces commutes with reduction after
adjoining the character values, so these spaces are reductions of
$V_{\theta,\ell}$; the rational identifications and independence of
semisimplified reduction of Section~\ref{sec:sys} show they give exactly
$\overline V_{\theta,\ell}$, even when the system is defined using $C_n$.

For decomposed genericity fix $(m,\ell)$ and a preimage $g\in G_D$ of
$h_{m,\ell}$. An element $\sigma\in G_\Q$ preserves or exchanges $C_m(t)$ and
$C_m^c(t^c)$ according to its restriction to $K$, and permutes rotation characters
through its action on roots of unity; so $\sigma g\sigma^{-1}$ on a character
space of the first cover is conjugate to $g$ on some character space of one of
the two covers. Regular semisimplicity was imposed on all these spaces, so it
holds for every conjugate.

The kernel of the surjection $G_\Q\surj H$ defines a finite Galois extension of
$\Q$ containing $D$. Chebotarev applied to the conjugacy class of $h_{m,\ell}$
gives rational primes $q$ whose Frobenius class is that class; they split
completely in $D$ because $h_{m,\ell}$ is trivial on $D$. Discard $q=\ell$ and
the finitely many primes where curves or residual representations ramify. For
every $v\mid q$ of $F$ the Frobenius is represented by a conjugate of the
selected element and acts regular semisimply on each character space; the
residue degree is one since $q$ splits in $F$, and $q\equiv1\pmod\ell$ since
$\zeta_\ell\in D$. Hence the ratio of the two Frobenius eigenvalues differs from
$q$ in characteristic $\ell$, at every $v\mid q$: this is decomposed genericity.
A different $q$ may be used for each pair. Assertion (2) follows from
Lemma~\ref{res:triv}.

To exclude CM for $A_t$: for $(m,\ell)=(3,p)$ the specialization places two
opposite nonidentity unipotents in the residual image of $V_{\eta,p}$, which by
Proposition~\ref{sys:elliptic} is the residual cohomology of $A_t$ (both are
irreducible, so their common semisimplification is the representation itself).
If $A_t$ had CM, all its endomorphisms would be defined over an extension of $F$
of degree at most two; over that extension the characteristic-zero Galois image
lies in the commutative centralizer of the CM algebra, so its reduction is
abelian, and the full residual image would have an abelian normal subgroup of
index at most two. Each opposite unipotent has odd prime order $p$, so both
would lie in that abelian subgroup; but
\[
 \begin{pmatrix}1&a\\0&1\end{pmatrix}\begin{pmatrix}1&0\\b&1\end{pmatrix}
 \ne\begin{pmatrix}1&0\\b&1\end{pmatrix}\begin{pmatrix}1&a\\0&1\end{pmatrix}
 \qquad(ab\ne0),
\]
a contradiction. This proves (3). Finally, choosing the prescribed opens as in
Proposition~\ref{loc:opens}, the Hilbert specialization was made inside them, so
the local conclusions of Corollary~\ref{loc:chars} also hold.
\end{proof}

\section{Propagation of automorphy, descent, and the CM case}\label{sec:auto}

Fix $t\in U(K)$ from Proposition~\ref{res:special}, taking the local opens of
Proposition~\ref{loc:opens}, and write $B=B_t$. The four systems
$V_\eta,V_\chi,V_\psi,V_1$ have the congruences of
Corollary~\ref{sys:congruences}, the local properties of
Corollary~\ref{loc:chars}, and the residual properties of
Section~\ref{sec:res}. We first explain how automorphy at one coefficient prime
gives what is needed at the next.

\subsection{Coefficient primes and local automorphic conditions}

\begin{lemma}\label{auto:transport}
Suppose a weight-zero regular algebraic cuspidal representation $\pi$ of
$\GL_2(\A_F)$ realizes one member of the compatible system $V_\theta$. Then it
realizes every member, with the coefficient embeddings fixed in
Section~\ref{sec:sys}.
\end{lemma}

\begin{proof}
Outside a finite set and away from the coefficient prime, the characteristic
polynomial of Frobenius on $V_\theta$ is independent of that prime, as an
algebraic polynomial with the fixed embeddings of character values; the
good-place characteristic polynomials of $r_\iota(\pi)$ have the same
compatibility. Equality at the original prime identifies the polynomials at
every other prime. Both representations are semisimple: for $V_\theta$ this is
Faltings' theorem as in Proposition~\ref{sys:properties}, and automorphic Galois
representations are semisimple. Chebotarev and Brauer--Nesbitt give the
isomorphism.
\end{proof}

The next lift also needs the local automorphic component at the new prime. We
recover it by applying Varma's comparison at a different coefficient prime,
where it applies directly.

