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\title{A geometric prime switch for elliptic curves\\over totally real and CM fields}
\author{Discussion draft}
\date{October 7, 2026}
\begin{document}
\maketitle

\begin{abstract}
In recent work \cite{OAI}, OpenAI established the modularity of elliptic
curves over imaginary quadratic fields. We show that their method also
establishes modularity over totally real fields and over CM fields not
containing $\zeta_5$.
\end{abstract}

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

One of the great achievements of modern number theory is the proof of
the Taniyama--Shimura conjecture: every elliptic curve $E/\Q$ is
modular, that is, $L(E,s)=L(f,s)$ for a newform $f$ of weight two and
level $\Gamma_0(N)$. Equivalently, the Tate modules of $E$ are
isomorphic to the $\ell$-adic representations attached to $f$ by
Eichler and Shimura \cite{Shimura}. The conjecture was the key to
Fermat's Last Theorem: a solution of the Fermat equation gives a
semistable elliptic curve \cite{Frey} which, as anticipated by Serre
\cite{SerreDuke}, cannot be modular by Ribet's level-lowering theorem
\cite{Ribet}. Wiles and Taylor--Wiles proved the conjecture for
semistable curves \cite[Theorem 0.4]{Wiles}, \cite{TW}, and after
further work \cite{Diamond,CDT} Breuil, Conrad, Diamond, and Taylor
proved it in general \cite{BCDT}.

Over a number field $K$, Langlands reciprocity \cite{Langlands} predicts
that every elliptic curve $E/K$ is \emph{modular}: the compatible system
$\rho_{E,\ell}$ given by $H^1(E)$ is attached to a regular algebraic automorphic
representation $\pi$ of $\GL_2(\A_K)$, cuspidal unless $E$ has complex
multiplication defined over $K$. This formulation, and essentially all
known approaches to modularity, require the construction of compatible systems of Galois representations attached
to regular algebraic cuspidal automorphic representations of
$\GL_2(\A_K)$. Such constructions are known when $K$ is totally real, by work of
Carayol, Wiles, Blasius--Rogawski, and Taylor
\cite{Carayol,WilesHilbert,BR,TaylorHilbert}, and when $K$ is CM: first
over imaginary quadratic fields under a condition on the central
character \cite{HST,TaylorIQ,BH}, and in general by
Harris--Lan--Taylor--Thorne \cite{HLTT}. Because of this, modularity
results to date are mostly confined to totally real or CM fields; a
notable exception is the work of Boxer, Calegari, Gee, and Pilloni on
elliptic curves over quadratic extensions of totally real fields that
need not be CM \cite{BCGP}.

The main approach to modularity follows the strategy, due to Wiles, of
propagating modularity along congruences. It has two steps:
\begin{itemize}
\item (Modularity lifting) If $\rho$ is an $\ell$-adic representation
satisfying suitable local conditions (including $\ell$-adic
Hodge-theoretic conditions at $\ell$) and $\bar\rho\simeq\bar\rho_{\pi,\ell}$
for the Galois representation $\rho_{\pi,\ell}$ of some automorphic
$\pi$, then $\rho$ is modular. This usually requires the image of
$\bar\rho$ to be sufficiently large.
\item (Chaining) Find a chain of congruences between $\rho_{E,\ell}$ and
some $\rho_{\pi,\ell'}$ with $\pi$ automorphic, such that the first step
applies at each link. Crucially, the prime at which the congruence holds
may change along the chain.
\end{itemize}
The strategy is most developed in the totally real case. Building on
Wiles's method, work of Kisin \cite{Kisin}, Gee \cite{Gee}, and
Barnet-Lamb--Gee--Geraghty \cite{BLGG1,BLGG2} led to essentially optimal
modularity lifting theorems in weight two; see
\cite[Th\'eor\`eme 3.2.2]{BD} for a convenient statement. Thus the
difficulty lies in the second step: finding enough congruences to
representations known to be modular. If one is willing to enlarge $K$,
that is, to prove only \emph{potential modularity}, this is resolved by
Taylor \cite{TaylorFM}; for elliptic curves see
\cite[Theorem A.1]{Wintenberger}, and \cite[Theorem 1.1.1]{Snowden} for
general weight-two representations. On the other hand, it seems very
hard to produce enough congruences while keeping $K$ fixed, as this is
related to finding points on complicated algebraic varieties. For this
reason, previous results restrict the field: to low degree, namely real
quadratic \cite{FLS}, totally real cubic \cite{DNS}, quartic fields not
containing $\sqrt5$ \cite{Box}, and all but finitely many quintic fields
\cite{IIY}; to fields with restricted ramification at small primes
\cite{LeHung,Yoshikawa}; or to special towers, such as the cyclotomic
$\Z_p$-extension of $\Q$ \cite{ThorneCyclotomic}. These hypotheses are
used to deal with exceptional collections of elliptic curves, given by
points on certain modular curves, which are not known to be chainable
to a modular one. Lifting theorems for small residual images
\cite{ThorneDihedral,Kalyanswamy} shrink these exceptional cases, though they fall short of getting rid of them altogether.

