Last time we discussed what it means for $\pi = \bigotimes^\prime \pi_v$ to be a cuspidal automorphic representation. We also had this strong multiplicity one theorem, which is that $\pi, \pi^\prime$ are two representations with $\pi_v \cong \pi_v^\prime$ for all but finitely many $v$, then $\pi = \pi^\prime$ (even as subspaces of $\mathcal{A}_ 0(\GL_n(F) \backslash \GL_n(\mathbb{A}_ F))$. We also discussed what it means for $\pi$ to be regular algebraic.
For $v$ a finite place of $F$, we have $$ 0 \to I_{F_v} \to G_{F_v} \twoheadrightarrow G_{k(v)} = \langle \mathrm{Frob}_ v \rangle \xrightarrow{v,\cong} \hat{\mathbb{Z}}, \quad \mathrm{Frob}_ v(\alpha)^{\lvert k(v) \rvert} = \alpha. $$ Then the Weil group is $W_{F_v} = \lbrace \sigma \in G_{F_v} : v(\sigma) \in \mathbb{Z} \rbrace$ with the topology so that $I_{F_v}$ is open. For each $\ell \neq p$ there is a map $t_\ell \colon I_{F_v} \twoheadrightarrow \mathbb{Z}_ \ell$ unique up to $\mathbb{Z}_ \ell^\times$-multiples.
Definition 1. A Weil–Deligne representation of $W_{F_v}$ over a field $L$ of characteristic $0$ is
- a finite-dimensional vector space $V/L$,
- a homomorphism $\rho \colon W_{F_v} \to \GL_L(V)$ with open kernel,
- an element $N \in \End(V/L)$ with the property that $\rho(\sigma) N \rho(\sigma)^{-1} = \lvert k(v) \rvert^{-v(\sigma)} N$ for all $\sigma \in W_{F_v}$.
We say that $(\rho, N)$ is Frobenius-semisimple if $\rho$ is semisimple, and just semisimple when $\rho$ is semisimple and $N = 0$.
This implies that $N$ is nilpotent.
Lemma 2. There exists a unique $u \in \GL_L(V)$ unipotent which commutes with $\im \rho$ and $N$ such that $\sigma \mapsto \rho(\sigma) u^{-v(\sigma)}$ is a semisimple representation.
So we can define $$ (\rho, N)^\mathrm{F-ss} = (\rho u^{-v(-)}, N), \quad (\rho, N)^\mathrm{ss} = (\rho u^{-v(-)}, 0). $$
Theorem 3. Let $v \nmid \ell$ and choose $t_\ell \colon I_{F_v} \twoheadrightarrow \mathbb{Z}_ \ell$ as well as a lift $\phi \in W_{F_v}$ with $v(\phi) = 1$. Then there exists a fully faithful functor $$ \begin{Bmatrix} \text{finite-dimensional continuous} \br \text{representations of } G_{F_v} \text{ over } \bar{\mathbb{Q}}_ \ell \end{Bmatrix} \to \begin{Bmatrix} \text{Weil–Deligne representations} \br \text{of } W_{F_v} \text{ over } \bar{\mathbb{Q}}_ \ell \end{Bmatrix}. $$
Theorem 4. For $v \mid \ell$, to a de Rham representation $r$ of $G_{F_v}$ on a finite-dimensional $\bar{\mathbb{Q}}_ \ell$-vector space we can associate a Weil–Deligne representation $\mathrm{WD}(r)$ of $W_{F_v}$ over $\bar{\mathbb{Q}}_ \ell$.
But this is not fully faithful. For $\tau \colon F_v \hookrightarrow \bar{\mathbb{Q}}_ \ell$ we have Hodge–Tate weights $\mathrm{HT}_ \tau(r) \subseteq \mathbb{Z}$.
Recall that we had $\bar{r} \colon G_{F_v} \to \GL_n(\bar{\mathbb{F}}_ \ell)$ that is Fontaine–Laffaille, so that $\mathbb{G}(M) = \bar{r}$ for $M / k(v) \otimes_{\mathbb{F}_ \ell} \bar{\mathbb{F}}_ \ell$. For each $k(v) \hookrightarrow \bar{\mathbb{F}}_ \ell$, we have a corresponding $M_\tau / \bar{\mathbb{F}}_ \ell$, so that $M = \bigoplus M_\tau$. Then we have a multiset $$ \mathrm{HT}_ \tau(\bar{r}) \subseteq \mathbb{Z} $$ where we declare that $i$ appear with multiplicity $\dim_{\bar{\mathbb{F}}_ \ell} \mathrm{gr}^i M_\tau$.
