We more or less proved this theorem.
Theorem 1 (Theorem B). Let $\mathbb{F}/\mathbb{F}_ 3$ be a finite extension and $\bar{r} \colon G_\mathbb{Q} \to \mathrm{GL}_ 2(\mathbb{F})$ such that
- $\det \bar{r} = \bar{\epsilon}_ 3^{-1}$,
- $\bar{r}$ is unramified outside $6$,
- $\bar{r} \vert_{G_{\mathbb{Q}_ 2}} \sim \begin{pmatrix} 1 & \ast \br 0 & \bar{\epsilon}_ 3^{-1} \end{pmatrix} \otimes \delta$ with $\delta^2 = 1$ unramified,
- $\bar{r} \vert_{G_{\mathbb{Q}_ 3}} = \mathbb{G}(M)$ for some $M \in \mathcal{MF}_ {\mathbb{Q}_ 3}$ with $\operatorname{rk}_ \mathbb{F} \mathrm{gr}^i M = 1$ for $i = 0, 1$ and $0$ otherwise.
Then $\bar{r}$ is of the form $\begin{pmatrix} 1 & \ast \br 0 & \bar{\epsilon}_ 3^{-1} \end{pmatrix}$.
There is a $3$-adic version of this theorem.
Theorem 2 (Theorem B'). Suppose $L / \mathbb{Q}_ 3$ be a finite extension with integers $\mathcal{O}$, and let $r \colon G_\mathbb{Q} \to \mathrm{GL}_ 2(\mathcal{O})$ be a continuous representation such that
- $\det r = \epsilon_3^{-1}$,
- $r$ is unramified outside $6$,
- $r \vert_{G_{\mathbb{Q}_ 2}} \sim \begin{pmatrix} 1 & \ast \br 0 & \epsilon_3^{-1} \end{pmatrix} \otimes \delta$ with $\delta^2 = 1$ unramified,
- $r \vert_{G_{\mathbb{Q}_ 3}} = \mathbb{G}(M)$ for $M \in \mathcal{MF}_ {\mathbb{Q}_ 3, \mathcal{O}}$, where $\mathrm{rk}_ \mathcal{O} \mathrm{gr}^i M = 1$ for $i = 0, 1$ and $0$ otherwise.
Then $(r \otimes L)^\mathrm{ss} = 1 \oplus \epsilon_3^{-1}$.
Proof.
Consider $G_\mathbb{Q}$-stable $\mathcal{O}$-modules $\Lambda \subseteq \mathcal{O}^{\oplus 2}$ with the property that $\im(\Lambda \to (\mathcal{O}/\lambda)^{\oplus 2})$ contains $((\mathcal{O}/\lambda)^{\oplus 2})^{G_\mathbb{Q}}$. There is a minimal such $\Lambda$, because if we have a chain $\lbrace \Lambda_i \rbrace$ then we can find lifts of a generator $\bar{e} \in ((\mathcal{O}/\lambda)^{\oplus 2})^{G_\mathbb{Q}}$ and then these $e_i$ has a limit $e$ by compactness.
Now we claim that this minimal $\Lambda$ cannot have rank $2$. If it had rank $2$, then we would have $$ \Lambda \to \Lambda \otimes \mathbb{F} \xrightarrow{\neq 0} (\mathcal{O}/\lambda)^{\oplus 2}. $$ The Galois action on $\Lambda \otimes_\mathcal{O} \mathbb{F}$ has a fixed line $L$ by Theorem B. Then $$ \Lambda^\prime = \ker(\Lambda \to (\Lambda \otimes \mathbb{F}) / L) \subsetneq \Lambda $$ is a small $G_\mathbb{Q}$-stable lattice, and this also contains the lift $e$.
This shows that $\operatorname{rk} \Lambda = 1$, and so we have $$ r \sim \begin{pmatrix} \chi_1 & \ast \br 0 & \chi_2 \end{pmatrix} $$ where $\chi_1, \chi_2$ are unramified at $2$. Similarly by studying the Fontaine–Laffaille modules we see that either $\chi_i$ or $\chi_i \otimes \epsilon_3$ is unramified at $3$. This shows that $\chi_i$ is either $1$ or $\epsilon_3^{-1}$.
Now we want to prove the following theorem.
Theorem 3 (Theorem A). Let $\ell \ge 3$ be a prime, and let $\bar{r} \colon G_\mathbb{Q} \to \mathrm{GL}_ 2(\mathbb{F}_ \ell)$ be a continuous representation. If
- $\det \bar{r} = \epsilon_\ell^{-1}$,
- $\bar{r}$ is unramified outside $2\ell$,
- $\bar{r} \vert_{G_{\mathbb{Q}_ 2}} = \begin{pmatrix} 1 & \ast \br 0 & \bar{\epsilon}_ \ell^{-1} \end{pmatrix} \otimes \delta$ for $\delta^2 = 1$ unramified,
- $\bar{r} \vert_{G_{\mathbb{Q}_ \ell}} = \mathbb{G}(M)$ for some $M \in \mathrm{MF}_ {\mathbb{Q}_ \ell}$ such that $\mathrm{rk}_ {\mathbb{F}_ \ell} \mathrm{gr}^i M = 1$ for $i = 0, 1$ and $0$ otherwise,
then $\bar{r}$ is reducible.