Compared to the totally real case, the CM case is more recent and
harder: automorphic forms for $\GL_2$ over a CM field occur in the
cohomology of locally symmetric spaces that are not algebraic varieties,
in several degrees, and are not directly related to geometry. As a
consequence, modularity lifting theorems are harder to establish. The
seminal work of Calegari--Geraghty \cite{CG} generalized the
Taylor--Wiles method to this setting, conditionally on conjectures on
torsion in the cohomology of locally symmetric spaces. Since then many
of these conjectures have been proven or bypassed \cite{Scholze,ACC},
leading to potential modularity results over CM fields
\cite[Theorem 1.0.1]{ACC} and, more recently, to modularity lifting
theorems in weight two of strength comparable to the totally real case,
due to Caraiani--Newton \cite[Theorem 5.2]{CN}. Combined with the
residual modularity results of Allen--Khare--Thorne \cite{AKT}, the
latter led to major progress over CM fields. For instance,
Caraiani--Newton showed that all elliptic curves over an imaginary
quadratic field $K$ with $X_0(15)(K)$ finite are modular, such as
$K=\Q(\sqrt{-d})$ with $d=1,2,3,5$ \cite[Theorem 1.1]{CN}, and more
generally proved modularity over CM fields not containing $\zeta_5$
under hypotheses on $\bar\rho_{E,3}$ or $\bar\rho_{E,5}$
\cite[Theorem 6.1]{CN}. The main thrust behind these results is the
modularity lifting step.

Most recently, OpenAI announced a proof that every elliptic curve over
every imaginary quadratic field is modular \cite[Theorem 1.1]{OAI}, by
introducing a clever new idea for the chaining step. Unlike previous
arguments, which exploit small primes such as $2$, $3$, $5$, $7$ and
special moduli problems, OpenAI's chain passes through a congruence
modulo an arbitrarily large auxiliary prime $p$, after a large solvable
base change. The use of such large primes is traditionally overlooked, since it is believed (e.g.\ by the Frey--Mazur conjecture, see \cite{KO}) that there are very few congruences between elliptic curves over a fixed field for large $p$.
Since its geometric ingredients work over any number
field, one expects OpenAI's argument to extend beyond imaginary
quadratic fields. This is indeed the case.

\begin{theorem}\label{thm:main}
Assume Inputs~\ref{in:geometry} and~\ref{in:monodromy} from
\cite{OAI}. Then every elliptic curve over every totally real number
field is modular. Assume in addition Input~\ref{in:local} from
\cite{OAI}. Then every elliptic curve over every CM number field $K$
with $\zeta_5\notin K$ is modular.
\end{theorem}

\begin{remark}
\begin{enumerate}
\item The inputs are the geometric results of \cite{OAI}.
Input~\ref{in:geometry} gives the basic properties of the family of
motives that carries the chain of congruences;
Input~\ref{in:monodromy} is the large monodromy of the family, which
gives large residual images along the chain; and Input~\ref{in:local}
concerns local properties of the family.
\item In the totally real case we use no modularity lifting theorem
over CM fields.
\end{enumerate}
\end{remark}

\subsection{Outline}
Let $E$ be an elliptic curve over $K$. Fix $n=3p$ for a large auxiliary prime $p$. OpenAI constructs a family of quadratic twists $B_t$ and cyclic degree $n$ covers $C_n(t)\rightarrow B_t$ over an open subset $U\subset \mathbb{A}^1_t$. $C_n(t)$ comes with an action of a dihedral group defined over $F=K(\mu_n)$. The key feature of the construction is that the integral $\ell$-adic cohomology $H^1(C_n(t))$ is free of rank $2$ over $\Z_\ell[\mu_n]$ for all $\ell$. In particular, it decomposes into rank-two pieces $V_\theta$ with Hodge--Tate weights $0,1$, indexed by characters $\theta$ of $\mu_n$. 
One has $V_1=H^1(B_t)$, and for $\eta$ of order three, $V_\eta=H^1(A_t)$ for an elliptic curve $A_t$. The reduction of $V_\theta$ modulo $\ell$ only
depends on $\theta\bmod\ell$. With $\psi$ of order $p$ this gives the
chain of congruences (for each $t$)
\[
 V_\eta\equiv_p V_{\eta\psi}\equiv_3 V_\psi\equiv_p V_1,
\]
and the idea is to propagate modularity from the left to the right.
In order to do this propagation, the essential point is to keep the involved residual image large at each step so that a modularity lifting theorem applies. As $p$ is chosen to be large, the key point is to arrange $t$ so that $V_\psi$ mod $3$ has large image. This is achieved by the fact that the family has two opposite transvections around two points of the parameter space where the cover degenerates, and thus Hilbert irreducibility guarantees there are many such choices of $t$. As a byproduct, one can also guarantee that the mod $5$ image of $V_\eta=H^1(A_t)$ is large, and hence that $A_t$ is modular. Importantly, this gives the starting point for the chain.


