We are now going to study a purely local problem. Let $R$ be a set of bad primes of $F$, such that for $v \in R$ we have $q_v = k(v) \equiv 1 \pmod{\ell}$. (The coefficient $\mathcal{O}$ where $L / \mathbb{Q}_ \ell$ and $\mathcal{O} = \mathcal{O}_ L$ and $\mathcal{O}/\lambda = \mathbb{F}$.) We had a $\ell$-power order character $\chi_v \colon k(v)^\times \to \mathcal{O}^\times$ for each $v$, and $\chi = \prod \chi_v$. We then looked at the universal lifting ring $R_{\chi_v}^\square$ over $\mathcal{O}$ for lifts $r \colon G_{F_v} \to \mathrm{GL}_ 2(T)$ (where $T$ is a test complete local Noetherian $\mathcal{O}$-algebra) such that $r \pmod \mathfrak{m}_ T = 1$ and $\det r = \epsilon_\ell^{-1}$ and $\operatorname{char}_ {r(\sigma)}(x) = (x - \chi_v(\sigma)) (x - \chi_v(\sigma)^{-1})$ for $\sigma \in I_{F_v}$. Then we looked at $$ R_\chi^\mathrm{loc} = \bigotimes_{v \in R}^\wedge R_{\chi_v}^\square. $$
Proposition 1. 1. If $\chi_v \neq 1$ for all $v \in R$ then $\operatorname{Spec} R_\chi^\mathrm{loc}$ is irreducible and its generic point has characteristic $0$ and Krull dimension $R_\chi^\mathrm{loc} = 3 \lvert R \rvert + 1$.
- Let $C_1, \dotsc, C_r$ denote the irreducible components of $\operatorname{Spec} R_1^\mathrm{loc}$. Then each $C_i$ has generic point in characteristic zero and $\dim C_i = 3 \lvert R \rvert + 1$, and the special fibers $C_i \times_{\mathcal{O}} \mathbb{F}$ remain irreducible and distinct.
We briefly discussed last time that these deformations are going to be trivial on wild inertia, hence tamely ramified. So this parametrizes matrices $$ \biggl\lbrace (\Phi_v, \Sigma_v) \in M_{2 \times 2}(T) : \begin{matrix} \Phi_v^{-1} \Sigma_v \Phi_v = \Sigma_v^{q_v}, \Phi_v, \Sigma_v \equiv 1 \pmod{\mathfrak{m}_ t}, \br \det \Phi_v = q_v, \operatorname{char}_ {\Sigma_v}(x) = (x - \zeta_v) (x - \zeta_v^{-1}) \end{matrix} \biggr\rbrace $$ for $\phi_v \in G_{F_v}$ a lift of $\mathrm{Frob}_ v$ and $\tau_ v \in I_{F_v}$ a generator of $I_{F_v} / P_{F_v}$ with $\zeta_v = \chi_v(\tau_v)$.
So let $\mathcal{M}_ \chi / \mathcal{O}$ be the moduli space of matrices satisfying these relations (minus the reduction) so that we have $$ M_\chi = \prod_{v \in R} M_{\chi_v} $$ and $R_\chi^\mathrm{loc}$ is just the formal completion of $M_\chi$ at $1$.
- If $\zeta_v \neq 1$, then the characteristic polynomial of $\Sigma_v$ divides $x^{q_v} - x$, and so $\Sigma_v^{q_v} = \Sigma_v$. Then $\Sigma_v$ and $\Phi_v$ commutes.
- If $\zeta_v = 1$ then $\Sigma_v$ is unipotent and so we might as well instead try to parametrize $N_v = \Sigma_v - 1$. Then we have $\det \Phi_v = q_v$ and $\operatorname{char}_ {N_v}(x) = x^2$ and $\Phi_v^{-1} N_v \Phi_v = q_v N_v$.
Lemma 2. (The second part of the proposition can be checked globally on $\mathcal{M}$.) Let $X/\mathcal{O}$ is a scheme of finite type, let $x \in X(\mathbb{F})$, and let $X_i$ be the irreducible components of $X$. Assume that $X_i$ all have generic points in characteristic zero, irreducible in dimension $d$ and the special fibers $X_i \times_{\mathcal{O}} \mathbb{F}$ are irreducible and distinct. Then $\operatorname{Spec} \mathcal{O}_ {X,x}^\wedge$ has the same property.
So we just need to check the corresponding property for $\mathcal{N} = \prod_v \mathcal{N}_ v$, and then we can even reduce it to checking the property for each $\mathcal{N}_ v$.
Lemma 3. Suppose $R$ is a complete Noetherian local $\mathcal{O}$-algebra. Then $R[1/\ell]$ is Jacobson (i.e., every prime ideal is the intersection of maximal ideals). Moreover, if $\mathfrak{p}$ is a maximal ideal of $R[1/\ell]$ then $k(\mathfrak{p})$ is a finite extension of $L$ and the image of $R$ in $k(\mathfrak{p})$ is a finite $\mathcal{O}$-module.
For the second part of this lemma, we can assume without loss of generality that $\mathfrak{p} \cap R = (0)$. Then $R[1/\ell]$ is a field, so $\dim(R/\ell) = 0$ and so $R/\ell$ is Artinian. This means that it is finite over $\mathbb{F}$ and so $R$ is finite over $\mathcal{O}$.