Now assume we have $r \colon G_{F_v} \to \GL_n(\mathcal{O}_ {\bar{\mathbb{Q}}_ \ell})$ is Fontaine–Laffille. Then to $\bar{r} = r \bmod \lambda$ we can associate $\mathrm{HT}_ \tau(\bar{r})$, and also $\mathrm{HT}_ \tau(r \otimes \bar{\mathbb{Q}}_ \ell$. These two agree.
Theorem 5. There exists a natural bijection $$ \mathrm{rec} \colon \begin{Bmatrix} \text{irreducible smooth} \br \text{representations} \br \text{of } \GL_n(F_v) \text{ over } \mathbb{C} \end{Bmatrix} \xrightarrow{\cong} \begin{Bmatrix} n\text{-dimensional Frobenius-semisimple} \br \text{Weil–Deligne representations} \br \text{of } W_{F_v} \text{ over } \mathbb{C} \end{Bmatrix}. $$
Theorem 6. Suppose $F$ is a CM field and let $\pi = \bigotimes_v^\prime \pi_v$ be a regular algebraic cuspidal automorphic representation of $\GL_n(\mathbb{A}_ F)$. Choose an isomorphism $i \colon \mathbb{C} \cong \bar{\mathbb{Q}}_ \ell$. Then there exists a continuous semisimple $r_{\ell,i}(\pi) \colon G_F \to \GL_n(\bar{\mathbb{Q}}_ \ell)$, unramified almost everywhere, such that for all $v \nmid \ell$ we have $$ \mathrm{WD}(r_{\ell,i}(\pi))^\mathrm{ss} \cong i \circ \mathrm{rec}(\pi_v)^\mathrm{ss}, $$ and for $v \mid \ell$ the restriction $r_{\ell,i}(\pi) \vert_{G_{F_v}}$ is de Rham.
This in particular over-determines $r_{\ell,i}(\pi)$ because all the Frobenius traces at unramified places are known. Often we know more. For example, for $n = 2$ and $F$ totally real, we know that $$ \mathrm{WD}(r_{\ell,i}(\pi))^\mathrm{F-ss} \cong i \circ \mathrm{rec}(\pi_v) $$ for all $v$. Moreover, for $\tau \colon F \hookrightarrow \bar{\mathbb{Q}}_ \ell$ we have $$ \mathrm{HT}_ \tau(r_{\ell,i}(\pi)) = \mathrm{HT}_ \tau(r_{\ell,i}(\pi) \vert_{G_{F_v(\tau)}}) = -\mathrm{HC}_ {i^{-1} \circ \tau}(\pi_\infty). $$
Definition 7. We say that $r \colon G_F \to \GL_n(\bar{\mathbb{Q}}_ \ell)$ is (regular) automorphic if it arises in this way for some $\pi, i$. We say that it has weight 0 when we have $\mathrm{HC}_ \tau(\pi) = \lbrace 0, -1, \dotsc, 1-n \rbrace$ for all $\tau$. We also say that it has level prime to $\ell$ when $\pi_v$ is unramified for all $v \mid \ell$.
We also say that $\bar{r} \colon G_F \to \GL_n(\bar{\mathbb{F}}_ \ell)$ is automorphic if there exists a lift $r \colon G_F \to \GL_n(\mathcal{O}_ {\bar{\mathbb{Q}}_ \ell})$ that is (regular) automorphic. We can also add the adjectives weight 0 and level prime to $\ell$.
Conjecture 8 (Fontaine–Mazur). If $r \colon G_F \to \GL_n(\bar{\mathbb{Q}}_ \ell)$ is continuous, and
- $r$ is unramified almost everywhere,
- $r \vert_{G_{F_v}}$ is de Rham for all $v \mid \ell$ and $\mathrm{HT}_ \tau(r)$ has $n$ distinct elements for all $\tau$,
- $r$ is irreducible,
then $r$ is automorphic.
Theorem 9. If $E \supseteq F$ is a soluble Galois extension of CM fields, and if $r \colon G_F \to \GL_n(\bar{\mathbb{Q}}_ \ell)$ is continuous with $r \vert_{G_E}$ irreducible, then $r$ is automorphic if and only if $r \vert_{G_E}$ is automorphic.
This is a consequence of the theory of cyclic base change, due to Langlands in the case of $n = 2$ and to Arthur–Clozel for general $n$. The point is that $\Gal(E/F) = \langle \sigma \rangle$ then for any $\pi$ an automorphic representation of $\GL_n(\mathbb{A}_ E)$ and $\pi \cong \pi \circ \sigma^{-1}$ then we get a representation $\pi_F$ such that $r(\pi_F) \vert_{G_E} = r(\pi)$.