For the case of totally real $K$, we first note that the $V_\theta$ in the construction are only defined over the CM field
$K(\mu_{3p})$. However, twisting the cover by the reflection descends them to
totally odd systems $W_\theta$ over $L=K(\zeta_{3p}+\zeta_{3p}^{-1})$
with the same congruences, and thus one can follow the above argument with the (less restrictive) modularity lifting theorems over totally real fields; thus one merely needs the existence of the chain of congruences to run the argument.

Over CM $K$, invoking the Caraiani--Newton modularity lifting theorem requires one to verify two additional hypotheses: that the residual representations are decomposed generic and that the motives in the chain have the same local $\ell$-adic Hodge-theoretic behavior at the coefficient prime $\ell$. The first is easily dealt with by applying Hilbert irreducibility to a Weil restriction, while the second is dealt with by imposing local conditions on $t$ at the places above $3p$, using an analysis of the reduction of the family near its compact-type degeneration.

\begin{remark} 
\begin{enumerate}
\item The main reason for the hypothesis $\zeta_5 \notin K$ in the CM case was so that one can apply the existing modularity lifting of Caraiani--Newton at the prime $5$ \cite[Theorem 6.1]{CN}. An improvement to this has been obtained by AI, as released in \cite{CMrepo}. With this improvement, OpenAI's argument now covers all CM fields. Nevertheless, we find it instructive to treat the totally real field case on its own, as it illustrates clearly the distinctive role of the two parts of the strategy.
\item It is essential that our chain connects to an elliptic curve $A_t$, which is the reason to work with the prime $3$ in the middle congruence of the chain. Replacing the prime $3$ with a larger prime will instead connect $H^1(B_t)$ to some other motive for which one does not know how to establish modularity. On the other hand, one could contemplate using the prime $2$ to establish the modularity of $A_t$ along the lines of \cite{AKT}. 
% There are many families
\item One might wonder whether it is possible to use a different prime to get the modularity of the starting motive $A_t$. The prime $3$ does not work, since $H^1(A_t)$ turns out to always be congruent to a quadratic twist of $E$ mod $3$.
% \item
\end{enumerate}
\end{remark}

\begin{remark}
The mechanism has antecedents in families of curves with real
multiplication \cite{Mestre,TTV,DM,Ellenberg}, and above all in
Darmon's work on rigid local systems \cite{DarmonFibres,DarmonRigid}.
There, hypergeometric families are related by reduction modulo primes
dividing the orders of local monodromy, and modularity is propagated
conditionally on a lifting conjecture
\cite[Conjecture~1.1]{DarmonFibres}.
\end{remark}

Section~\ref{sec:geometry} sets up the geometry and the congruences,
Sections~\ref{sec:totally-real} and~\ref{sec:cm} treat the totally real
and CM cases, and Section~\ref{sec:descent} contains descent and
local--global compatibility.

\paragraph{AI disclosure.}
The first draft of this note, apart from the introduction, was
generated by the AI system Astra. The introduction was written by the
author. The whole text was proofread and edited, and its references
checked, by Claude (Anthropic; model Claude Opus 5.5).

\paragraph{Conventions.}
Write $G_k=\Gal(\overline k/k)$ and $H^1(X)=H^1_{\et}(X_{\overline
k},\Q_\ell)$, extending scalars to $\overline\Q_\ell$ when needed. The
cyclotomic character $\eps_\ell$ has Hodge--Tate weight $-1$, so
$\det H^1(E)=\eps_\ell^{-1}$ and $\mathrm{HT}_\tau(H^1(E))=\{0,1\}$ for
every $\tau$. Bars denote semisimplified reductions. We fix embeddings
of $\overline\Q$ into $\C$ and into every $\overline\Q_\ell$. When
applying \cite{BD,FLS} we dualize to the Tate-module convention.

