# Lecture 12, \(\ell\dash \)adic Representations

## Families of \(\ell\dash \)adic reps

:::{.remark}
See Néron-Shafarevich criterion, Tate modules of AVs.
:::

:::{.example title="?"}
Let $K\in \NF$, say $K=\QQ$, and let $E/K$ be an elliptic curve.
Let $S_0$ be a finite set of places of $K$, so points in $\mspec \OO_K$, where $E$ has bad reduction.
For $\ell$ a prime, consider the \(\ell\dash \)adic Tate module: since $E[\ell^n](\bar K)$ receives an action of $G_K$, setting $T_\ell E \da \cocolim_n E[\ell^n]$ produces a representation 
\[
\rho_{E, \ell}: G_K\to \GL_2(\ZZladic)
,\]
which is well-defined up to conjugation that factors through $G(K^{\tilde S_0}/K)$ where $\tilde S_0 \da S_0 \union \ts{p\st p\divides \ell}$.

If $p\not\in S_0$ and $p\notdivides \ell$,
\[
\charpoly \rho_{E, \ell} \Frob_p = x^2 - a_p x - \Norm(p) \in \QQ[x] \embeds \QQladic[x], \qquad a_p \da 1 + \Norm(p) - \size E(\kappa(p))
.\]
Note that this no longer depends on $\ell$, so 
\[
\Tr \rho_{E, \ell} \Frob_p = a_p
,\]
independent of $\ell$.
However, $\rho_{E, \ell} \not\cong \rho_{E. p}$ for $\ell\neq p$, since $\rho_{E, \ell}$ is infinitely ramified at $\ell$ and has wild inertia, while $\rho_{E, p}$ will have finite wild inertia at $\ell$.
:::

:::{.definition title="$\ell\dash$adic representations of absolute Galois groups"}
Setup: 

- $K\in \NF$, 
- $E/\QQladic$ a finite extension,
- $S$ a finite set of places in $\mspec \OO_K$

Then a morphism of topological groups
\[
\rho\in \Top\Grp(G(K^S/K) \to \GL_n(E))
\]
where the LHS has the profinite topology and the RHS has the \(\ell\dash \)adic topology is an **$\ell\dash$adic representation** of $G_K$.
In this situation we say $\rho$ is *unramified outside of $S$*.

:::

:::{.remark}
Note that we regard this as $G_K$ representation despite having $G(K^S/K)$ as the source, since we implicitly require reps to be unramified away from $S$ since $G_K\surjects G(K^S/K)$.
:::

:::{.definition title="Rational representations"}
We say $\rho$ is **rational over $E_0$** for $E_0 \subseteq E$ if $\charpoly \rho \Frob_p \in E[x]$ in fact lies in $E_0[x]$.
:::

:::{.example title="?"}
\envlist

- For $\rho$ the cyclotomic character $\rho: G_\QQ\to \GL_1(\QQladic)$ satisfies $\rho \Frob_r = r$ for all $r\neq \ell$ and is thus rational over $\QQ$.
- $T_\ell E$ for $E$ an elliptic curve is rational over $\QQ$.
- $\rho = H^1_\et(X_{\bar K}; \QQladic)$ the \(\ell\dash \)adic etale cohomology of $X\in \smooth\Alg\Var\slice K$ with $X$ proper is rational over $\QQ$.
:::

:::{.definition title="Pure reps and their weights"}
The representation $\rho$ is **pure of weight $w$** iff $\rho$ is rational over some $E_0 \in \NF$ and for all embeddings $i\in \Field(E_0, \CC)$ and for all eigenvalues $\alpha$ of $\rho\Frob_p$, the magnitude satisfies
\[
\abs{i(\alpha)} = q_p^{-{w\over 2}}, \qquad q_p \da \size \OO_K/p
.\]
:::

:::{.theorem title="Deligne"}
For $X\in \smooth\proj\Var\slice K$ proper, $H^i(X_{\bar K}; \QQladic)$ is pure of weight $i$.
:::

:::{.example title="?"}
\envlist

- $H^2(\PP^1\slice K; \QQladic) = \omega_\ell\inv$ is the inverse of the cyclotomic character, making it pure of weight $-2$.
- $T_\ell E$ is pure of weight -1, so the roots of $x^2-a_px + \Norm(p)$ are complex conjugates, which yields the Hasse bound
\[
\abs{a_p} \leq 2\sqrt{\Norm(p)}
.\]


:::

:::{.remark}
We'll now let $\ell$ vary, and formalize the notion that the cyclotomic characters for $\ell\neq p$ are distinguished.
:::

:::{.definition title="Compatible Systems"}
Setup:

