# Lecture 10: Part 2, Global Langlands

:::{.remark}
Global means number fields, which are harder than local fields but perhaps more familiar.
From now on, $K\in \Number\Field$, so $K/\QQ$ is finite.
We'll study the global groups $\Gal(L/K)$ and its relation to local Galois groups -- taking limits will produce structure on $G_K$.
There is an analog of the Weil group in the global case, but it's much more complicated -- similar to how we needed to replace local Weil groups with WD reps, we'll need to replace global Weil groups with "global Langlands groups", which won't quite be defined.

The same machines for producing $\ell\dash$adic representations of $G_K$ in the local setting will work here:

- $T_\ell X$ the $\ell\dash$adic Tate module of $X$ an elliptic curve or abelian variety,
- $H_*^\et(X; \QQladic)$ for $X\in \smooth\Proj\Var\slice K$,

yielding a $\rho$ side.
The $\pi$ side will be **automorphic representations**, which we'll define.
:::

:::{.conjecture title="Global Langlands"}
Every automorphic representation of $\GL_2(K)$ corresponds to a 2-dimensional representation of the global Langlands group.
:::

:::{.remark}
Locally, given an \(\ell\dash \)adic representation of $G_K$, a proposition of Grothendieck let us construct a WD rep, but we don't get all such WD reps this way since there was an $\ell\dash$adic unit issue.
Here, they'll give us a source of automorphic reps, but perhaps not all of them.
However, reps that are "algebraic" (so those with a good \(p\dash \)adic Hodge theory) should match on both sides.
:::

:::{.conjecture}
There is a correspondence between automorphic reps $\pi$ of $\GL_n(K)$ and motivic $n\dash$dimensional representations $\rho: G_K\to \GL_n(\bar \QQladic)$ which are unramified away from a finite set of places which support a good de Rham and \(p\dash \)adic Hodge theory.
Both should come from motives, and this should be compatible with the LLC in the following sense: global $\ell\dash$adic reps $\rho$ of $G_K$ should restrict to local \(\ell\dash \)adic representations, and global algebraic automorphic representations $\pi$ should be related to local $\pi$s.
:::

:::{.remark}
Note that the global Langlands conjecture above is essentially uncheckable, except perhaps for $\GL_1$ where it suffices to understand the abelianization of the global Langlands group $(?)^\ab \cong \dcosetl{K\units}{\AA_K\units}$.
This can be handled with global class field theory.
:::

## Galois Groups

:::{.remark}
Let $K/\QQ$ be a finite Galois extension, then $K \contains\OO_K$ -- note that when $K$ is local, $\OO_K$ is a DVR and has a unique prime ideal, but e.g. if $K=\QQ$ then $\OO_K = \ZZ$ has infinitely many primes.
Pick $\mfp\in \spec K \subseteq \mspec K$ nonzero, so $\kappa(\mfp) \da \OO_K/\mfp$ is a finite field.
Take the $\mfp\dash$adic completion of $K$:
\[
\OO\complete{K, \mfp} \da \cocolim_n \OO_K/\mfp^n, \qquad K\complete{\mfp} \da \ff(\OO_{K, \mfp})
.\]
Equivalently, pick $\lambda \in K\units$ and consider the fractional ideal $\lambda \OO_K$ it generates, which is a finitely generated $\OO_K\dash$submodule of $\OO_K$.
It factors into principal fractional ideals:
\[
\lambda \OO_K = \mfp^{v_\mfp(\lambda)} \times \prod_{\mfq_i \neq \mfp}\mfq_i^{e_i}
.\]
where $v_\mfp: K\units \surjects \ZZ$ and $\norm{\lambda}_{\mfp} \da \eps^{-v_\mfp( \lambda)}$ where $\eps \da q_\mfp \da \size \kappa(\mfp)$.
This norm induces a metric $d(x,y) \da \norm{x-y}_\mfp$, and one can take the Cauchy completion to define the local field $K\complete\mfp$.
Note that $K\complete\mfp/\QQpadic$ is a finite extension where $\mfp \intersect \ZZ =\gens{p}$.
:::