\section{The geometric construction}\label{sec:geometry}

Let $k$ be a number field, $E:y^2=h(x)$ an elliptic curve over $k$, and
$n\geq3$ odd. Put
\[
 B_t:h(t)y^2=h(x),\qquad R_t=(t,1),\qquad S_t=[n]R_t,
\]
and let $U\subset\A^1_k$ be the open set where $h(t)\neq0$ and
$O,\pm R_t,\pm S_t$ are distinct. The divisor
$(S_t)-(-S_t)-n(R_t)+n(-R_t)$ is principal; let $f_t$ be its function
with $f_t(O)=1$, so that $f_t(-Z)=f_t(Z)^{-1}$. For $m\mid n$ let
$C_m(t)$ be the normalization of $B_t$ in
\begin{equation}\label{eq:cover}
 Y^m=f_t(Z),
\end{equation}
with rotations $u_\zeta(Z,Y)=(Z,\zeta Y)$, $\zeta\in\mu_m$, and
reflection $w(Z,Y)=(-Z,Y^{-1})$, so that $wu_\zeta w=u_{\zeta^{-1}}$.
The curve and $w$ are defined over $k$, the rotations over $k(\mu_m)$.
The map $C_n\to C_m$ is $Y\mapsto Y^{n/m}$, and $C_m(t)/\mu_m=B_t$.

\begin{geominput}[{Cover and lattice; \cite[Lemma 3.1, Propositions 3.3
and 4.2, Remark 4.3]{OAI}}]\label{in:geometry}
The curve $C_m(t)$ is smooth of genus $m$, and for every prime $\ell$,
including $\ell\mid m$,
\begin{equation}\label{eq:free}
 H^1_{\et}(C_{m,\overline k},\Z_\ell)\simeq\Z_\ell[\mu_m]^2
\end{equation}
as modules over the group ring of the rotation group. After pullback
along $E\to\A^1$, $(t,s)\mapsto t$, over which $B_t\simeq E$ and $R_t$
becomes the moving point $R=(t,s)$, the Jacobian family extends across
$R=O$ with fiber $E^m$, whose factors the rotations permute regularly
over $k(\mu_m)$.
\end{geominput}

\begin{geominput}[{Monodromy; \cite[Theorem 4.5]{OAI}}]
\label{in:monodromy}
After any complex embedding of $k$, geometric monodromy contains, in one
basis of the free module \eqref{eq:free}, two opposite transvections
with off-diagonal entries $\pm(2-u-u^{-1})$, where $u$ is
$Y\mapsto e^{2\pi i/m}Y$. On the eigenspace where $u$ acts by
$\lambda\neq1$ they are
\begin{equation}\label{eq:transvections}
 T_a=\begin{pmatrix}1&\nu_a c_\lambda\\0&1\end{pmatrix},\quad
 T_b=\begin{pmatrix}1&0\\\nu_b c_\lambda&1\end{pmatrix},\quad
 c_\lambda=2-\lambda-\lambda^{-1},\quad \nu_a,\nu_b\in\{\pm1\}.
\end{equation}
\end{geominput}

The common basis matters: triangular matrices in unrelated bases would
not give irreducibility. Both inputs hold over any number field.

\subsection{Rank-two systems}

Let $F\supset k(\mu_n)$. For a character $\theta$ of $\mu_n$ let
$V_\theta=V_{\theta,\ell}\subset H^1_{\et}(C_{n,\overline
F},\overline\Q_\ell)$ be the $\theta$-eigenspace of the rotations; it is
$G_F$-stable, of dimension two by \eqref{eq:free}. Cup product
$\langle\ ,\ \rangle$ is rotation-invariant, so it pairs $V_\theta$
perfectly with $V_{\theta^{-1}}=w^*V_\theta$, and
\begin{equation}\label{eq:pairing}
 [x,y]_\theta=\langle x,w^*y\rangle
\end{equation}
is a nondegenerate $G_F$-equivariant pairing on $V_\theta$ with values
in $\overline\Q_\ell(-1)$. It is alternating, as
$\langle y,w^*x\rangle=\langle w^*y,x\rangle=-\langle x,w^*y\rangle$.
Hence $\det V_\theta=\eps_\ell^{-1}$. Since $w^*$ exchanges the
$\theta$- and $\theta^{-1}$-parts of $H^{1,0}$, and complex conjugation
exchanges the $\theta^{-1}$-part of $H^{1,0}$ with the $\theta$-part of
$H^{0,1}$, each $\theta$-part of $H^{1,0}$ is a line; so the labelled
Hodge--Tate weights are $\{0,1\}$. The $V_\theta$ are semisimple,
potentially semistable, and form compatible systems (use the character
projector and traces of rotations composed with Frobenius).

The trivial character gives $V_1=H^1(B_t)$. Put
\begin{equation}\label{eq:seed}
 D_t=C_3(t)/\langle w\rangle,\qquad A_t=\Jac(D_t),
\end{equation}
both defined over $k$. On $H^1(C_3)$, $w$ acts by $-1$ on $H^1(B_t)$,
since it induces inversion, and swaps the other two character spaces;
so $D_t$ has genus one. For $\eta$ of order three, projecting the
$w$-invariants to the $\eta$-space gives
\begin{equation}\label{eq:endpoints}
 H^1(A_t)|_{G_F}\simeq V_\eta,\qquad H^1(B_t)|_{G_F}\simeq V_1.
\end{equation}

