We had $F/\mathbb{Q}$ some even degree totally real and $R$ a finite set of primes of $F$ away from $\ell$ with the property that $q_v \equiv 1 \pmod{\ell}$ and also characters $\chi_v \colon k(v)^\times \to \mathcal{O}^\times$ of $\ell$-power order for each $v \in R$. (Here for the coefficients we have $L/\mathbb{Q}_ \ell$ is finite and $\mathcal{O} = \mathcal{O}_ L$ with $\mathbb{F} = \mathcal{O}/\lambda$.) We had the space of modular forms and a corresponding Hecke algebra. We were looking at $\mathfrak{m}$ a non-Eisenstein maximal ideal and enlarge $F$ is necessary so that $\bar{r}_ \mathfrak{m} \vert_{G_{F_v}} = 1$.
We had an additional finite set $Q$ disjoint from $R \cup \lbrace v \mid \ell \rbrace$ such that $q_v \equiv 1 \pmod{\ell}$ for $\ell \in Q$ and $\bar{r}_ \mathfrak{m}(\mathrm{Frob}_ v)$ having distinct eigenvalues $\alpha_v \neq \beta_v$ in $\mathbb{F}$. Then we defined this space $$ S(Q, \chi) $$ of modular forms localized at $\mathfrak{m}_ q$ with level $Q$ and some conditions on the $U_a$ operator for $a \in \mathcal{O}_ {F_v} - \lbrace 0 \rbrace$ if $v \in Q$. This had the action of the Hecke algebra $$ \mathbb{T}(Q, \chi). $$ For each $v \in Q$ let $H_v$ be the maximal $\ell$-power order quotient of $k(v)^\times$ and write $H_Q = \prod_{v \in Q} H_v$. Then we have a map $$ \mathcal{O}[H_Q] \to \mathbb{T}(Q, \chi) $$ that realizes the action of $H_Q$ on $S(Q, \chi)$. We stated that
- $S(Q, \chi)$ is a finite free module over $\mathcal{O}[H_Q]$,
- $S(Q, \chi) / \mathfrak{a}_ Q S(Q, \chi) \cong S(\emptyset, \chi)$.
There is a corresponding picture on the Galois side. We consider $R_{Q, \chi}^\mathrm{univ}$ the universal deformation ring for lifts $r$ of $\bar{r}_ \mathfrak{m}$ such that
- $\det r = \epsilon_\ell^{-1}$,
- $r$ is unramified away from $R \cup Q \cup \lbrace v \mid \ell \rbrace$,
- if $v \mid \ell$ then $r \vert_{G_{F_v}}$ is Fontaine–Laffaille with Hodge–Tate weights $0, 1$,
- if $v \in R$ and $\sigma \in I_{F,v}$ then $r(\sigma)$ has characteristic polynomial $(x - \chi_v(\sigma)) (x - \chi_v(\sigma)^{-1})$.
Let us call $r_{Q,\chi}^\mathrm{univ}$ the corresponding Galois representation. Because there is a Galois representation valued in $\mathbb{T}(Q, \chi)$, we have a map $$ R_{Q,\chi}^\mathrm{univ} \twoheadrightarrow \mathbb{T}(Q, \chi); \quad \tr r_{Q,\chi}^\mathrm{univ}(\mathrm{Frob}_ v) \mapsto T_v. $$
When $v \in Q$ we have $$ r_{Q,\chi}^\mathrm{univ} \vert_{G_{F_v}} \sim \chi_{\alpha_v} \oplus \chi_{\beta_v}, $$ where $\chi_{\alpha_v}$ and $\chi_{\beta_v}$ are characters that are unramified modulo $\mathfrak{m}$. On the inertia, the character recovers $$ \chi_{\alpha_v} \vert_{I_{F_v}} \colon H_v \to (R_{Q,\chi}^\mathrm{univ})^\times, $$ and so the locus where this is unramified is exactly $$ R_{\chi,Q}^\mathrm{univ} / \mathfrak{a}_ Q \cong R_{\chi,\emptyset}^\mathrm{univ}. $$
Theorem 1. The ring homomorphism $R_{\emptyset,\chi}^\mathrm{univ} \twoheadrightarrow \mathbb{T}(\emptyset, \chi)$ has nilpotent kernel.
What we really need is the case $\chi = 1$, but the point is that once we have this we can lift a ring homomorphism $R_{\emptyset,\chi}^\mathrm{univ} \to \mathcal{O}^\prime$ to $\mathbb{T}(\emptyset, \chi) \to \mathcal{O}^\prime$.
