Are there any simple, interesting consequences to motivate the local Langlands correspondence?

Question

Let's pretend that we know local Langlands at a fairly high level of generality... i.e. we know something along the lines of:

Let $G=\mathbf{G}(F)$ be the group of $F$-points of a connected reductive algebraic group $\mathbf{G}$ defined over a nonarchimedean local field $F$ with separable algebraic closure $\bar{F}$. Let $W_F'=W_F\times SL_2(\Bbb{C})$ be the Weil-Deligne group of $\bar{F}/F$ and let $^LG={}^LG^0\rtimes\mathrm{Gal}(\bar{F}/F)$ be the Langlands dual of $G$, where $^LG^0$ is the connected reductive complex algebraic group with root datum dual to that of $G$.

Then there exists a natural surjective map

$$\mathrm{Irr}(G)\twoheadrightarrow\mathrm{Hom}(W_F',{}^LG),$$

where $\mathrm{Irr}(G)$ is the set of equivalence classes of smooth irreducible complex representations of $G$. This map has finite fibres (the $L$-packets), is "compatible" with a list of operations: parabolic induction, twisting, etc, and is the unique such map.

As far as I know, this is a theorem (or maybe very close to being one for the latter two?) for $GL_N$, $SL_N$, $Sp_{2N}$ and $SO_N$.

When I'm trying to explain why I'm interested in this to someone I'll give the usual explanation along the lines of "we want to understand $\mathrm{Gal}(\bar{F}/F)$, local class field theory lets us understand the abelianisation of it in the form $F^\times\simeq W_F^{\mathrm{ab}}$, LLC generalises the dual form of this to a nonabelian setting and should tell us an awful lot about $\mathrm{Gal}(\bar{F}/F)$". Obviously you aren't going to hit them with the correspondence as stated above, but you can usually get away with saying "smooth irreps of $GL_N(F)$ naturally correspond to $N$-dim complex reps of $W_F'$, and that should generalise in a reasonable way to other groups".

At this point, I'll usually have either satisfied my questioner, or they'll ask me if I can give an example of what the LLC should let us do. That's when I run in to trouble -- I don't know of a single, reasonably simple, appealing application of it. In the global case people often bring up the proof of FLT. This isn't exactly "simple", but it's at least well known and can be summarised as "if FLT doesn't hold we have a non-modular semistable elliptic curve. Wiles then uses Langlands-Tunnell as a starting point, does a lot of work and eventually shows that every semistable elliptic curve is modular, hence FLT".

So... are there any such good examples of applications of the local correspondence?

Answer 1

In the case of $GL_{N}$, the $L$-packets are a non-issue, and the surjective map in the local Langlands correspondence becomes a bijection. At that point, we can think of allowing the information to flow the other way. Here's a simple application.

Let $f(z)$ be a classical modular form of weight $4k+2$ for the group $\Gamma_{0}(4)$ (that is also a cusp form, in the new subspace, and is an eigenform of all the Hecke operators). If $L(f,s)$ is the $L$-function for $f(z)$, what is the sign of the functional equation for $L(f,s)$?

The sign of the functional equation is always $1$, for the following reason. It is determined by the local components of the automorphic representation $\pi$ attached to $f$, and we only have to worry about the local components at $\infty$ (which is a discrete series representation that contributes a factor of $1$ to the sign because the weight is $\equiv 2 \pmod{4}$), and the local representation $\pi_{2}$ at $2$. The fact that the level of the modular form is $4$ shows that $\pi_{2}$ corresponds (under local Langlands) to a representation $\rho : W_{\mathbb{Q}_{2}} \to GL_{2}(\mathbb{C})$ that comes from a character $\chi$ of $W_{K}$, where $K = \mathbb{Q}_{2}(\omega)$ is the unramified quadratic extension of $\mathbb{Q}_{2}$, and that this character has order $6$. It follows that $\rho$ comes from an $S_{3}$ extension of $\mathbb{Q}_{2}$, and it turns out that there is a unique $S_{3}$ extension of $\mathbb{Q}_{2}$. From this, $\rho$ and hence $\pi_{2}$ is uniquely determined, and it turns out that the local root number of $\pi_{2}$ is also $1$.

(This fact was also observed by Atkin and Lehner in 1970, but the explanation above gives a more conceptual reason for it to be true, in my opinion.)

Answer 2

That's how I tend to motivate things, not really a concrete application, just shifting focus...

The local Galois group is a profinite, hence compact group. Picturing this group is difficult. Two ways are known:

1. Understanding it in terms of generators and relations
2. Understanding it in terms of its representation category (Tannaka-Krein)

Classification 1 is known in some cases, 2 is probably difficult, so having a different "equivalent" category to work with seems desirable.

Now, you have reduced the question why we should bother about the local Galois group (see discussion in the comments).