\subsection{Congruences}\label{sec:congruences}

Let $p>100$ be prime, $n=3p$, let $\eta,\psi$ be characters of orders
$3,p$, and $\chi=\eta\psi$. For $\cO$ the integers of a large enough
finite extension of $\Q_\ell$, the lattice
\[
 T_{\theta,\ell}=H^1_{\et}(C_{n,\overline F},\Z_\ell)
 \otimes_{\Z_\ell[\mu_n],\theta}\cO\subset V_{\theta,\ell}
\]
is $G_F$-stable and free of rank two by \eqref{eq:free}. Its reduction
depends only on $\theta\bmod\ell$, and roots of unity of $\ell$-power
order reduce to $1$, so
\begin{equation}\label{eq:chain}
 V_\eta\equiv_p V_\chi\equiv_3 V_\psi\equiv_p V_1,
\end{equation}
where $\equiv_\ell$ means isomorphic semisimplified reductions modulo
$\ell$. No division by $3$ or $p$ occurs.

\subsection{Residual irreducibility}\label{sec:common-residual}

Take $k=K$ and consider the pairs
\begin{equation}\label{eq:pairs}
 (m,\ell)=(3,5),\ (3,p),\ (p,3),
\end{equation}
with $m$ the order of the character and $\ell$ the residual
characteristic. As $\ell\nmid m$, $c_\lambda=-(\lambda-1)^2/\lambda$ is
nonzero modulo $\ell$, so $T_a$ and $T_b$ have distinct fixed lines and
geometric monodromy acts absolutely irreducibly on the reduction of
each character space of order $m$. For $m=3$ we have $c_\lambda=3$, so
modulo $5$ these matrices generate $\SL_2(\F_5)$.

\subsection{Changing the coefficient prime}\label{sec:transport}

If one member of the compatible system $V_\theta$, or $W_\theta$ of
Section~\ref{sec:real-twist}, is realized by an automorphic
representation $\pi$, then so is every member: compare Frobenius
polynomials and apply Chebotarev and Brauer--Nesbitt. This lets us
change the coefficient prime along \eqref{eq:chain}.

\section{The totally real case}\label{sec:totally-real}

Let $K$ be totally real.

\subsection{Modularity lifting}\label{sec:real-lifting}

The following is a consequence of  \cite[Theorem 3]{FLS},  \cite[Th\'eor\`eme 3.2.2]{BD}:
\begin{proposition}\label{prop:real-transfer}
Let $L$ be totally real, $\ell\neq2,5$ a prime, and $A/L$ an abelian
variety. Let $r:G_L\to\GL_2(\overline\Q_\ell)$ be the image of an
idempotent of $\mathrm{End}_L(A)\otimes\overline\Q$ on
$H^1(A_{\overline L},\overline\Q_\ell)$, with $\det r=\eps_\ell^{-1}$
and labelled weights $\{0,1\}$. If $\bar r$ is modular and
$\bar r|_{G_{L(\zeta_\ell)}}$ is absolutely irreducible, then $r$ is
modular.
\end{proposition}

We remark that in the above Proposition, there is no need to arrange any local conditions for $r$ to match with a modular lift.
\subsection{The reflection twist}\label{sec:real-twist}

Let $M=K(\mu_{3p})$ and $L=K(\zeta_{3p}+\zeta_{3p}^{-1})$. Then $L/K$ is
abelian and totally real, and $M/L$ is quadratic with character
$\delta=(-1)^e$, $e:G_L\to\Z/2\Z$. Twisting $C_m$ by the cocycle
$g\mapsto w^{e(g)}$ (a cocycle since $w$ is defined over $K$) gives a
curve $C_m^\delta/L$, whose cohomology is that of $C_m$ with
\begin{equation}\label{eq:twist-action}
 \rho^\delta(g)=(w^*)^{e(g)}\rho(g).
\end{equation}

\begin{lemma}\label{lem:real-twist}
The rotations and $w$ of $C_m^\delta$ are defined over $L$. The
$\theta$-eigenspaces $W_\theta$ of $H^1(C^\delta_{n,\overline L})$ are
totally odd $G_L$-representations with determinant $\eps_\ell^{-1}$ and
labelled weights $\{0,1\}$, and
\begin{equation}\label{eq:real-chain}
 W_\eta\equiv_p W_\chi\equiv_3 W_\psi\equiv_p W_1,
\end{equation}
\begin{equation}\label{eq:real-endpoints}
 W_\eta\simeq H^1(A_t)|_{G_L},\qquad
 W_1\simeq H^1(B_t)|_{G_L}\otimes\delta.
\end{equation}
\end{lemma}

\begin{proof}
An element $g\in G_L$ acts on $\mu_m$ by $\zeta\mapsto\zeta^{\pm1}$
according to $e(g)$, and $w^{e(g)}u_{g(\zeta)}w^{e(g)}=u_\zeta$, so the
twisted descent datum fixes each $u_\zeta$ and $w$. The integral lattice
is unchanged and commutes with Galois, which gives \eqref{eq:real-chain}
as in Section~\ref{sec:congruences}. Since $w$ and cup product are
defined over $L$, \eqref{eq:pairing} gives the determinant, which is
$-1$ on complex conjugations. The quotient $C_3^\delta/w$ is $D_t$,
giving the first endpoint; $w$ acts by $-1$ on $H^1(B_t)$, so
\eqref{eq:twist-action} gives the second.
\end{proof}

