Theorem 1. Let $\ell \gt 3$ and $\bar{r} \colon G_{\mathbb{Q},S} \to \GL_2(\mathbb{F}_ \ell)$ for $S = \lbrace 2, \ell \rbrace$ satisfy $\det \bar{r} = \bar{\epsilon}_ \ell^{-1}$ and $\bar{r} \vert_{G_{\mathbb{Q}_ 2}} \sim \begin{pmatrix} 1 & \ast \br 0 & \epsilon_\ell^{-1} \end{pmatrix} \otimes \delta$ with $\delta$ unramified and $\bar{r} \vert_{G_{\mathbb{Q}_ \ell}}$ Fontaine–Laffille with Hodge–Tate weights $0, 1$. Then there exists $F/\mathbb{Q}$ a totally real finite extension such that $2$ and $\ell$ are unramified in $F$ and $F \cap \bar{\mathbb{Q}}^{\ker \bar{r}}(\zeta_\ell) = \mathbb{Q}$ such that $\bar{r} \vert_{G_F}$ is automorphic with weight $0$ and $U_0$-level prime to $\ell$.
We choose $M/\mathbb{Q}$ an imaginary quadratic field, a prime $\ell^\prime \neq \ell, 2$ that splits in $M$, and a character $\theta_{\lambda^\prime} \colon G_M \to \mathbb{Z}_ {\ell^\prime}^\times$ that ramifies only at $\ell^\prime$, $p$, primes that ramify in $F/\mathbb{Q}$, with the property that $\operatorname{HT}_ \tau(\theta_{\lambda^\prime}) = \lbrace 0 \rbrace$ and $\operatorname{HT}_ {\bar{\tau}}(\theta_{\lambda^\prime}) = \lbrace 1 \rbrace$. Then we checked that $$ \operatorname{Ind}_ {G_M}^{G_\mathbb{Q}} \theta_{\lambda^\prime} $$ satisfy the property that its determinant $\epsilon_{\lambda^\prime}^{-1}$ and its reduction restricted to $G_{\mathbb{Q}(\zeta_{\ell^\prime})}$ is irreducible.
Now we looked at the moduli $T/\mathbb{Q}$ of elliptic curves $E$ together with isomorphisms $E[\ell] \cong \bar{r}$ and $E[\ell^\prime] \cong \operatorname{Ind}_ {G_M}^{G_\mathbb{Q}} \bar{\theta}_ {\lambda^\prime}$ taking the Weil pairing to fixed pairings. This $T$ was geometrically connected and smooth and $T(\mathbb{R}) \neq \emptyset$. We also saw that there exist $L/\mathbb{Q}_ \ell$ and $L^\prime / \mathbb{Q}_ {\ell^\prime}$ unramified such that we have points $(E, i) \in T(L)$ and $(E^\prime, i^\prime) \in T(L^\prime)$, where $E, E^\prime$ both have good reduction.
Claim 2. There exists an unramified extension $L^{\prime\prime} / \mathbb{Q}_ 2$ with $T(L^{\prime\prime}) \neq \emptyset$.
Proof.
We look at Tate elliptic curves $E_q / \mathbb{Q}_ 2$ for $q \in \mathbb{Q}_ 2^\times$ with $v(q) \gt 0$. Then $E_q(\bar{\mathbb{Q}}_ 2) = \bar{\mathbb{Q}}_ 2^\times / q^\mathbb{Z}$. At $\ell$, we have to look for $q$ such that the image of $q$ in $$ \mathbb{Q}_ 2^\times / (\mathbb{Q}_ 2^\times)^\ell \cong H^1(G_{\mathbb{Q}_ 2}, \mathbb{F}_ \ell(\bar{\epsilon}_ \ell)) $$ is $\ast$. Here only the valuation of $q$ matter. Similarly, because we need $E_q[\ell^\prime]$ to be unramified, so we can take $q \in (\mathbb{Q}_ 2^\times)^{\ell^\prime}$. Once we choose $q$ appropriately, we can choose a sufficiently large unramified extension $L^{\prime\prime}$ such that $\delta = 1$ and $E_q[\ell^\prime]$ and $\operatorname{Ind}_ {G_M}^{G_\mathbb{Q}} \bar{\theta}_ {\lambda^\prime}$ are both trivial.
Now we look at this moduli $T/\mathbb{Q}$ and the set $$ S = \lbrace \infty, 2, \ell, \ell^\prime, p, \text{primes that ramify in } F/\mathbb{Q} \rbrace. $$ We now choose the data
- for $v = \infty$, we take $L_\infty^\prime = \mathbb{R}$ and $\Omega_\infty = T(\mathbb{R})$,
- for $v = 2$, we take $L_2^\prime = L^{\prime\prime}$ and $\Omega_2 = T(L_2^\prime)$,
- for $v = \ell$, we take $L_\ell^\prime = L$ and $\Omega_\ell$ the good reduction locus of $T(L_\ell^\prime)$,
- for $v = \ell^\prime$, we take $L_{\ell^\prime}^\prime = L^\prime$ and $\Omega_{\ell^\prime}$ the good reduction locus of $T(L_{\ell^\prime}^\prime)$,
- for $v = p$ or $v$ ramified in $F/\mathbb{Q}$, we take $L_v^\prime / \mathbb{Q}_ v$ finite Galois so that $\operatorname{Ind}_ {G_M}^{G_\mathbb{Q}} \theta_{\lambda^\prime} \vert_{G_{L_v^\prime}}$ is unramified and $T(L_v^\prime) \neq \emptyset$.
