Recall we fixed a finite Galois extension $F/\mathbb{Q}$ with $F \cap \bar{\mathbb{Q}}^{\ker \bar{r}} = \mathbb{Q}$ and unramified at $\ell$. If we had a representation $r \colon G_{\mathbb{Q}} \to \GL_2(\mathbb{F})$, we looked at the deformations $$ R_F^\mathrm{univ} \to R^\mathrm{univ} $$ satisfying $\det r = \epsilon_\ell^{-1}$ with some local conditions. We calculated that $\dim R^\mathrm{univ} \ge 1$ and wanted to prove that $R^\mathrm{univ}$ is finite over $\mathcal{O}$. We then wanted to prove that $R_F^\mathrm{univ} \to R^\mathrm{univ}$ is finite, which reduced to $R^\mathrm{univ} / \mathfrak{m}_ F R^\mathrm{univ}$ is finite over $\mathbb{F}$. We saw that
- $r^\mathrm{univ} \colon G_\mathbb{Q} \to \GL_2(R^\mathrm{univ} / \mathfrak{m}_ F R^\mathrm{univ})$ factors through some finite Galois group $\Gal(E/\mathbb{Q})$, where $E = F \bar{\mathbb{Q}}^{\ker \bar{r}}$,
- the ring $R^\mathrm{univ} / \mathfrak{m}_ F R^\mathrm{univ}$ is generated by $\tr r^\mathrm{univ}(\sigma)$ for $\sigma \in \Gal(E/\mathbb{Q})$.
So it suffices to prove for all $\sigma \in \Gal(E/\mathbb{Q})$ that $\mathbb{F}[\tr r^\mathrm{univ}(\sigma)] \subseteq R^\mathrm{univ} / \mathfrak{m}_ F R^\mathrm{univ}$ is finite over $\mathbb{F}$. Because $\sigma$ has order divisible by $m = [E:\mathbb{Q}]$, this is some linear algebra question.
Lemma 1. If $A$ is the ring $\mathbb{F}[x_{ij}]$ quotiented out by the equation $\begin{pmatrix} x_{11} & x_{12} \br x_{21} & x_{22} \end{pmatrix}^m = 1$ then inside $A$ the subalgebra $\mathbb{F}[T = x_{11} + x_{22}]$ is finite over $\mathbb{F}$.
Proof.
If we consider the polynomial $$ f(T) = \prod_{\zeta_1, \zeta_2 \in \bar{\mathbb{F}}, \zeta_1^m = \zeta_2^m = 1} (T - (\zeta_1 + \zeta_2)), $$ then for all $\mathfrak{p} \subseteq A$ a prime we have $f(T) \in \mathfrak{p}$. This shows that $f(T)$ is nilpotent in $A$ and so $f(T)^n = 0$ in $A$.
Automorphic forms
At this point, it becomes increasingly contrived to not introduce automorphic forms. We were looking at three conditions on our Galois representation $\bar{r} \colon G_{\mathbb{Q},S} \to \GL_2(\mathbb{F}_ \ell)$, namely
- (A) $\det \bar{r} = \bar{\epsilon}_ \ell^{-1}$,
- (B) $\bar{r} \vert_{G_{\mathbb{Q}_ 2}} \sim \begin{pmatrix} 1 & \ast \br 0 & \bar{\epsilon}_ \ell^{-1} \end{pmatrix} \otimes \delta$ for $\delta^2 = 1$ unramified,
- (C) $\bar{r} \vert_{G_{\mathbb{Q}_ \ell}}$ is Fontaine–Laffaille with graded pieces in degree $0, 1$.
What we want to do is to
- show that for some $F/\mathbb{Q}$ finite Galois that $R_F^\mathrm{univ}$ is finite over $\mathcal{O}$,
- then $r$ lifts to $G_{\mathbb{Q},S} \to \GL_2(\bar{\mathbb{Q}}_ \ell)$, and then $r$ fits into a compatible system.
Now we look at the $3$-adic part and get a contradiction. But both of these come from automorphy.
Let $F$ be a number field. We look at the space $$ \mathcal{A}_ 0(\GL_n(F) \backslash \GL_n(\mathbb{A}_ F)) = \lbrace \varphi \colon \GL_n(F) \backslash \GL_n(\mathbb{A}_ F) \to \mathbb{C} \rbrace $$ satisfying
- smoothness, for all $x \in \GL_n(\mathbb{A}_ F)$ there exists a neighborhood $x^\infty \in W \subseteq \GL_n(\mathbb{A}_ F^\infty)$ such that $\varphi \vert_{W \times \GL_n(F_\infty)}$ factors as $W \times \GL_n(F_\infty) \twoheadrightarrow \GL_n(F_\infty) \to \mathbb{C}$ where the latter map is smooth in the usual sense,
- finiteness, $\varphi$ is finite under the action of $\GL_n(\hat{\mathcal{O}}_ F) \times U_\infty$ in the sense $\varphi(- \cdot u)$ over $u \in \GL(\hat{\mathcal{O}}_ F \times U_\infty)$ spans a finite-dimensional space, where $U_\infty$ is the product of $\operatorname{O}(n)$ for real places and $\operatorname{U}(n)$ for complex places,
- writing $\mathfrak{g}_ 0 = \mathfrak{gl}(F_\infty)$ and $\mathfrak{g} = \mathfrak{g}_ 0 \otimes_\mathbb{R} \mathbb{C}$, the action of $x \in \mathfrak{g}_ 0$ given by $$ (x\varphi)(g) = \frac{d}{dt} \varphi(g \exp(tx)) \vert_{t=0} $$ extend $\mathbb{C}$-linearly to $\mathfrak{g}$,
- for the corresponding $U(\mathfrak{g})$-action, $\varphi$ is finite under the action of its center $\mathfrak{Z} \cong \bigotimes_{\tau \colon F \hookrightarrow \mathbb{C}} \mathfrak{Z}_ \tau$,
- $\varphi$ is slowly increasing, there exist constants $C, r$ such that $\lvert \varphi(g) \rvert \le C \lvert g \rvert^r$, where $$ \lVert g \rVert = \prod_v \max(\lvert g_{ij} \rvert_ v, \lvert (g^{-1})_ {ij} \rvert_ v), $$
- for every unipotent radical $N_m \subseteq G$ (block matrices of the form $\begin{pmatrix} 1_m & \ast \br 0 & 1_{n-m} \end{pmatrix}$) we have $$ \int_{N_m(F) \backslash N_m(\mathbb{A}_ F)} \varphi(ng) dn = 0 $$ for all $g \in \GL_n(\mathbb{A}_ F)$.