\subsection{The parameter}\label{sec:real-specialization}

Let $E/K$ be non-CM. By Serre's open image theorem
\cite[\S4.2, Th\'eor\`eme 2]{SerreOI}, choose $p>100$ with
$\bar\rho_{E,p}(G_K)\supset\SL_2(\F_p)$. As $\SL_2(\F_p)$ is perfect
and $D=K(\mu_{15p})$ is abelian over $K$, $\bar\rho_{E,p}(G_D)=\SL_2(\F_p)$.

The local systems $H^1(C_m,\F_\ell)$ over $U_D$, for the pairs
\eqref{eq:pairs}, cut out a finite quotient $Q$ of $\pi_1(U_K)$ mapping
onto $\Gal(D/K)$ (take the normal core of their common kernel). By
Hilbert irreducibility \cite[Proposition 3.3.1 and Theorem
3.4.1]{SerreTopics} there is $t\in U(K)$ with $G_K\to Q$ surjective;
then $G_D$ maps onto $\ker(Q\to\Gal(D/K))$, which contains geometric
monodromy. By Section~\ref{sec:common-residual}, the reductions of
$V_\eta$ modulo $5$ and $p$ and of $V_\psi$ modulo $3$ are absolutely
irreducible on $G_D\subset G_{L(\zeta_\ell)}$, and so are those of the
$W_\theta$, since $\delta|_{G_D}=1$. The mod-$5$ image of $A_t$ contains
two noncommuting elements of order $5$, so $A_t$ is non-CM.

\subsection{Proof for totally real fields}

By \cite[Theorem 3]{FLS}, $A_t/L$ is modular, hence so is $W_\eta$.
Apply Proposition~\ref{prop:real-transfer} as follows, changing the
coefficient prime after each row (Section~\ref{sec:transport}):
\begin{center}
\begin{tabular}{@{}cccc@{}}
\toprule
Modular & Prime & Target & Residual irreducibility from\\
\midrule
$W_\eta$ & $p$ & $W_\chi$ & order-three piece modulo $p$\\
$W_\chi$ & $3$ & $W_\psi$ & order-$p$ piece modulo $3$\\
$W_\psi$ & $p$ & $W_1$ & $E$ modulo $p$\\
\bottomrule
\end{tabular}
\end{center}
By \eqref{eq:real-endpoints}, $B_t/L$ is modular. Descending along $L/K$
(Lemma~\ref{lem:descent}) and twisting back to $E$ proves the theorem
for totally real $K$.

\section{The CM case}\label{sec:cm}

Let $K$ be CM with $\zeta_5\notin K$, $E/K$ non-CM, and
$F=K(\mu_{3p})$.

\subsection{The lifting theorem}\label{sec:cm-lifting}

The seed is \cite[Theorem 6.1]{CN}: an elliptic curve $A/K$ is modular
if $\bar\rho_{A,5}$ is decomposed generic and absolutely irreducible on
$G_{K(\zeta_5)}$. For the chain we use \cite[Theorem 5.2]{CN} at $\ell=3,p$. Besides the
determinant and weight conditions of Section~\ref{sec:geometry}, it
requires $\bar\rho$ to be decomposed generic and irreducible on
$G_{F(\zeta_\ell)}$, and the following match at each $v\mid\ell$ with the
automorphic source $\pi$: either $\rho$ is potentially crystalline,
$r_\iota(\pi)$ is potentially ordinary exactly when $\rho$ is, and
$\rec(\pi_v)$ has zero monodromy; or $\rho$ and $r_\iota(\pi)$ are not
potentially crystalline and $\pi$ is $\iota$-ordinary at $v$. Here
$\bar\rho$ is \emph{decomposed generic} if some prime $q\neq\ell$ splits
completely in $F$ and, at each $v\mid q$, $\bar\rho$ is unramified with no
ratio of Frobenius eigenvalues equal to $q$.