\begin{lemma}\label{auto:autolocal}
Suppose a weight-zero regular algebraic cuspidal representation $\pi$ of
$\PGL_2(\A_F)$ realizes $V_\theta$. Let $l\in\{3,p\}$ and $v\mid l$, with
$J_t=\Jac(C_n(t))$.
\begin{enumerate}[label=(\roman*)]
\item If $J_t$ has potentially good reduction at $v$, then
$\rec_{F_v}(\pi_v)$ has zero monodromy.
\item If $J_t$ has semistable reduction of toric rank $n$ at $v$, then $\pi_v$
is a quadratic or trivial twist of Steinberg and $\pi$ is $\iota_l$-ordinary of
weight zero at $v$.
\end{enumerate}
\end{lemma}

\begin{proof}
Choose a coefficient prime $a$ different from the residue characteristic $l$.
Lemma~\ref{auto:transport} identifies $r_{\iota_a}(\pi)$ with $V_{\theta,a}$.
Varma's local-global compatibility identifies the underlying semisimplified
Weil representations of its local Weil--Deligne parameter and
$\rec^T_{F_v}(\pi_v)$ \cite[Theorem~1]{Varma}; we use this for the Weil
representations first, not assuming equality of monodromy operators.

In the potentially good case, pass to an extension where the Jacobian has good
reduction. All Frobenius eigenvalues on its $H^1$, hence on the character
summand, have complex absolute value the square root of the residue
cardinality; taking a positive power of Frobenius shows this also before the
extension. A rank-two Weil--Deligne representation with nonzero monodromy has
Frobenius eigenvalues differing by the norm, which cannot have equal absolute
value, so the automorphic parameter has zero monodromy. The Tate twist in
\eqref{eq:tate} does not change this.

In the totally toric case, Corollary~\ref{loc:chars} gives nonzero monodromy on
$V_{\theta,a}$, so its semisimplified Weil representation is a sum of two
characters differing by the norm. Under local Langlands for $\GL_2$ the only
possibilities are the special parameter with nonzero monodromy and the
zero-monodromy parameter of a one-dimensional local representation; the latter is
not generic, and every local component of a cuspidal representation is generic.
So $\pi_v$ is a twist of Steinberg, and trivial central character makes the
twisting character have square one.

The local computation in the proof of \cite[Lemma~6.1.3(4)]{CN}, invoking
\cite[Lemma~5.6]{Geraghty}, shows a quadratic twist of Steinberg is
$\iota_l$-ordinary of weight zero. It uses only the stated local representation
and trivial central character, so it applies also when the quadratic character is
ramified: the ordinary condition is imposed at a deep level, and a finite-order
twist multiplies normalized Hecke eigenvalues by units.
\end{proof}

\subsection{The three lifts}

\begin{proposition}\label{auto:overF}
The elliptic curve $B/F$ is modular: $H^1(B)$ is realized by a cuspidal
weight-zero regular algebraic representation of $\PGL_2(\A_F)$.
\end{proposition}

\begin{proof}
The order-three system $V_\eta$ is $H^1(A)$ for the elliptic curve $A/F$ of
Proposition~\ref{sys:elliptic}. It is non-CM by Proposition~\ref{res:special},
its mod-$5$ representation is decomposed generic and absolutely irreducible
over $F(\zeta_5)$ by the $(m,l)=(3,5)$ part of that proposition, and
$\zeta_5\notin F$ by \eqref{sys:nozeta}. So Theorem~\ref{aut:seed} supplies a
cuspidal weight-zero representation of $\PGL_2(\A_F)$ realizing $V_\eta$.

Apply Theorem~\ref{aut:lift} in the order
\begin{center}
\begin{tabular}{@{}cccc@{}}
\toprule
Input & Coefficient prime & Lifted & Residual input\\
\midrule
$V_\eta$ & $p$ & $V_\chi$ & $(m,l)=(3,p)$\\
$V_\chi$ & $3$ & $V_\psi$ & $(m,l)=(p,3)$\\
$V_\psi$ & $p$ & $V_1$ & $H^1(B)\bmod p$\\
\bottomrule
\end{tabular}
\end{center}
In each row Corollary~\ref{sys:congruences} identifies the residual
representations and Proposition~\ref{res:special} gives decomposed genericity and
absolute irreducibility over the cyclotomic extension;
Proposition~\ref{sys:properties} supplies the determinant, finite ramification,
potential semistability, and labeled Hodge--Tate weights; and
Lemma~\ref{auto:transport} makes the previous automorphy available at the next
coefficient prime.