Framing
The way we are going to approach this is to view $R_{\emptyset,\chi}^\mathrm{univ}$ as some sort of local deformation ring at $v \in R$. There exists a local lifting ring $R_v^\square$ that parametrizes lifts with no equivalence relation. On the global side, we look at the space of (where $T$ is the test ring)
- $r \colon G_F \to \mathrm{GL}_ 2(T)$ satisfying the conditions listed before,
- for each $v \in R$ an element $\alpha_v \in \ker(\mathrm{GL}_ 2(T) \to \mathrm{GL}_ 2(\mathbb{F}))$,
with the equivalence relation $$ (r, \lbrace \alpha_v \rbrace) \sim (\beta r \beta^{-1}, \lbrace \beta \alpha_v \rbrace) $$ for $\beta \in \ker(\mathrm{GL}_ 2(T) \to \mathrm{GL}_ 2(\mathbb{F}))$. This is representable by some $R_{Q,\chi}^\square$. Here, the point is that the elements $$ (\alpha_v^\mathrm{univ})^{-1} r^\mathrm{univ} \alpha_v^\mathrm{univ} \vert_{G_{F_v}} $$ are well-defined for $v \in R$, and so we get a map $$ R_{Q,\chi}^\mathrm{loc} = \bigotimes_{v \in R}^\wedge R_v^\square \to R_{Q,\chi}^\square. $$
Non-canonically, we just have $$ R_{Q,\chi}^\square = R_{Q,\chi}^\mathrm{univ}[[A_{i,j,v} : v \in R, 1 \le i, j \le 2 ]] / (A_{1,1,v_0}) $$ is a power series ring in $4\lvert R \rvert - 1$ variables. So if we similarly define $$ \Lambda_Q = \mathcal{O}[H_Q] [[A_{i,j,v}]] / (A_{1,1,v_0}) \supset \mathfrak{a}_ Q = \langle h-1, A_{i,j,v} \rangle, $$ then we can write $$ R_{Q,\chi}^\square = R_{Q,\chi}^\mathrm{univ} \hat{\otimes}_ {\mathcal{O}[H_Q]} \Lambda_Q, \quad S^\square(Q, \chi) = S(Q, \chi) \otimes_{\mathcal{O}[H_Q]} \Lambda_Q. $$
To summarize, for every $Q$ as above we have $$ R_\chi^\mathrm{loc}[[x_1, \dotsc, x_{d_Q}]] \twoheadrightarrow R_{Q,\chi}^\square \curvearrowright S^\square(Q, \chi), $$ where $S^\square(Q, \chi)$ is finite free over $\Lambda_Q$, and quotienting by $\mathfrak{a}_ Q$ gives $$ R_{\emptyset,\chi} \curvearrowright S(\emptyset, \chi). $$
Deformation theory
What is the smallest value of $d_Q$ we can take? This is given by $$ d_Q = \dim H_{\mathcal{L}_ Q^\perp}^1 (G_F, \mathrm{ad}^0 \bar{r}(1)) + \lvert R \rvert + \lvert Q \rvert - 1. $$
To see this, we need to calculate the dimension of $$ \mathfrak{m}_ {Q,\chi}^\square / ((\mathfrak{m}_ {Q,\chi}^\square)^2 + \mathfrak{m}_ \chi^\mathrm{loc}). $$ These correspond to deformations $R_{\chi,Q}^\square \to \mathbb{F}[\epsilon]/(\epsilon^2)$ for which the restriction to $R_\chi^\mathrm{loc}$ factors through $\mathbb{F}$. These are given by liftings $(1 + \epsilon \phi) \bar{r}_ \mathfrak{m}$ plus elements $1 + \epsilon \alpha_v$ such that
- it is cocycle $\phi \in Z^1(G_F, \mathrm{ad} \bar{r})$,
- its determinant is fixed, $\tr \phi = 0$,
- it is unramified away from $Q \cup R \cup \lbrace v \mid \ell \rbrace$,
- for $v \mid \ell$ it is Fontaine–Laffaille with correct weights, $[\phi \vert_{G_{F_v}}] \in H_f^1(G_{F_v}, \mathrm{ad}^0 \bar{r})$,
- for $v \in R$ the deformation is trivial, $\phi \vert_{G_{F_v}} = \partial \alpha_v$.