Strategy of the proof of Theorem A
Here are the steps.
Show that there exists $L / \mathbb{Q}_ \ell$ a finite extension with $\mathcal{O} = \mathcal{O}_ L \supseteq \lambda$ and a continuous $r \colon G_\mathbb{Q} \to \mathrm{GL}_ 2(\mathcal{O})$ such that
- $\det r = \epsilon_\ell^{-1}$,
- $r$ is unramified outside $2\ell$,
- $r \vert_{G_{\mathbb{Q}_ 2}} \sim \begin{pmatrix} 1 & \ast \br 0 & \epsilon_\ell^{-1} \end{pmatrix} \otimes \delta$ with $\delta^2 = 1$ unramified,
- $r \vert_{G_{\mathbb{Q}_ \ell}} = \mathbb{G}_ M$ for $\operatorname{rk}_ \mathcal{O} \mathrm{gr}^i M = 1$ for $i = 0, 1$ and $0$ otherwise,
- $r \bmod{\lambda} \cong \bar{r}$.
Show that there exists a number field $M$ and for each prime $\mu$ of $M$ a continuous representation $r_\mu \colon G_\mathbb{Q} \to \mathrm{GL}_ 2(M_\mu)$ such that
- each $r_\mu$ satisfies (1)–(4) of above,
- $r \cong r_\lambda$ for some $\lambda \mid \ell$,
- if $p \nmid 2N(\mu \mu^\prime)$ then $\mathrm{tr} r_\mu(\mathrm{Frob}_ p) = \mathrm{tr} r_{\mu^\prime}(\mathrm{Frob}_ b)$ for all $\mu, \mu^\prime$.
If you believe in the Fontaine–Mazur conjecture, this you would think that $r$ has some geometric origin, and then $r_\mu$ are the corresponding family of representations.
Choose $\mu \mid 3$ so that we have $r_\mu^\mathrm{ss} \sim 1 \oplus \epsilon_3^{-1}$. Then $\tr r_\mu(\mathrm{Frob}_ p) = 1 + p$ and then for $p \nmid 6\ell$ we have $$ \mathrm{tr}(r = r_\lambda)(\mathrm{Frob}_ p) = 1 + p = \mathrm{tr}(1 + \epsilon_\ell^{-1}) (\mathrm{Frob}_ p). $$ Now the Chebotarev theorem implies that if $\mathrm{tr}(r) = \mathrm{tr} (1 \oplus \epsilon_\ell^{-1})$ for almost all primes $p$ then $r^\mathrm{ss} \cong 1 \oplus \epsilon_\ell^{-1}$.
Deformations of Galois representations
We are going to look at the category $\mathcal{C}_ \mathcal{O}$ of Noetherian complete local $\mathcal{O}$-algebras $R$ such that $\mathbb{F} \cong R/\mathfrak{m}$. We are going to look at the functor $$ D \colon \mathcal{C}_ \mathcal{O} \to \mathsf{Sets} $$ sending $R$ to the set of $r \colon G_\mathbb{Q} \to \mathrm{GL}_ 2(R)$ such that
- $r \bmod{\mathfrak{m}} = \bar{r}$,
- $\det r = \epsilon_\ell^{-1}$,
- $r$ is unramified outside $2, \ell$,
- $r \vert_{G_{\mathbb{Q}_ 2}} \sim \begin{pmatrix} 1 & \ast \br 0 & \epsilon_\ell^{-1} \end{pmatrix} \otimes \delta$ with $\delta^2 = 1$ unramified,
- for all $J \subseteq R$ open we have $(R/J)^2 = \mathbb{G}(M_J)$
modulo conjugation by elements of $\ker(\mathrm{GL}_ 2(R) \to \mathrm{GL}_ 2(\mathbb{F}))$.
Proposition 4. If $\bar{r}$ is absolutely irreducible then $D$ is representable by some $(R^\mathrm{univ}, [r^\mathrm{univ}])$.
There are some subtleties in this fact, which we will discuss next time. The way we will use it is that we will have $$ \dim R^\mathrm{univ} \ge H^1(G_\mathbb{Q}, \cdots) - H^2(G_\mathbb{Q}, \cdots) = 1. $$ The harder thing to see is that $R^\mathrm{univ}$ deforms outside characteristic $\ell$. To see this, we will show that $R^\mathrm{univ}$ is a finitely generated $\mathcal{O}$-module. Then there will be a ring homomorphism $$ R^\mathrm{univ} \xrightarrow{\phi} \mathcal{O}_ {L^\prime} $$ for some $L^\prime / L$ a finite extension. Then $\phi \circ r^\mathrm{univ}$ will do.