The basic automorphy lifting theorem we proved was the following form.
Theorem 1. Let $r_1, r_2 \colon G_F \to \mathrm{GL}_ 2(\mathcal{O})$ be two Galois representations. If $\bar{r}_ 1 \cong \bar{r}_ 2$ and $r_1$ is automorphic and $r_1 \vert_{G_{F_v}}$ and $r_2 \vert_{G_{F_v}}$ lie on the same irreducible component of the local deformation ring $R_{\bar{r}_ 1 \vert_{G_{F_v}}}^\square$ then $r_2$ is also automorphic.
For $v \nmid \ell$, we showed that there is actually this trick that allows us to deal with the case when $r_1, r_2$ are on different components. But if $v \mid \ell$, this is a serious problem. We avoided this issue because we looked at Fontaine–Laffaille deformation, so that the local deformation ring is smooth. You can use various tricks to ensure that you are in the same component, and that can get you a long way, but this is not ideal.
When $v \mid \ell$ and $F_v = \mathbb{Q}_ \ell$ with $G = \mathrm{GL}_ 2$, there are two ideas to proceed. There is the idea of Kisin which uses the Breuil–Mezard conjecture, and there is the idea of Emerton which uses completed cohomology. But both of them fundamentally depend on $\ell$-adic local Langlands, which we only understand for $\mathrm{GL}_ 2(\mathbb{Q}_ \ell)$. Nonetheless, there is a generalization of Emerton’s approach due to Pan that solves some of these issues.
What else can go wrong?
When we did this Galois cohomology computation we had $R^\mathrm{loc}_ {\chi,\infty}$ and $\Lambda_\infty$. The Krull dimensions of these were equal, and this was lucky. But it is possible that $\dim R_{\chi,\infty}^\mathrm{loc} \lt \dim \Lambda_\infty$. In this case, Calegari–Geraghty understood what to do. This discrepancy is reflected in the fact that automorphic forms occur in more than one degree in cohomology. Then we can take some derived approach.
Another thing is that the Cheboratev argument for finding these suitable auxiliary primes $Q_N$ can fail. In this case, we are pretty ignorant. The only argument we have is Skinner–Wiles, generalized by Pan. For example, let $\pm \ell$ be $1 \bmod{4}$ and consider $\mathbb{Q}(\sqrt{\pm \ell}) \subseteq \mathbb{Q}(\zeta_\ell)$ and a character $\bar{\chi} \colon G_{\mathbb{Q}(\sqrt{\pm \ell})} \to \mathbb{F}^\times$. Let $\bar{\chi}^\prime$ be the composition of $\chi$ with conjugation by some $\tau \in G_{\mathbb{Q}} - G_{\mathbb{Q}(\sqrt{\pm \ell})}$ so that $\bar{\chi} \neq \bar{\chi}^\prime$. Let us use $\bar{r} = \operatorname{Ind}_ {G_{\mathbb{Q}(\sqrt{\pm \ell})}} ^{G_{\mathbb{Q}}} \bar{\chi}$. We wanted to find $\sigma \in G_\mathbb{Q}$ such that
- $\sigma \in G_{\mathbb{Q}(\zeta_\ell)}$,
- $\bar{r}(\sigma)$ has distinct eigenvalues,
- $\sigma$ has eigenvalue $1$ on $\operatorname{Ind}_ {G_{\mathbb{Q}(\sqrt{\pm \ell})}} ^{G_{\mathbb{Q}}} (\bar{\chi}/\bar{\chi}^\prime)$.
One can check that it is impossible to satisfy all three conditions.
Theorem 2 (Pan). Let $F$ be a totally real field of even degree. Let $\bar{r} \colon G_F \to \mathrm{GL}_ 2(\mathbb{F})$ be modular and irreducible, and let $r \colon G_F \to \mathrm{GL}_ 2(\mathcal{O})$ be a crystalline lift of any weight. Then $r$ is modular.
There is a number of tools we are going to need. So let’s start with some of them.