Once we do that, the theorem gives us $F/\mathbb{Q}$ linearly disjoint from $$ \mathbb{Q}^\mathrm{av} = \bar{\mathbb{Q}}^{\ker \bar{r} \times \operatorname{Ind}_ {G_M}^{G_\mathbb{Q}} \bar{\theta}_ {\lambda^\prime}}(\zeta_{\ell \ell^\prime}), $$ so that
- $F$ is totally real,
- $2, \ell, \ell^\prime$ is unramified in $F$,
- $\operatorname{Ind}_ {G_M}^{G_\mathbb{Q}} \theta_{\lambda^\prime} \vert_{G_F}$ is unramified away from $\ell^\prime$,
and an elliptic curve $E/F$ with good reduction $\ell$ and $\ell^\prime$, such that we have isomorphisms $$ E[\ell]^\vee \cong \bar{r} \vert_{G_F}, \quad E[\ell^\prime]^\vee \cong \operatorname{Ind}_ {G_M}^{G_\mathbb{Q}} \bar{\theta}_ {\lambda^\prime}. $$ Because $\GL_2(\mathbb{Z}_ {\ell^\prime})$ has no $\ell^\prime$-torsion as $\ell^\prime \gt 3$, we see that the action of $I_v$ for $v \nmid \ell^\prime$ on $T_{\ell^\prime} E$ is unipotent. It follows that $E$ has good reduction or multiplicative reduction everywhere.
On the other hand, we have by construction that $\operatorname{Ind}_ {G_M}^{G_\mathbb{Q}} \theta_{\lambda^\prime} \vert_{G_F}$ is automorphic of weight $0$ and level $1$ (meaning that it comes from $\pi$ unramified everywhere). So by definition $E[\ell^\prime]^\vee \cong \operatorname{Ind}_ {G_M}^{G_\mathbb{Q}} \bar{\theta}_ {\lambda^\prime} \vert_{G_F}$ is automorphic of weight $0$ and level $1$. By automorphy lifting, we then have that $(T_{\ell^\prime} E)^\vee$ is automorphic of weight $0$ and $U_0$-level prime to $\ell$. Here, we do have by linear disjointness of $F$ and $\mathbb{Q}^\mathrm{av}$ that $$ (\operatorname{Ind}_ {G_M}^{G_\mathbb{Q}} \bar{\theta}_ {\lambda^\prime})(G_{F(\zeta_{\ell^\prime})}) = (\operatorname{Ind}_ {G_M}^{G_\mathbb{Q}} \bar{\theta}_ {\lambda^\prime})(G_{\mathbb{Q}(\zeta_{\ell^\prime})}) $$ is irreducible. So we see that $(T_\ell E)^\vee$ is also automorphic of weight $0$ and $U_0$-level prime to $\ell$.
Theorem 3. Suppose that $\ell \gt 3$ and $\bar{r} \colon G_{\mathbb{Q},S} \to \GL_2(\mathbb{F}_ \ell)$ with $S = \lbrace 2, \ell \rbrace$ be such that
- $\det \bar{r} = \bar{\epsilon}_ \ell^{-1}$,
- $\bar{r} \vert_{G_{\mathbb{Q}_ \ell}}$ is Fontaine–Laffille with Hodge–Tate numbers $0, 1$,
- $\bar{r} \vert_{G_{\mathbb{Q}_ 2}} \sim \begin{pmatrix} 1 & \ast \br 0 & \bar{\epsilon}_ \ell^{-1} \end{pmatrix} \otimes \delta$ with $\delta$ unramified,
- $\bar{r}$ is irreducible.
Then there exists $L/\mathbb{Q}_ \ell$ finite and $r \colon G_{\mathbb{Q},S} \to \GL_2(\mathcal{O}_ L)$ continuous lifting $\bar{r}$ such that
- $\det r = \epsilon_\ell^{-1}$,
- $r \vert_{G_{\mathbb{Q}_ \ell}}$ is Fontaine–Laffille with Hodge–Tate numbers $0, 1$,
- $r \vert_{G_{\mathbb{Q}_ 2}} \sim \begin{pmatrix} 1 & \ast \br 0 & \epsilon_\ell^{-1} \end{pmatrix} \otimes \delta$ with $\delta$ unramified.
Moreover, for $F$ as in the last theorem, the representation $r \vert_{G_F}$ is automrophic of weight $0$ and $U_0$-level prime to $\ell$.
The automorphy lifting theorem does give this lift $r$ because it tells us that the universal deformation ring $R^\mathrm{univ}_ {\bar{r} \vert_{G_F}}$ is finite over $\mathbb{Z}_ \ell$. Applying automorphy lifting again, we do get the automorphy of $r \vert_{G_F}$. Here, we do need to check the irreducibility of $\bar{r} \vert_{G_{F(\zeta_\ell)}}$. By linear disjointness, this would be implied by the irreducibility of $\bar{r} \vert_{G(\zeta_\ell)}$. If $E \subseteq G(\zeta_\ell)$ is the quadratic extension $E/\mathbb{Q}$, it suffices to check that $\bar{r} \vert_{G_E}$ is irreducible. If not, we would have $\bar{r} \cong \operatorname{Ind}_ {G_E}^{G_\mathbb{Q}} \bar{\phi}$ for some $\bar{\phi} \colon G_E \to \mathbb{F}_ {\ell^2}^\times$. But using the Fonatine–Laffille condition we can rule this out.
Remark 4. If $F \supseteq E \supseteq \mathbb{Q}$ where $F \supseteq E$ is soluble, then $r \vert_{G_E}$ is automorphic.