As we did in the winter we can then define these local conditions $\mathcal{L}_ Q = \lbrace L_v \rbrace$ where
- if $v \mid \ell$ then $L_v = H_f^1$,
- if $v \in R$ then $L_v = 0$,
- if $v \in Q$ then $L_v = H^1$,
- if $v \notin Q \cup R \cup \lbrace v \mid \ell \rbrace$ then $L_v = H^1(G_{k(v)}, \mathrm{ad}^0 \bar{r})$.
Then we can summarize the condition above as $\phi \in Z_{\mathcal{L}_ Q}^1$ and $\alpha_v \in \mathrm{ad} \bar{r}$ for $v \in R$ with $\phi \vert_{G_{F_v}} = \partial \alpha_v$. Then we have $$ d_Q = \dim H_{\mathcal{L}_ Q}^1(G_F, \mathrm{ad}^0 \bar{r}) + (3 - \dim H^0(G_F, \mathrm{ad}^0 \bar{r})) + \sum_{v \in R} H^0(G_F, \mathrm{ad} \bar{r}) - 4,$$ where we add the $3 - \dim H^0$ to get from $\dim H^1$ to $\dim Z^1$, and then add in the dimensions of $\alpha_v$, and the subtract the relation $(\phi, \lbrace \alpha_v \rbrace) \sim (\phi + \partial \beta, \lbrace \beta + \alpha_v \rbrace)$. Using Tate duality, we can compute this to be $$ \begin{align} &\dim H_{\mathcal{L}_ Q^\perp}^1(G_F, (\mathrm{ad}^0 \bar{r})(1)) - \dim H^0(G_F, (\mathrm{ad}^0 \bar{r})(1)) + \sum_{v \mid \infty} (3 - \dim H^0(G_{F_v}, \mathrm{ad}^0 \bar{r})) \br &- \sum_{v \mid \ell} 3 [F_v : \mathbb{Q}_ \ell] + \sum_{v \mid Q \cup R \cup \lbrace v \mid \ell \rbrace} (\dim L_v - \dim H^0(G_{F_v}, \mathrm{ad}^0 \bar{r})) + \sum_{v \in R} H^0(G_{F_v}, \mathrm{ad} \bar{r}) - 1. \end{align} $$ We can now compute everything (where we use local duality to compute the contribution of $v \in Q$).
What is $\mathcal{L}_ Q^\perp$? At $v \mid \ell$ it is still the $L_v^\perp = H_f^1$ and now for $v \in R$ we have $L_v^\perp = H^1$ while for $v \in Q$ we have $L_v^\perp = 0$. For other $v$, this is still the unramified part $L_v^\perp = H_1(G_{k(v)})$. In particular, we have by definition $$ 0 \to H_{\mathcal{L}_ Q^\perp}^1 \to H_{\mathcal{L}_ \emptyset^\perp}^1 \to \bigoplus_{v \in Q} H^1(G_{k(v)}, (\mathrm{ad}^0 \bar{r})(1)). $$ This $H^1$ is just now the coinvariants $(\mathrm{ad}^0 \bar{r})(1) / (\mathrm{Frob}_ v - 1)$. Let us write $$ r = \dim H^1_{\mathcal{L}_ \emptyset^\perp}. $$
Proposition 2. For every $N \in \mathbb{Z}_ {\gt 0}$ there exists a finite set $Q_N$ of places of $F$ such that
- $Q_N$ is disjoint form $R \cup \lbrace v \mid \ell \rbrace$,
- $\lvert Q_N \rvert = r$,
- if $v \in Q_N$ then $q_v \equiv 1 \pmod{\ell^N}$,
- if $v \in Q_N$ then $\bar{r}_ \mathfrak{m}(\mathrm{Frob}_ v)$ has distinct eigenvalues $\alpha_v \neq \beta_v$,
- $H^1_{\mathcal{L}_ {Q_N}^\perp}(G_F, \mathrm{ad}^0 \bar{r}) = 0$ (and therefore $d_{Q_N} = \lvert R \rvert - 1 + r$).
It will suffice to prove that for every $0 \neq \phi \in H_{\mathcal{L}_ \emptyset^\perp}^1$ there exists a $v \notin R \cup \lbrace v \mid \ell \rbrace$ such that
- $v$ splits completely in $F(\zeta_{\ell^N})$,
- $\bar{r}_ \mathfrak{m}(\mathrm{Frob}_ v)$ does not have $\ell$-power order,
- $\phi(\mathrm{Frob}_ v) \notin (\mathrm{Frob}_ v - 1)(\mathrm{ad}^0 \bar{r})(1)$.