Remark 2. We note that this space has a right translation action by $\GL_n(\mathbb{A}_ F^\infty)$, but it doesn’t have an action of $\GL_n(F_\infty)$ because we made a choice of maximal compact $U_\infty$. But there is still an action of $\GL_n(\mathbb{A}_ F^\infty) \times U_\infty$ and also an action of $\mathfrak{g}$. There is an obvious compatibility between the two actions of $\mathfrak{g}$ and $U_\infty$.
Now it is known that there is a decomposition $$ \mathcal{A}_ 0(\GL_n(F) \backslash \GL_n(\mathbb{A}_ F)) = \bigoplus \pi $$ where $\pi$ are irreducible $(\mathfrak{g}, \GL_n(\mathbb{A}_ F^\infty) \times U_\infty)$-modules. These $\pi$ are called cuspidal automorphic representations, and $\pi \ncong \pi^\prime$ unless $\pi = \pi^\prime$, i.e., everything occurs with multiplicity $1$. Moreover, each $\pi$ can be decomposed as $$ \pi = \bigotimes_v^\prime \pi_v. $$ For $v \nmid \infty$ each $\pi_v$ is a smooth representation of $\GL_n(F_v)$, and for $v \mid \infty$ each $\pi_v$ is a $(\mathfrak{g}_ v, U_\infty)$-module. Here, for all but finitely many $v$, this $\pi_v$ is unramified, i.e., $\dim \pi_v^{\GL_n(\mathcal{O}_ {F_v})} = 1$, and so we can choose bases $0 \neq e_v \in \pi_v^{\GL_n(\mathcal{O}_ {F_v})}$ for all but finitely $v$, and now we define the restricted tensor product as $$ \pi = \varinjlim_{S \text{ finite}} \bigotimes_{v \in S} \pi_v, $$ where the transition maps are given by tensoring with $e_v$. This is independent of the choice of $e_v$ (and only dependent on just the lines).
By some version of Schur’s lemma, the center $\mathfrak{Z}$ acts on $\pi$ by a homomorphism $\mathfrak{Z} = \bigotimes_{\tau \colon F \hookrightarrow \mathbb{C}} \mathbb{C}[x_1, \dotsc, x_n]^{S_n} \to \mathbb{C}$, and so this corresponds to, for each $\tau$ we have a multiset $\mathrm{HC}_ \tau(\pi_\infty)$ of $n$ complex numbers.
Remark 3. There is a relation between $\mathrm{HC}_ \tau$ and $\mathrm{HC}_ {c\tau}$.
Definition 4. We say that $\pi$ is algebraic if $\mathrm{HC}_ \tau(\pi_\infty)$ consist of integer, and say that $\pi$ is regular if $\mathrm{HC}_ \tau(\pi_\infty)$ has $n$ distinct elements for all $\tau$.
The algebraic ones are what we believe to come from algebraic geometry, and we don’t really know what the non-algebraic ones are there for.
Theorem 5. Suppose that $F$ is a CM field (i.e., there is a $c \in \Aut(F)$ for which $\tau c = c \tau$ for all $\tau \colon F \hookrightarrow \mathbb{C}$), and let $\pi$ be regular algebraic. Fix an isomorphism $\iota \colon \mathbb{C} \cong \bar{\mathbb{Q}}_ \ell$. Then there exists a continuous representation $$ r_{\ell,\iota}(\pi) \colon G_F \to \GL_n(\bar{\mathbb{Q}}_ \ell) $$ such that for all $v \mid \ell$ we have $$ r_{\ell,\iota}(\pi) \vert_{W_{F_v}}^\mathrm{ss} = \operatorname{rec}(\pi_v)^\mathrm{ss}, $$ and if $\pi_v$ is unramified then $r_{\ell,\iota}(\pi)$ is unramified, where $\mathrm{rec}$ is the local Langlands reciprocity map.
Remark 6. The point is to look at the cohomology of of the real manifold $$ X_U = \GL_n(F) \backslash \GL_n(\mathbb{A}_ F) / U \times U_\infty(F_\infty^\times)^0 $$ and compute in terms of automorphic representations.