It remains to check the local hypotheses at that prime. At a place of potentially
good ordinary type, both systems are potentially crystalline and potentially
ordinary of weight zero; at a place of potentially good supersingular type, both
are potentially crystalline and neither is potentially ordinary of weight zero
(these are Corollary~\ref{loc:chars}). In either case
Lemma~\ref{auto:autolocal}(i) gives zero monodromy in the input automorphic
parameter. At a place of potentially totally toric type, neither Galois
representation is potentially crystalline and Lemma~\ref{auto:autolocal}(ii)
gives automorphic ordinarity. So the relevant alternative of
Theorem~\ref{aut:lift} holds at every place above the coefficient prime.

The theorem thus applies in all three rows. Its conclusion preserves cuspidality,
weight zero, and trivial central character, so each row feeds the next, and the
last gives automorphy of $V_1=H^1(B)$.
\end{proof}

\subsection{Descending to the original field}

The character adjustment in the next argument identifies the descended
representation with the prescribed one.

\begin{lemma}\label{auto:descent}
Let $L'/L$ be a cyclic extension of prime degree between CM fields and
$r:G_L\to\GL_2(\overline\Q_l)$ continuous with absolutely irreducible restriction
to $G_{L'}$. If this restriction is realized by a cuspidal weight-zero regular
algebraic representation $\Pi$ of $\GL_2(\A_{L'})$, then $r$ is realized by a
cuspidal weight-zero regular algebraic representation of $\GL_2(\A_L)$.
\end{lemma}

\begin{proof}
For $\sigma\in\Gal(L'/L)$ the representations $\Pi^\sigma$ and $\Pi$ have the
same good-place Hecke polynomials, since the Galois representation extends to
$G_L$; by strong multiplicity one $\Pi$ is Galois invariant, and the cyclic
base-change theorem gives a cuspidal $\pi$ of $\GL_2(\A_L)$ with base change
$\Pi$ \cite{LanglandsBC,ArthurClozel}. Every archimedean place is complex, so
local extensions are trivial and $\pi$ is regular algebraic of weight zero.

Good-place compatibility and Chebotarev give
$\bigl(r_\iota(\pi)|_{G_{L'}}\bigr)^{\ss}\simeq r|_{G_{L'}}$. The right side is
absolutely irreducible, so the left restriction is already irreducible and
isomorphic to it. After choosing this isomorphism the two representations agree
on $G_{L'}$; at $g\in G_L$ their matrices differ by a scalar (same conjugation on
the absolutely irreducible $G_{L'}$-representation), and Schur's lemma makes
these scalars a character $\delta:\Gal(L'/L)\to\overline\Q_l^\times$ with
$r_\iota(\pi)\simeq r\otimes\delta$. The finite-order Hecke character
$\delta^{-1}$ removes the discrepancy, preserving cuspidality and weight zero,
with Galois-twisting compatibility at good places and hence everywhere by
Chebotarev.
\end{proof}

Since $F/K$ is abelian, choose a tower from $K$ to $F$ with cyclic prime-degree
steps; each intermediate field is abelian over $\Q$ and contains the imaginary
quadratic $K$, hence is CM. The representation $H^1(B,\overline\Q_p)$ stays
absolutely irreducible over $F$ because its residual representation is
absolutely irreducible there. Apply Lemma~\ref{auto:descent} down the tower to
Proposition~\ref{auto:overF}: this gives automorphy of $B/K$. Finally $B$ is the
quadratic twist of $E$ by $h(t)$, so twisting the automorphic representation by
the same quadratic Hecke character gives a cuspidal weight-zero $\pi_E$
realizing $H^1(E)$.

\subsection{Local parameters and the CM case}

For non-CM $E$, Theorem~\ref{aut:compat} shows $\pi_E$ has trivial central
character, realizes $H^1(E)$ at every coefficient prime, and has the required
Frobenius-semisimplified Weil--Deligne parameter at each finite place (choosing
a coefficient prime different from the residue characteristic). At infinity a
cuspidal representation is generic, so the archimedean classification with
weight-zero infinitesimal character gives exactly \eqref{eq:inf}; the
finite-dimensional alternative is not generic. Equations \eqref{eq:tate} and
\eqref{eq:Lshift} give the stated normalization.

Suppose now $E$ has complex multiplication by an imaginary quadratic $M$. Its
endomorphisms are defined over $KM$ (their faithful action on invariant
differentials has eigenvalues in $M$ fixed by $G_{KM}$). The main theorem of
complex multiplication expresses the two one-dimensional constituents of
$H^1(E)|_{G_{KM}}$ as realizations of algebraic Hecke characters with infinity
types given by the CM type \cite{ShimuraCM}.

If $M\not\subset K$, Galois exchanges the two characters and $H^1(E)$ is induced
from either over $KM/K$; automorphic induction gives the required representation
of $\GL_2(\A_K)$. If $M=K$, use the isobaric sum of the two characters. In either
case normalize by $|\det|^{1/2}$ so the Tate-normalized parameter realizes
$H^1(E)$; complex multiplication and local class field theory, together with
local compatibility of quadratic automorphic induction in the first case, give
the finite parameters and infinity types. Their central character is trivial:
the determinant of $H^1(E)$ is $\eps_l^{-1}$, so undoing the Tate normalization
gives determinant one, hence trivial central character.

