# Lecture 7: Local Langlands ## Statement of LLC :::{.conjecture title="The Local Langlands Correspondence for $\GL_n$"} There is a canonical bijective correspondence of sets \[ \Disjoint_{n\geq 1} F\dash\ss\mathsf{WD}\Rep\modiso &\mapstofrom \Disjoint_{n\geq 1} \smooth\Irr\Adm\Rep(\GL_n(K))\modiso \\ \rho &\mapstofrom \pi \] The objects on the left are $F\dash$semisimple Weil-Deligne reps, on the right are *admissible* reps. Note that we've not yet seen interesting objects showing up naturally on either side! Both sides are somewhat pathological at this point. Compare to the Taniyama-Shimura conjecture, where elliptic curves come from modular forms, and both show up naturally. See also the BSD conjecture -- if you don't know an elliptic curve is modular, you can't meromorphically extend the $L$ function. ::: :::{.remark} Here "canonical" means that the bijection satisfies a long list of properties. Some examples of what should match up: - Duality. A representation $V$ will have a left $G\dash$action, so $V\dual$ will have a right $G\dash$action, and composing with inversion makes it a left action again -- typically this yields a different representation. - Conductors. The conductor on the RHS is harder to define. - $L\dash$functions. - $\eps$ factors. - $\eps$ factors of pairs -- these turn out to be crucial! - $\cdots$ This list of properties became so long that there is now a theorem showing that there is at most one bijection satisfying all of them. Another theorem shows that there is *at least* one bijection. - In the function field case, this was established in the 80s (Laumm, Rapoport, Stuhler). - In the \(p\dash \)adic field case, this was proved in 2000 (Harris-Taylor) All of the proofs are global, i.e. use number fields, despite the fact that Galois groups of number fields are more complicated than Galois groups of local fields. ::: :::{.remark} Philosophy: this gives a direction to go after class field theory, and highlights that $G_K$ isn't the fundamental object, but rather the category $\Rep(G_K)$. In analogy, note that $\pi_1 X \in \Grp$ is not canonically defined, but $\Pi_1 X\in \Grpd$ is. Defining the group depends on choice of a base point, and a path $x_0\to x_1$ produces an isomorphism $\pi_1(X, x_0)\to \pi_1(X, x_1)$, but e.g. on a torus the paths can wind around genus so that two choices of path need not be homotopic. However, there is a theorem that says $\Rep(\pi_1 X) \iso \Locsys(X)$. In this analogy, choosing a point for $\pi_1$ is like choosing an algebraic closure $\bar K/K$. ::: :::{.remark} Next goal: finding natural examples of the LHS of the correspondence, and at least some global occurrences on the RHS, and show that the LLC is useful. We'll consider - The $n=1$ case, recovering CFT, - Weil-Deligne representations showing up naturally, - Examples of smooth irreducible admissible reps $\pi$, e.g. in the cohomology of Shimura varieties. ::: ## The $n=1$ case :::{.conjecture} If $G\in\Alg\Grp\slice K$ is connected and reductive, i.e. over $\bar K$, $G$ is isomorphic to one of \[ \GL_n, \SL_n, \PGL_n, \Orth_n, \Sp_{2g}, E_6, E_7, E_8, F_4, G_2 ,\] then there is a LLC for $G$: there is a canonical surjection \[ F \text{ finite fibers} \injects \correspond{ \text{Certain Weil-Deligne reps} (\rho, N): W_K \to {}^L G(\CC) } &\surjects \smooth\Irr\Rep( G(K) ) \] where ${}^LG$ is an $L\dash$group for $G$ and the fibers are referred to as **$L\dash$packets**. This surjection is similarly supposed to satisfy a big list of properties, but it is not known if these uniquely characterize the surjection. ::: :::{.remark} Consider $n=1$ dimensional reps, so on the LHS we have pairs $(\rho, N): W_K\to \GL_1(\CC)$ where $N$ being $1\times 1$ nilpotent forces $N=0$. Note that $\ker \rho_0$ is closed and the quotient is Hausdorff and abelian, so $\rho_0$ factors through $W_K^\ab$. So the LHS reduces to 1-dimensional continuous reps of $W_K^\ab$ over $\CC$. A coincidence in dimension 1: the RHS reads smooth admissible irreducible reps of $K\units$, and one can show that admissible and irreducible implies $\dim \pi < \infty$ and further $\dim \pi = 1$. This needs that we have compact open normal subgroups, and this fails quite seriously for $\GL_2$. In the $n=1$ case, continuity is equivalent to smooth and admissible, so we're considering \[ \pi \in \Top\Grp(K\units, \CC\units), \qquad \rho \in \Top\Grp(W_L^\ab, \CC\units) .