The local condition at ℓ

• The local condition at ℓ
• Obstruction theory

We were looking at $S = \lbrace 2, \ell \rbrace$ and $L / \mathbb{Q}_ \ell$ finite and $\mathcal{O} = \mathcal{O}_ L$ with $\mathcal{O} / \lambda = \mathbb{F}$. We had a fixed representation $\bar{r} \colon G_{\mathbb{Q},S} \to \mathrm{GL}_ 2(\mathbb{F}_ \ell)$ and we saw that there is a universal lift $$ r^\mathrm{univ} \colon G_{\mathbb{Q},S} \to \mathrm{G}_ 2(R^\mathrm{univ}) $$ satisfying these conditions (A), which says that $\det r = \epsilon_\ell^{-1}$, (B), which is some ramification at $2$, and (C), which is some ramification at $\ell$.

We were trying to understand its tangent space. We want to say that there is an isomorphism $$ \Hom_\mathbb{F}(\mathfrak{m} / (\lambda, \mathfrak{m}^2), \mathbb{F}) \cong H_{\lbrace L_ 2, L_\ell \rbrace}^1(G_{\mathbb{Q},S}, \mathrm{ad}^0 \bar{r}). $$ We haven’t talked about $L_\ell$ yet. We saw that $$ \dim L_2 = \dim H^0(G_{\mathbb{Q}_ 2}, \mathrm{ad}^0 \bar{r}). $$

The local condition at ℓ

We wanted $r \vert_{G_{\mathbb{Q}_ \ell}}$ to be in the essential image of the Fontaine–Laffaille functor $\mathbb{G}$. First of all, we need $\bar{r} \vert_{G_{\mathbb{Q}_ \ell}} = \mathbb{G}(\bar{M})$ where $\bar{M}$ is a $2$-dimensional $\mathbb{F}$-vector space with filtration $\mathrm{Fil}^0 \bar{M} = \bar{M}$ and $\dim \mathrm{Fil}^1 \bar{M} = 0$ and $\mathrm{Fil}^2 \bar{M} = 0$. We also have linear maps $$ \bar{\Phi}^1 \colon \mathrm{Fil}^1 \bar{M} \to \bar{M}, \quad \bar{\Phi}^0 \colon \bar{M} / \mathrm{Fil}^1 \bar{M} \to \bar{M} $$ because the Frobenius is trivial on $\mathbb{F}_ \ell$, where we also have the condition that $$ \bar{\Phi} = \bar{\Phi}^0 + \bar{\Phi}^1 \colon \mathrm{gr}^0 \bar{M} \xrightarrow{\cong} \bar{M}. $$

Now what happens if we look at its lift to $\mathbb{F}[\epsilon]$? If we write $r_{G_{\mathbb{Q}_ \ell}} = \mathbb{G}(M)$ where $M$ is a free $\mathbb{F}[\epsilon]$-module, then because $\mathcal{MF}$ is an abelian category we have exactness of $\mathrm{Fil}^1$ and hence $\mathrm{Fil}^1 M$ is free of rank $1$. Now we see that $$ \theta \colon \mathrm{Ext}_ {\mathcal{MF}, \mathbb{F}}^1(\bar{M}, \bar{M}) \hookrightarrow \Ext_{\mathbb{F}[G_{\mathbb{Q}_ \ell}]}^1(\bar{r}, \bar{r}) = H^1(G_{\mathbb{Q}_ \ell}, \mathrm{ad} \bar{r}). $$ So we define $$ L_\ell = (\im \theta) \cap H^1(G_{\mathbb{Q}_ \ell}, \mathrm{ad}^0 \bar{r}) \subseteq H^1(G_{\mathbb{Q}_ \ell}, \mathrm{ad} \bar{r}). $$