In both the CM and non-CM cases the finite factors also agree for the
residue-characteristic coefficient prime. For good reduction this is crystalline
comparison. For bad potentially good reduction, the Weil--Deligne inertia
action is finite and nontrivial with zero monodromy and determinant one; its
invariant space cannot be one-dimensional (a semisimple two-dimensional
determinant-one representation with a nonzero fixed vector is trivial), so it has
no invariants and local factor one, nontriviality of inertia following from the
crystalline good-reduction criterion \cite{ColemanIovita}. In the potentially
multiplicative case the Tate-curve description, with a quadratic twist, gives the
same special Weil--Deligne parameter at every coefficient prime. This completes
the proof of Theorem~\ref{thm:main}.

\begin{thebibliography}{99}
\bibitem{ArthurClozel} J.~Arthur and L.~Clozel, \emph{Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula}, Ann. of Math. Stud. 120, Princeton Univ. Press, 1989.
\bibitem{Bianchi} L.~Bianchi, Sui gruppi di sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginar\^{\i}, Math. Ann. 40 (1892), 332--412.
\bibitem{BCDT} C.~Breuil, B.~Conrad, F.~Diamond, R.~Taylor, On the modularity of elliptic curves over $\Q$: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), 843--939.
\bibitem{BLR} S.~Bosch, W.~L\"utkebohmert, M.~Raynaud, \emph{N\'eron Models}, Ergeb. Math. Grenzgeb. 21, Springer, 1990.
\bibitem{CN} A.~Caraiani and J.~Newton, On the modularity of elliptic curves over imaginary quadratic fields, preprint arXiv:2301.10509v3 (2025).
\bibitem{ColemanIovita} R.~Coleman and A.~Iovita, The Frobenius and monodromy operators for curves and abelian varieties, Duke Math. J. 97 (1999), 171--215.
\bibitem{Faltings} G.~Faltings, Endlichkeitss\"atze f\"ur abelsche Variet\"aten \"uber Zahlk\"orpern, Invent. Math. 73 (1983), 349--366.
\bibitem{Geraghty} D.~Geraghty, Modularity lifting theorems for ordinary Galois representations, Math. Ann. 373 (2019), 1341--1427.
\bibitem{HLTT} M.~Harris, K.-W.~Lan, R.~Taylor, J.~Thorne, On the rigid cohomology of certain Shimura varieties, Res. Math. Sci. 3 (2016), Paper No. 37.
\bibitem{LanglandsBC} R.~Langlands, \emph{Base Change for $\GL(2)$}, Ann. of Math. Stud. 96, Princeton Univ. Press, 1980.
\bibitem{Scholze} P.~Scholze, On torsion in the cohomology of locally symmetric varieties, Ann. of Math. 182 (2015), 945--1066.
\bibitem{SerreHilbert} J.-P.~Serre, \emph{Topics in Galois Theory}, 2nd ed., Res. Notes Math. 1, A K Peters, 2008.
\bibitem{SerreOpenImage} J.-P.~Serre, Propri\'et\'es galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259--331.
\bibitem{SGA7I} A.~Grothendieck, M.~Raynaud, D.~S.~Rim, \emph{Groupes de monodromie en g\'eom\'etrie alg\'ebrique I (SGA 7 I)}, Lecture Notes in Math. 288, Springer, 1972.
\bibitem{SGA7II} P.~Deligne, N.~Katz (eds.), \emph{Groupes de monodromie en g\'eom\'etrie alg\'ebrique II (SGA 7 II)}, Lecture Notes in Math. 340, Springer, 1973.
\bibitem{ShimuraCM} G.~Shimura and Y.~Taniyama, \emph{Complex Multiplication of Abelian Varieties and Its Applications to Number Theory}, Publ. Math. Soc. Japan 6, Math. Soc. Japan, 1961.
\bibitem{Tsuji} T.~Tsuji, $p$-adic \'etale cohomology and crystalline cohomology in the semi-stable reduction case, Invent. Math. 137 (1999), 233--411.
\bibitem{Varma} I.~Varma, Local-global compatibility for regular algebraic cuspidal automorphic representations when $\ell\ne p$, Forum Math. Sigma 12 (2024), Paper No. e21.
\bibitem{BlochKato} S.~Bloch and K.~Kato, $L$-functions and Tamagawa numbers of motives, in \emph{The Grothendieck Festschrift I}, Progr. Math. 86, Birkh\"auser, 1990, 333--400.
\end{thebibliography}

\end{document}