\] and these are equal by local class field theory. ::: :::{.remark} A source of Weil-Deligne reps: **$\ell\dash$adic representations**. Let $K/\QQpadic$ be finite, and suppose $\rho\in \Top\Grp(G_K, \GL_n(\QQladic))$ where $G_K$ is a profinite topological group with the profinite topology, and $\QQladic$ has $\ell\dash$adic topology. Assume also $\ell \neq p$. Note that the target is now *not* discrete, so there won't be finite inertia. The wild part of inertia will be pro$\dash p$ and end up being finite, while the tame part will have a large $\ell\dash$adic component. In the discrete case, we knew the image of inertia was finite by a compactness argument, but that may not hold here. ::: :::{.example title="of where Weil-Deligne reps show up"} These show up in nature, e.g. - In the $\ell\dash$adic Tate module $T_\ell(E/K)$ of an elliptic curve $E/K$. - In $\ell\dash$adic etale cohomology $H^i_\et(X_{\bar K}; \QQladic)$. The vast majority of Weil-Deligne representations come from here. - In $\ell\dash$adic deformations of the above, i.e. taking an $\ell\dash$adic representation, reducing $\mod \ell$, and try to deform back into characteristic zero. Some families come reps of $\GL_n(R)$ for $R$ an affinoid object. See also eigencurves. These are slightly more difficult because we can no longer control inertia. ::: :::{.remark} Given a $\rho$, one can construct a WD rep. One example of where $T_\ell(E/K)$ is easy to compute: if $E$ is an elliptic curve with split (bad) multiplicative reduction, then $E$ can be uniformized to obtain \[ E(\bar K) \cong \bar{K}\units / q^\ZZ \qquad q\in K, \abs{q} < 1 .\] This \[ \ro{T_\ell E}{E_{\bar K /K}} = \matt {\chi} {*} 0 {\id} ,\] where $\chi$ can be a nontrivial infinite cyclotomic character. Note that this yields infinite inertia. ::: :::{.remark} Recall that if $\rho$ is an $\ell\dash$adic representation as above, $\rho(I_{\bar K/K})$ can be infinite but can't be too bad, e.g. $\rho(I_{\bar K /K}^\eps)$ must be finite for $\eps>0$ since it is pro$\dash p$. The tame inertia isn't so bad: we have \[ \Gal(K^t/K^\unram) \cong \prod_{r\neq p\text{ prime}} \ZZ_r ,\] and we should worry about the $\ZZladic$ component. To isolate this part, fix $t\in \Gal(K^t/K^\unram) \surjects \ZZladic$ and $\phi\in G_K$ lifting $\Frob\in G_{K^\unram}$. The full $G_K$ now breaks into three stages: - Unramified: controlled by what happens to $\phi$, - Tamely ramified: controlled by $t$, - Wildly ramified: pro$\dash p$ and hence finite. ::: :::{.proposition title="Grothendieck"} If $\rho\in \Top\Grp(G_K, \GL_n(E))$ with $E/\QQladic$ is finite, then there exists a unique Weil-Deligne representation $(\rho, N) \in \Top\Grp(W_K, \GL_n(E)^\disc)$. This satisfies the following: if $\sigma\in I_{\bar K/K}$, for every $m\in \ZZ_{\geq 0}$, \[ \rho(\phi^m \sigma) = \rho_0(\phi^m \sigma) \exp\qty{N\cdot t(\sigma)} ,\] where nilpotence of $N$ guarantees that the series expansion for $\exp(\wait)$ is finite. ::: :::{.remark} Note that we can always restrict $\rho$ using that $W_K \leq G_K$, but this may not be continuous. Since we dropped the topology on $\GL_n(E)$, it makes it harder for $\rho_0$ to be continuous. ::: :::{.remark} Not all WD reps show up this way, since $\rho(\phi)$ has restrictions on its eigenvalues -- if $(\rho_0, N)$ arises this way, eigenvalues of $\rho(\phi)$ will be in $\QQladic\units$. However, this is essentially the only obstruction: given $(\rho_0, N)$ with $\lambda_i \in \QQladic\units$, then it comes from a $\rho$. ::: ## Smooth admissible reps $\pi$ of $\GL_n(K)$ :::{.remark title="on conductors"} Given $\pi: K\units\to \CC\units$ smooth admissible irreducible, define \[ f(\pi) = \begin{cases} 0 & \ro{\pi}{\OO_K\units} = 1 \\ r & \text{ if $r \in \ZZ_{>0}$ is minimal such that } \ro{\pi}{1+ \mfp_K^r \OO_K}=1. \end{cases} .\] As $r$ increases, this gives a basis of neighborhoods of the identity. In the LLC for $n=1$, if $\rho_0 = (\rho_0, N=0) \mapstofrom \pi$, it turns out that $f(\rho_0) = f(\pi)$. ::: ## The $n=2$ case :::{.remark} Constructing a $\pi$: given a pair of characters, one can construct a representation of $\GL_2(K)$. Let $\chi_1, \chi_2\in \Top\Grp(K\units\to\CC\units)$ and define \[ I(\chi_1, \chi_2) \da \ts{\phi: \GL_2(K) \to \CC \st \phi \text{ is locally constant}, \phi\qty{ \matt a b 0 d g } = \chi_1(a) \chi_2(d) \norm{a\over d}^{1\over 2} \phi(g) } .\] :::{.remark} The norm term is a fudge factor! ::: Note that since the source is totally disconnected, there can be many locally constant but non-constant functions, e.g. $\chi_{\ZZpadic}(x)$ on $\QQpadic$. This is a vector space under pointwise operations in the target, and is supposed to look like $\Ind_B^G V$ for $V$ a 1-dimensional representation of $B = \matt * * 0 *$. Now define \[ \pi: \GL_2(K) &\to \Aut_{\mods{\CC}} (I(\chi_1, \chi_2)) \\ g &\mapsto \phi\mapsto (h\mapsto \phi(hg)) ,\] so that $(\pi(g)\phi)(h) = \phi(hg)$ for all $g,h\in \GL_2(K)$. One can check that this defines an action -- write this as $g.\phi(h) = \phi(hg)$, then one needs to check that $(g_1 g_2)\phi = g_1(g_2\phi)$. Consider instead writing $(x) f \da f(x)$, then the equation reads $(h)(g\phi) = (hg)\phi$ and the check is $(h)(g_1 g_2(\phi)) = (hg_1)(g_2\phi)$, and one can rearrange the brackets by definition. Thus this yields a representation of $\GL_2(K)$. :::