\subsection{Local reduction}\label{sec:cm-local}

\begin{geominput}[{Local reduction; \cite[Propositions 7.3 and 7.5,
Corollary 7.6]{OAI}}]\label{in:local}
For each $v_0\mid3p$ of $K$ there is a nonempty open subset of
$U(K_{v_0})$ such that, for $t$ in it, the Jacobian of $C_n(t)$ has at
every $v\mid v_0$ of $F$ the same potential reduction type as $E^n$:
good ordinary, good with all slopes $1/2$, or semistable of toric rank
$n$.
\end{geominput}

In \cite{OAI} this is stated over imaginary quadratic fields, but the
argument is general. On each
character space the three types give, respectively, potentially
crystalline and ordinary, potentially crystalline and non-ordinary, and
monodromy of rank one, for every coefficient prime
\cite[Corollary 7.6]{OAI}.

\subsection{Local matching}\label{sec:cm-local-matching}

Let $\pi$ realize the source of a row of \eqref{eq:chain}, and let
$v\mid\ell$. The source and the target are summands of the same
Jacobian, so by Input~\ref{in:local} they are both potentially
crystalline or neither, with the same potential ordinarity. It remains
to check $\pi_v$, using semisimplified local--global compatibility
\cite[Theorem 1]{Varma} at another prime. In the potentially good case
the two Frobenius eigenvalues have equal absolute value, so
$\rec(\pi_v)$ has zero monodromy. In the toric case they differ by the
norm; as $\pi_v$ is generic, it is a twist of Steinberg, by a quadratic
character since the central character is trivial. Then $\pi$ is
$\iota$-ordinary at $v$ by the argument of \cite[Lemma 6.1.3(4)]{CN}.

\subsection{Decomposed genericity}\label{sec:cm-residual}

\paragraph{Regular semisimple elements.}
For a pair in \eqref{eq:pairs},
$\tr(T_aT_b^j)=2+\nu_a\nu_bjc_\lambda^2$, and an element of $\SL_2$ is
regular semisimple if and only if its trace is not $\pm2$. For $m=3$
take $j=2$: the traces $20$ and $-16$ differ from $\pm2$ by $18$, $22$,
or $14$, so they are not $\pm2$ modulo $5$ or $p$. For $(m,\ell)=(p,3)$
take $j=1$: a bad trace forces $(\lambda-1)^4=\pm4\lambda^2$, so
$\lambda\in\F_{3^a}$ with $a\leq4$, impossible since $\lambda$ has
order $p>80\geq3^a-1$.

\paragraph{All conjugates.}
Let $d=[K:\Q]$, $K^{\mathrm{nor}}$ the Galois closure of $K$, $D$ a
finite Galois extension of $\Q$ containing $K^{\mathrm{nor}}(\mu_{15p})$,
and $X=\Res_{K/\Q}U\subset\A^d_\Q$, so that
$X_D\simeq\prod_{\sigma:K\hookrightarrow D}U^\sigma_D$. The local systems
$H^1(C^\sigma_m,\F_\ell)$ on the factors, for the pairs
\eqref{eq:pairs}, cut out a finite quotient $Q$ of $\pi_1(X)$ mapping
onto $\Gal(D/\Q)$. By Hilbert irreducibility with weak approximation
\cite[Lemma 8.3]{OAI} (see also \cite[Proposition 3.3.1, Theorems 3.4.1
and 3.5.3]{SerreTopics}) there is $t\in X(\Q)=U(K)$, in any prescribed
nonempty open subsets of finitely many $U(K_{v_0})$, with $G_\Q\to Q$
surjective. Then $G_D$ maps onto $\ker(Q\to\Gal(D/\Q))$, which contains
geometric monodromy, a product over the factors. So for each pair some
$g\in G_D$ acts regularly semisimply on every nontrivial character
space in every factor. By Chebotarev, infinitely many primes $q$ have
Frobenius conjugate to $g$. They split in $D$, so $q\equiv1\bmod\ell$,
and every Frobenius above $q$ is regular semisimple on every character
space, since conjugation permutes factors and characters. This is
decomposed genericity: over $K$ for $A_t[5]$, and over $F$ for the
pairs $(3,p)$ and $(p,3)$. As in Section~\ref{sec:real-specialization},
$A_t[5]$ is absolutely irreducible on $G_{K(\zeta_5)}$, $A_t$ is
non-CM, and the residual representations are absolutely irreducible on
$G_D\subset G_{F(\zeta_\ell)}$. This replaces the two conjugates of
\cite[Proposition 8.4]{OAI}; no Galois assumption on $K$ is needed.