Pseudo-representations
Let $R$ be a topological ring, and let $\Gamma$ be topologically finitely generated profinite group.
Definition 3. A ($2$-dimensional) pseudo-representation is a continuous function $T \colon \Gamma \to R$ such that
- $T(1) = 2$,
- $T(\sigma \tau) = T(\tau \sigma)$,
- $T(\sigma \tau \rho) + T(\sigma \rho \tau) - T(\sigma) T(\rho \tau) - T(\tau) T(\sigma \rho) - T(\rho) T(\sigma \tau) + T(\sigma) T(\rho) T(\tau) = 0$,
for all $\sigma, \tau, \rho \in \Gamma$.
Lemma 4. If $r \colon \Gamma \to \mathrm{GL}_ 2(R)$ is a continuous representation, then $\tr(r) \colon \Gamma \to R$ is a pseudo-representation.
Where does this last formula come from? Note that $\det(A) = \tfrac{1}{2}(\tr(A)^2 - \tr(A^2))$. Now if we expand the Cayley–Hamilton formula for $\lambda A + \mu B$ and look at the coefficient of $\lambda \mu$, we get $$ AB + BA - \tr(A) B - \tr(B) A - \tr(AB) + \tr(A) \tr(B) = 0. $$ Now we can multiply this by $C$ and take the trace.
Facts 5. 1. If $R$ is an algebraically closed field of characteristic not $2$ (or $n! \neq 0$ in $R$ if we are working with $n$-dimensional ones) and if $T \colon \Gamma \to R$ is a pseudo-representation, then there exists a (semi-simple) representation $r \colon \Gamma \to \mathrm{GL}_ 2(R)$ with $T = \tr(r)$.
- Given $\Gamma$ there exists a finite subset $S \subseteq \Gamma$ such that any pseudo-representation $T \colon \Gamma \to R$ is determined by its values on $S$.
It is easier to deform pseudo-representations. Consider the deformation functor $\mathcal{F}$ on local Artinian $\mathcal{O}$-algebras with residue field $\mathbb{F}$.
Proposition 6. Let $\mathcal{F}$ be a functor from the category of local Artinian $\mathcal{O}$-algebras with residue field $\mathbb{F}$ to sets. This functor $\mathcal{F}$ is pro-representable by a complete Noetherian local $\mathcal{O}$-algebra if and only if
- $\mathcal{F}(\mathbb{F}) = \ast$,
- for every $A \to B \leftarrow C$ we have $\mathcal{F}(A \times_B C) = \mathcal{F}(A) \times_{\mathcal{F}(B)} \mathcal{F}(C)$,
- $\mathcal{F}(\mathbb{F}[\epsilon] / (\epsilon^2))$ is finite(-dimensional over $\mathbb{F}$).
For (2), it is actually sufficient to consider certain special cases.
So now if $\bar{T} \colon \Gamma \to \mathbb{F}$ is a pseudo-representation, then consider the deformation functor $\mathcal{F}$. This obviously satisfies both (1) and (2), and (3) is also true because we can just test on the subset $S \subseteq \Gamma$. So we get a universal deformation ring $R_{\bar{T}}^\mathrm{ps}$.
If we consider $$ (\det T)(\sigma) = \frac{1}{2} (T(\sigma)^2 - T(\sigma^2)) $$ then this is actually a homomorpism $\Gamma \to R^\times$. Then we can impose this determinant condition and get $R_{\bar{T},\chi}^\mathrm{ps}$ as well.
Let $\bar{r} \colon \Gamma \to \mathrm{GL}_ 2(\mathbb{F})$. We then have $$ R_{\tr(\bar{r})}^\mathrm{ps} \to R_{\bar{r}}^\mathrm{univ} \to R_{\bar{r}}^\square $$ if the centralizer of $\bar{r}$ in $\mathrm{GL}_ 2(\mathbb{F})$ is $\mathbb{F}^\times$. In this case, we have a non-canonical isomorphism $$ R_{\bar{r}}^\square \cong R_{\bar{r}}^\mathrm{univ}[[A_1, A_2, A_3]]. $$ We also have that the image of $R_{\tr(\bar{r})}^\mathrm{ps} \to R_{\bar{r}}^\mathrm{univ}$ is generated by the elements $\tr r^\mathrm{univ}(\sigma)$ for $\sigma \in \Gamma$.