- $K\in \NF$,
- $E_0\in \NF$ (e.g. $\QQ$),
- $S_0 \subseteq \Places(K)$ a finite set of places,
- For all $p\not\in S_0$, a polynomial $F_p(x)\in E_0[x]$ (e.g. primes of good reduction and $F_p(x) = x-\Norm(p)$),
- For all finite places $\lambda \in \OO_{E_0}$, an \(\ell\dash \)adic representation 
\[
\rho_\lambda: G(K^{\tilde S_0}/K) \to \GL_n\qty{ \cl_\alg\qty{ (E_0)\complete\lambda} }, \qquad \tilde S_0 \da S_0 \union \ts{p\in \Places(K) \st p\divides \ell}
.\]

We'll say $\rho_\lambda$ is a **compatible system** of $\lambda\dash$adic representations iff for all $\lambda\divides \ell$ and for all $p\not\in \tilde S_0$, so $p\not\in S_0$ and $p\notdivides \ell$, 
\[
\charpoly \rho_\lambda \Frob_p
\]
is independent of $\lambda$.

:::

:::{.remark}
Note that $(E_0)\complete{\lambda}/\QQladic$ is a finite extension for any $\lambda \divides \ell$.
If one assumes $E_0 = \QQ$, the above is the data of an \(\ell\dash \)adic representation for every rational prime $\ell$ whose traces all agree.
:::

:::{.example title="?"}
Examples that are known to be compatible systems:

- The cyclotomic characters $F_p(x) = x-p$,
- $T_\ell E$ for all $\ell$, where $E_0 = \QQ$ and $F_p(x) = x^2 - a_p x - \Norm(p)$, where $S_0$ are the primes of bad reduction,
- $H^i_\et(X; \QQladic)$.

:::

:::{.remark}
Instead of asking for traces to agree, one can apply LLC: the $\rho_\lambda$ are **strongly compatible** iff for all $p\in S_0$ with $p\divides P$, for all $\lambda \in E_0$ with $\lambda\not\divides P$, consider the restricted representation $\ro{\rho_\lambda}{G(\bar K_p/ K_P) }$ yields a WD representation by Grothendieck, and by LLC a representation $\pi$ of $\GL_n(K_p)$ on some infinite dimensional vector space.
One could then define them to be *strongly compatible* iff $\pi$ only depends on $\lambda$.
This is unknown for $H^i_\et$.

> See the **Weight-Monodromy conjecture**.

:::

## Adeles and Global CFT

:::{.question}
Motivating question: what is $G_K^\ab$?
:::

:::{.definition title="Infinite Places"}
Let $K\in \NF$ with $[K:\QQ] = d$, so there are $d$ field embeddings $K \mapsvia{\sigma} \CC$.
These split into two types:

- Let $r_1$ be the number of $\sigma$ such that $\sigma(K) \subseteq \RR$, i.e. the number of totally real embeddings.
- If $\sigma(K) \not\subseteq \RR$, writing $\cong(z) \da \bar{z}$ for conjugation in $\CC$, $\sigma$ and $\conj\circ \sigma$ are distinct embeddings that induce the same norm since $\abs{z} = \abs{\bar z}$.
  These non-real embeddings come in pairs, so let $2r_2$ be the number of such embeddings.

Then
\[
r_1 + 2r_2 = d
.\]
An **infinite place** $v$ of $K$ is either 

- A real place $v = \sigma\in \Field(K\to \RR)$, or
- A complex place, which is the pair of maps $\ts{\sigma, (z\mapsto \bar z)\circ \sigma}\in \Field(K\to \CC)$ whose image isn't totally real.

Further define
\[
K_\infty \da \prod_{v \divides \infty } K\complete{v}
,\]
the product of all infinite places,
noting that $K\complete{v} \cong \RR$ or $\CC$ so that $K \embeds K\complete v$.
:::

:::{.example title="?"}
For $K = \QQ(2^{1\over 3})$, note that

- $r_1 = 1$
- $r_2 = 1$

:::

:::{.remark}
It turns out that 
\[
K_\infty \cong \RR^{r_1} \times \CC^{r_2} = K\tensor_\QQ \RR,
\quad
K_\infty\units \cong (\RR\units)^{r_1} \times (\CC\units)^{r_2}
,\]
and $K_\infty\units$ is generally disconnected since $\RR\units$ is disconnected.
The connected component of the identity satisfies
\[
(K_\infty\units)^0 \cong (\RR_{\geq 0})^{r_1} \times (\CC\units)^{r_2}
.\]

We'll define the **adeles** as a restricted product in the category of topological rings:
\[
\AA_L = \prod_{p<\infty} K\complete p \times K_\infty
,\]
and the **ideles** as $\AA_L\units$.
It will be locally compact, has an explicit compact subspace, and is related to class groups and unit groups.
:::