We also calculate its dimension. Let us write $M = \mathbb{F}[\epsilon]^2$ and $\mathrm{Fil}^1 M = \mathbb{F}[\epsilon]$ and use the basis elements $e_1$ and $e_2$ and $\epsilon e_1$ and $\epsilon e_2$ to write the matrix $$ \Phi = \begin{pmatrix} \bar{\Phi} & \alpha \br 0 & \bar{\Phi} \end{pmatrix}, \quad \alpha \in M_{2 \times 2}(\mathbb{F}) = \Hom(\mathrm{gr} \bar{M}, \bar{M}). $$ Given $\beta \in \End_\mathbb{F}(\bar{M})$ satisfying $\beta \mathrm{Fil}^1 \bar{M} \subseteq \mathrm{Fil}^1 \bar{M}$, we can change $e_i \mapsto e_i + \epsilon \beta e_i$ and we have $$ \alpha \to \alpha - \beta \bar{\Phi} + \bar{\Phi} \mathrm{gr} \beta. $$ This modification is trivial if and only if $\beta \in \Hom_{\mathcal{MF}}(\bar{M}, \bar{M})$. So we have an exact sequence $$ 0 \to \Hom_{\mathcal{MF}}(\bar{M}, \bar{M}) \to \mathrm{Fil}^0 \Hom_\mathbb{F}(\bar{M}, \bar{M}) \to \Hom_\mathbb{F}(\mathrm{gr} \bar{M}, \bar{M}) \to \Ext_{\mathcal{MF},\mathbb{F}}^1(\bar{M}, \bar{M}). $$ Then we have $$ \dim \Ext_{\mathcal{MF},\mathbb{F}}^1(\bar{M}, \bar{M}) = 1 + \dim H^0(G_{\mathbb{Q}_ \ell}, \mathrm{ad} \bar{r}). $$

But we wanted to look at those things with fixed determinant. Here, we have $\wedge_{\mathbb{F}[\epsilon]}^2 M \cong \mathbb{F}[\epsilon]$ in degree $1$, and $\Phi$ acts by $a \in \mathbb{F}[\epsilon]^\times$. The corresponding Galois representation is $\mathbb{G}(\wedge_{\mathbb{F}[\epsilon]}^2 M) = \epsilon_ell^{-1} \chi_a$ where $\chi_a$ is an unramified character with $\mathrm{Frob}_ \ell \mapsto a$. Here $a \in 1 + \epsilon \mathbb{F}$ and so we can modify $\Phi$ by multiplying $a^{-1}$ to one of its factors. This shows that $$ \dim L_\ell = \dim \Ext_{\mathcal{MF},\mathbb{F}}^1(\bar{M}, \bar{M}) - 1 = 1 + \dim H^0(G_{\mathbb{Q}_ \ell}, \mathrm{ad}^0 \bar{r}). $$

Obstruction theory

Now we have for $d = \dim H_\mathscr{L}^1(G_{\mathbb{Q},S}, \mathrm{ad}^0 \bar{r})$ we have a surjective $$ \mathcal{O}[[x_1, \dotsc, x_d]] \twoheadrightarrow R^\mathrm{univ} $$ with kernel $I$. Now $I$ is generated by $\dim_\mathbb{F} I/\mathfrak{m}I$ elements, and we will show that there is an embedding $$ \Hom_\mathbb{F}(I/\mathfrak{m}I, \mathbb{F}) \hookrightarrow H_\mathscr{L}^2(G_{\mathbb{Q},S}, \mathrm{ad}^0 \bar{r}). $$

Here is the basic idea. Given any nonzero $\lambda \colon I/\mathfrak{m}I \to \mathbb{F}$, we have some other ideal $J \subseteq \mathcal{O}[[x_1, \dotsc, x_d]]$ where $J / \mathfrak{m}I = \ker \lambda$.

Proposition 1. The map $\mathcal{O}[[x_1, \dotsc, x_d]] / J \twoheadrightarrow \mathcal{O}[[x_1, \dots, x_d]] / I = R^\mathrm{univ}$ doesn’t have a section.

We will understand this as the obstructions to lifting Galois representations. In particular, there is a continuous set-theoretic lift $$ \tilde{r} \colon G_{\mathbb{Q},S} \to \mathrm{GL}_ 2(\mathcal{O}[[x_1, \dotsc, x_n]] / J) $$ of $r^\mathrm{univ}$. If we define $$ \phi \colon G_{\mathbb{Q},S}^2 \to 1 + M_{2 \times 2}(I/J) \cong M_{2 \times 2}(I/J), \quad \phi(\sigma, \tau) = \tilde{r}(\sigma) \tilde{r}(\tau) \tilde{r}(\sigma \tau)^{-1} $$ then we have $$ \mathrm{ad} \bar{r}(\sigma) (\phi(\tau, \rho)) \phi(\sigma, \tau \rho) = \phi(\sigma, \tau) \phi(\sigma\tau, \rho). $$ So we have $Z^2(G_{\mathbb{Q},S}, \mathrm{ad}^0 \bar{r})$. We can also see that the ambiguity lives in the coboundary, and so we get a class in $H^2(G_{\mathbb{Q},S}, \mathrm{ad}^0 \bar{r})$.

But we also have to feed in that there are local conditions. We will do this next time.