The first part of the proposition now reduces to
- $\operatorname{Spec} R_\chi[1/\ell]$ is connected,
- if $\mathfrak{p}$ is a maximal ideal of $R_\chi[1/\ell]$ then $R_\chi[1/\ell]_ \mathfrak{m}^\wedge$ is a power series ring over $k(\mathfrak{p})$,
- $M_{\chi_v}^\mathrm{red}$ is flat over $\mathcal{O}$,
- $M_{\chi_v}$ has dimension $4$.
Let us now actually prove these properties. Let us look at the second part, and try to study $\mathcal{N}_ v$ parametrizing $(\Phi, N)$ satisfying $\det \Phi = q_v$ and $\Phi^{-1} N \Phi = q_v N$ and $\operatorname{char}_ N(x) = x^2$. There is a locus $\mathcal{N}_ 0 \subseteq \mathcal{N}_ v$ where $N = 0$ and this is isomorphic to $\mathrm{SL}_ 2$. There are two other components $$ \mathcal{N}_ \pm \subseteq \mathcal{N}_ v $$ where $\tr \Phi = \pm (1 + q_v)$. If we analyze the field-valued points if $\mathcal{N}$, we see that either $N = 0$ or $N \sim (\begin{smallmatrix} 0 & 1 \br 0 & 0 \end{smallmatrix})$ and then $\Phi \sim (\begin{smallmatrix} \beta & \gamma \br 0 & q_v \beta \end{smallmatrix})$. But $\det \Phi = q_v$ so $b^2 = 1$ implies $\beta = \pm 1$, so that $\tr \Phi = \pm (1 + q_v)$. In fact, we have two maps $$ (\mathrm{PGL}_ 2 \times \mathbb{A}^1) / \mathbb{G}_ a \xrightarrow{\cong} (\mathcal{N}_ \pm - \mathcal{N}_ 0)^\mathrm{red}; \quad (g, a) \mapsto \biggl( \pm g \begin{pmatrix} 1 & a \br 0 & q_v \end{pmatrix} g^{-1}, g \begin{pmatrix} 0 & 0 \br 0 & 1 \end{pmatrix} g^{-1} \biggr), $$ where the $\mathbb{G}_ a$-action is given by sending $(g, a)$ to $(g, (\begin{smallmatrix} 1 & x \br 0 & 1 \end{smallmatrix}), a + (q_v - 1) x)$. We can also check that $\mathcal{N}_ \pm^\mathrm{red}$ is the closure of $\mathcal{N}_ \pm - \mathcal{N}_ 0$ in $\mathcal{N}_ v$ (by writing down a surjective map $\mathrm{PGL}_ 2 \times \mathbb{A}^1 \to (\mathcal{N}_ \pm \cap \mathcal{N}_ 0)^\mathrm{red}$ and checking that all of them generalize to $\mathcal{N}^\pm - \mathcal{N}_ 0$). So these three are the irreducible components of $\mathcal{N}_ v$, all of dimension $4$ and generically characteristic zero, with fibers over $\mathcal{O}$ being geometrically irreducible and distinct.
Let us now move on to studying $\mathcal{M}_ {\chi_v}$ when $\chi_v \neq 1$. For $T \subseteq \mathrm{PGL}_ 2$ the diagonal torus, we have a map $$ \mathrm{PGL}_ 2 / T \times \mathbb{G}_ m; \quad (g, a) \mapsto \biggl( g \begin{pmatrix} a & 0 \br 0 & q_v a^{-1} \end{pmatrix} g^{-1}, g \begin{pmatrix} \zeta_v & 0 \br 0 & \zeta_v^{-1} \end{pmatrix} g^{-1} \biggr) $$ that is an isomorphism on the reduction of the generic fiber. In particular, the generic fiber is geometrically irreducible of dimension $3$.
On the other hand, on the special fiber we have an isomorphism $$ (\mathcal{M}_ {\chi_v} \times \mathbb{F})^\mathrm{red} \cong (\mathcal{N}_ v \times \mathbb{F})^\mathrm{red}. $$ We already know that this has three components of dimension $3$. So we will be done with the analysis once we check that on each irreducible there is a point not in the other components that lifts to characteristic zero. We check that in $\mathcal{N}_ 0 \times \mathbb{F}$ we have $$ \biggl( \begin{pmatrix} \alpha & 0 \br 0 & q_v/\alpha \end{pmatrix}, \begin{pmatrix} 0 & 0 \br 0 & 0 \end{pmatrix} \biggr) \rightsquigarrow \biggl( \begin{pmatrix} \tilde{\alpha} & 0 \br 0 & q_v / \tilde{\alpha} \end{pmatrix}, \begin{pmatrix} \zeta_v & 0 \br 0 & \zeta_v^{-1} \end{pmatrix} \biggr) $$ where $\alpha \neq \pm 1$ and $\tilde{\alpha}$ lifts $\alpha$. We also have $$ \biggl( \pm \begin{pmatrix} 1 & 0 \br 0 & q_v \end{pmatrix}, \begin{pmatrix} 0 & 1 \br 0 & 0 \end{pmatrix} \biggr) \rightsquigarrow \biggl( \begin{pmatrix} 1 & (1 - q_v) / (\zeta_v - \zeta_v^{-1}) \br 0 & q_v \end{pmatrix}, \begin{pmatrix} \zeta_v & 1 \br 0 & \zeta_v^{-1} \end{pmatrix} \biggr). $$