Fact 7 (Nyssen). If $\bar{r}$ is absolutely irreducible, then $R_{\tr(\bar{r})}^\mathrm{ps} \cong R_{\bar{r}}^\mathrm{univ}$.
Example 8 (Bellaïche). Let $F/\mathbb{Q}_ \ell$ be finite and $\zeta_\ell \notin F$. Let $\bar{\chi}_ 1, \bar{\chi}_ 2 \colon G_F \to \mathbb{F}^\times$ and assume $\bar{\chi}_ 1 / \bar{\chi}_ 2 \neq 1, \bar{\epsilon}_ \ell^{\pm 1}$. Then $\bar{T} = \bar{\chi}_ 1 + \bar{\chi}_ 2$ is a pseudo-representation, and we consider some nontrivial extension $$ \bar{r} = \begin{pmatrix} \bar{\chi}_ 1 & \ast \br 0 & \bar{\chi}_ 2 \end{pmatrix}. $$ For $R = R_\bar{T}^\mathrm{ps}$ and $R_{\bar{r}}^\mathrm{univ}$ and $R_{\bar{r}}^\square$, the dimension of the tangent space is going to be $$ (1 + [F:\mathbb{Q}_ \ell])^2 + 1, \quad 1 + 4[F:\mathbb{Q}_ \ell], \quad 4(1 + [F:\mathbb{Q}_ \ell]). $$
Mod-ℓ representations
Let $(A, \mathfrak{m})$ be a complete Noetherian local $\mathcal{O}$-algebra with residue field $\mathbb{F}$. We consider $G = \mathrm{GL}_ 2(\mathbb{Q}_ \ell)$ and $Z = \mathbb{Q}_ \ell^\times$ and $G_0 = \mathrm{GL}_ 2(\mathbb{Z}_ \ell)$ and $B \subseteq G$ the upper-triangular elements. (We can also have $G = \mathrm{GL}_ 2(\mathbb{Q}_ \ell)^n$ and $Z = (\mathbb{Q}_ \ell)^n$ and $G_0 = \mathrm{GL}_ n(\mathbb{Z}_ \ell)^n$ and $B = (\operatorname{up-tri})^n$.)
We consider the category $\mathsf{Mod}_ G^\mathrm{sm}(A)$ of smooth representations $M$ of $G$ on $A$-modules. This means that for all $m \in M$ there exists an $r$ such that $\mathfrak{m}^r m = (0)$ and $\operatorname{Stab}_ G(m)$ is open. Inside here there is the locally finite full subcategory $$ \mathsf{Mod}_ G^{\mathrm{lfin}}(A) \subseteq \mathsf{Mod}_ G^\mathrm{sm}(A) $$ of those $M$ such that for all $m \in M$ there exists a $m \in N \subseteq M$ such that $N$ is a finite length object (as an object of $\mathsf{Mod}_ G^\mathrm{sm}(A)$). We also fix the central action so that $\mathsf{Mod}_ {G,\eta}^\mathrm{sm}(A)$ is the full subcategory on which $Z$ acts via a character $\eta \colon Z \to A^\times$.
Facts 9. 1. These are all abelian categories.
- These admit exact small inductive limits.
- They have a single generator $X$, i.e., for all $f, g \colon M \to N$ with $f \neq g$, there exists a $h \colon X \to M$ such that $f \circ h \neq g \circ h$.
Concretely, this generator is $$ X = \bigoplus_{r,U} \operatorname{smInd}_ U^G (A/\mathfrak{m}^r) $$ or $$\operatorname{smInd}_ {UZ}^G$ if we are fixing the central character.