\paragraph{The final switch.}
Choose $p$ so that $\bar\rho_{E^\sigma,p}(G_{K^{\mathrm{nor}}})\supset
\SL_2(\F_p)$ for every $\sigma:K\hookrightarrow K^{\mathrm{nor}}$
\cite[\S4.2, Th\'eor\`eme 2]{SerreOI}, and
\begin{equation}\label{eq:p-bound}
 p>\max\{100,2d+1\}.
\end{equation}
Then $G_{D_0}$, for $D_0=K^{\mathrm{nor}}(\mu_{15p})$, has image
$\SL_2(\F_p)$ on each $E^\sigma[p]$. Exactly $2p^2$ elements of
$\SL_2(\F_p)$ have trace $\pm2$, so the proportion of the joint image
that is not regular semisimple in some factor is at most
\begin{equation}\label{eq:union-bound}
 d\,\frac{2p^2}{p(p^2-1)}=\frac{2dp}{p^2-1}<1,
\end{equation}
as $p(p-2d)>1$; for $d=2$ this is \cite[Lemma 8.2]{OAI}. Chebotarev for
a good element, avoiding the primes of bad reduction of $B_t$ once $t$
is fixed, gives decomposed genericity of $\bar\rho_{E,p}$, hence of the
reductions of $V_\psi$ and $V_1$, over $F$.

\subsection{The parameter}\label{sec:cm-specialization}

Choose $p$ as in \eqref{eq:p-bound}, and by the specialization above a
single $t\in U(K)$ in the open sets of Input~\ref{in:local} for all
$v_0\mid3p$. Then:
\begin{enumerate}
\item $\bar\rho_{A_t,5}$ is decomposed generic over $K$ and absolutely
irreducible on $G_{K(\zeta_5)}$, and $A_t$ is non-CM;
\item each residual representation in \eqref{eq:chain} is decomposed
generic over $F$ and absolutely irreducible on $G_{F(\zeta_\ell)}$;
\item above $3p$, all pieces have the reduction type of
Input~\ref{in:local}, so Section~\ref{sec:cm-local-matching} applies.
\end{enumerate}

\subsection{Proof for CM fields}

By \cite[Theorem 6.1]{CN}, $A_t$ is modular over $K$, hence over $F$ by
solvable base change, cuspidally since $A_t$ is non-CM; so $V_\eta$ is
automorphic. Apply \cite[Theorem 5.2]{CN} as follows, changing the
coefficient prime after each row:
\begin{center}
\begin{tabular}{@{}cccc@{}}
\toprule
Automorphic & Prime & Target & Residual input\\
\midrule
$V_\eta$ & $p$ & $V_\chi$ & $(m,\ell)=(3,p)$\\
$V_\chi$ & $3$ & $V_\psi$ & $(m,\ell)=(p,3)$\\
$V_\psi$ & $p$ & $V_1$ & \eqref{eq:union-bound}\\
\bottomrule
\end{tabular}
\end{center}
Thus $V_1=H^1(B_t)|_{G_F}$ is automorphic. Descending along $F/K$
(Lemma~\ref{lem:descent}) and twisting back gives modularity of $E/K$.

\begin{remark}
We do not claim $\zeta_5\notin F$. For example,
$K=\Q(\zeta_{15}+\zeta_{15}^{-1},i)$ is CM and does not contain
$\zeta_5$, but $K(\zeta_3)=\Q(\zeta_{60})$ does. This is why $A_t$ is
made modular over $K$ before base change, and why the
ramification argument of \cite[(5.2)]{OAI} is not used. The later lifts
are at $3$ and $p$, so the exceptional case at $5$ does not arise. For
CM fields containing $\zeta_5$ we have no seed.
\end{remark}

\section{Descent and local--global compatibility}\label{sec:descent}

\begin{lemma}\label{lem:descent}
Let $F/K$ be an abelian extension of totally real fields or of CM
fields, and $r$ a rank-two representation of $G_K$ with $r|_{G_F}$
absolutely irreducible and automorphic. Then $r$ is automorphic.
\end{lemma}

This is standard: descend one cyclic step at a time by solvable base
change \cite[Chapter~3, Theorems 4.2(d) and 5.1]{AC}, and adjust by a
character using Schur's lemma.

For non-CM $E$, the resulting $\pi_E$ realizes $H^1(E)$ at every
coefficient prime, with the expected local parameters at finite places
by local--global compatibility (over CM fields, \cite[Lemma
6.1.3]{CN}); in particular $L(E/K,s)=L(\pi_E,s-1/2)$.

If $E$ has CM by an imaginary quadratic field $T$, then $H^1(E)|_{G_{KT}}$
is a sum of two Hecke-character systems; use automorphic induction if
$T\not\subset K$ (always the case for totally real $K$), and the
isobaric sum if $T\subset K$.

\subsection{Source and status}\label{sec:status}

We use \cite{OAI} at commit \texttt{adc7f12}. The new arguments here are the reflection twist, the $d$-conjugate
specialization with \eqref{eq:union-bound}, the order of seed modularity
and base change in the CM case, and the two applications of existing
lifting theorems. The geometric inputs are taken from \cite{OAI}, and
Theorem~\ref{thm:main} is stated conditionally on them.
The construction uses special features of elliptic curves: the group
law gives the two-branch-point cover, every character space has rank
two, the order-three quotient is an elliptic curve, and one integral
lattice gives congruences at several primes.

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\end{document}