:::{.remark}
Let $L/K$ be a finite Galois extension of number fields and $\mfp\in \spec \OO_K$ nonzero as above.
Then $\Gal(L/K) \in\Fin\Grp$ is the global Galois group we'll study.
We can then extend $\mfp$:

\begin{tikzcd}
	L && {\OO_L} && {\mfp\OO_L} \\
	\\
	K && {\OO_K} && \mfp
	\arrow[hook', from=3-1, to=1-1]
	\arrow[hook', from=3-3, to=1-3]
	\arrow[dashed, from=3-5, to=1-5]
	\arrow[hook', from=3-3, to=3-1]
	\arrow[hook', from=1-3, to=1-1]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=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)


:::{.warnings}
The ideal $\mfp \OO_L \in \Id(L)$ need not be prime!
E.g. take $L/K = \QQ(\sqrt p)/\QQ$, then $p$ becomes a square in $L$.
:::

We can factor 
\[
\mfp \OO_L = \prod_{1\leq i\leq g} P_i^{e_i}, \qquad P_i\in \spec \OO_L
.\]

Since $\Gal(L/K)\actson L$, it acts on $\OO_L$ and fixes $\mfp \subseteq \OO_K \subseteq K$.
This action fixes $\mfp \OO_L$ as a set and permutes its elements, and permutes the $P_i\in \spec \OO_L$ appearing in the factorization above since $\sigma(P_i)$ is again prime and divides $\mfp$.
This is clearly true by transport of structure, since $\sigma: L\to L$ is an isomorphism of fields.
:::

:::{.fact}
Galois acts transitively, so there is only one orbit and thus all of the $e_i$ are preserved by $\sigma$ and all of the $P_i$ are isomorphic to $\sigma(P_i)$.
As a corollary, all of the completions are isomorphic: 
\[
L\complete{P_1} \cong L\complete{P_2} \cong \cdots L\complete{P_g}
.\]
:::

:::{.remark}
Let $P \da P_1\in \spec L$ be a fixed choice of a prime above $\mfp$.
Note that if $\mcp \da \ts{P_1, P_2,\cdots, P_g}$ then by Orbit-Stabilizer,
\[
\Gal(L/K)/ \Stab(P_1) \iso \Orb(\Gal(L/K) \actson \mcp) = \mcp
.\]
Define the **decomposition group**
\[
D_P \da \ts{\sigma\in \Gal(L/K) \st \sigma(P) = P} \implies \Gal(L/K)/D_P \cong \mcp
.\]
Note that this need not be a normal subgroup.
Now $\sigma: L\to L$ descends to a map $\sigma: (\OO_L, P) \to (\OO_L, P)$ and thus to the completions $\sigma: L\complete{P} \to L\complete{P}$ by transport of structure.
:::

:::{.fact}
$L\complete P / \kappa(\mfp)$ is Galois and $D_P \cong \Gal(L\complete P/\kappa(\mfp))$, which is the local Galois group, and $D_P \embeds \Gal(L/K)$.
:::

:::{.remark}
So given $\Gal(L/K)$, choose $\mfp\in \spec \OO_K$ and $P\divides \mfp \OO_L$ with $P\in \spec \OO_L$ to produce $D_P \subseteq \Gal(L/K)$ with $D_P\cong \Gal(L\complete P/\kappa(\mfp))$.
Note that there is an inertia subgroup $I \subseteq \Gal(L\complete P/\kappa(\mfp))$, and the quotient by $I$ is generated by $\Frob$.
It turns out that $I$ is almost always trivial:
\[
\mfp\notdivides \disc(L/K) \implies I = 1
,\]
so $L\complete P$ is an unramified extension of $\kappa(\mfp)$ and there is a distinguished element $\Frob_P \in D_P \subseteq \Gal(L/K)$.
Note that changing $L$ might change $\Frob_P$, so this depends on $\mfp$, the extension $L$, and $P\divides \mfp \OO_L$.
Consider choosing a different $P'$: since Galois acts transitively, there is some $\sigma\in \Gal(L/K)$ with $\sigma(P) = P'$ and thus 
\[
D_{P'} = \sigma D_P \sigma\inv, \qquad
\Frob_{P'} = \sigma \Frob_P \sigma\inv
\]
by transport of structure.
So this yields a well-defined *conjugacy class*
\[
\Frob_\mfp \da \ts{\Frob_{P'} \st P'\divides \mfp\OO_L}
,\]
which works for all $\mfp\notdivides \disc(L/K)$.
:::