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The Langlands Program was launched almost fifty years ago, and progress has been made gradually, much of it hard earned. Langlands himself wrote a survey on the functoriality conjecture in 1997, Where Stands Functoriality Today? I am familiar with the following:

  • Lafforgue proved the Langlands correspondence for $GL_n$ for function fields.
  • Laumon, Rapoport and Stuhler proved local Langlands for $GL_n$ in characteristic $p>0$.
  • Henniart, Harris and Taylor, and Scholze each proved local Langlands for $GL_n$ characteristic 0.

As a young researcher, I am interested to know what else is now known? (including significant partial results, particularly after the stabilization of the trace formula thanks to the fundamental lemma.) And perhaps, though this may be too broad, what remains to be shown. Indeed, the proper formulation may not even be known in some cases.

On a related note, the geometric Langlands is an active area of research today. In what way is it related to the classical Langlands conjectures?

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Vincent Lafforgue proved the automorphic $\to$ Galois direction of function field Langlands for an arbitrary reductive group. –  Will Sawin Mar 29 at 20:32
    
There have been many similiar questions in the past. –  Marc Palm Mar 29 at 20:41
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@plusepsilon.de: Dear plusepsilon.de, Since many results for $GL(n)$ are proved via transfer to unitary groups, stabilization is in fact an issue, even in the theory for $GL(n)$. Regards, –  Emerton Apr 30 at 1:53
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@plusepsilon.de: Dear plusepsilon.de, Yes, that's right. But to actually construct/study Galois reps. attached to $GL(n)$ automorphic form, you have to work with unitary groups, for which stable conjugacy and conjugacy are distinct. For example, the proof of Sato--Tate is prima facie about proving cases of symmetric power functoriality from $GL(2)$ to $GL(n)$, and so one could imagine that it doesn't involve stabilization or the fundamental lemma. But in fact, the proof for elliptic curves with integral $j$-invariant depended on the fundamental lemma for unitary groups. Regards, –  Emerton Apr 30 at 11:28

2 Answers 2

up vote 14 down vote accepted

As said in comments, the question has already been discussed on MO, see the links given there.

To summarize and complete, it is important to remember that in the case of number fields the correspondence between Galois representations and certain automorphic representations (the ones which are algebraic at infinity in the case of number fields) is only a part of the Langlands program, essentially because it concerns just certain automorphic representations, and Langlands functoriality is about all of them.

In the function field case, for $\operatorname{GL}_n$ over an arbitrary field, the correspondence has been completely done 15 years ago by Laurent Lafforgue. More recently (last year) Vincent Lafforgue (younger brother of Laurent) has done the sense Automorphic $\rightarrow$ Galois for an arbitrary reductive group (that is to an automorphic form he attaches a suitable $L$-parameter as conjectured, that is a kind of twisted Galas representation). In the introduction of his paper he seems optimistic about his prospect to solve (with Genestier) the converse direction, which entails carefully regrouping the automorphic representations into classes called $L$-packets.

In the number field case, much less is known, but much more than 20 years ago. In the sense Automorphic $\Rightarrow$ Galois, we can now do the case of automorphic representations for $\operatorname{GL}_n$ over a field with is either totally real or CM, and which are not only algebraic but regular at infinity. We can also do this for other groups (unitary, orthogonal, symplectic) but that gives no new Galois representations so I don't dwell on it (though in the proof, we need to do the case of these groups before going to $\operatorname{GL}_n$). The great final stone was put, after a huge collective effort leading to the case of self-dual or conjugate slef-dual representations, by Harris-Lan-Taylor-Thorne and then by another method by Scholze, which also deals with the case of torsion automorphic forms, not part of the initial Langlands program. The next main frontier seems to me to be able to deal with non-regular algebraic automorphic representations, the simplest case of which being algebraic Maass form for $\operatorname{GL}_2$.

Much progress has been made since Wiles and Taylor's proof of FLT on the converse direction Galois $\Rightarrow$ Automorphic, but it definitely lags behind the other sense. Essentially, due to work of Harris-Taylor and many others, the case of almost all conjugate self-dual Galois representation is done. Work in progress involving a numb or people could do many new cases, perhaps all where we know the automorphic to Galois sense.

This is essentially what is known about the Global Galois/Automorphic correspondence. For the functoriality for (non necessarily algebraic) automorphic representations, much less is known (Solvable base change for $\operatorname{GL}_n$ by Langlands and Arthur-Clozel, some transfer between classical groups, some low degree exterior powers) but much remains to be done.

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You mention "not part of the initial Langlands program" regarding Scholze's work on torsion classes. Not being an expert on the geometric Langlands program, I am curious: is there any similar "surprise" in the case of functions fields? –  tkr Apr 1 at 3:12
    
@tkr I don't know. Interesting question. About the number field case, I just wanted to precise that though not part of the initial Langlands program, the conjectural existence of Galois representations attached to torsion classes was added to it twenty years ago, due to work of Ash and Stevens. –  Joël Apr 1 at 13:10
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Dear Joel, This is an impressive summary of the current status! Cheers, –  Emerton Apr 30 at 1:57
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Dear Emerton (or Matt), thanks. It's too bad you don't write here anymore, except for short comments. Why don'y you come back. Cheers. –  Joël May 1 at 14:42

I believe that this question is too broad to be answered completly, at least for me. I start with a few points you don't address:

  • Deligne and Deligne-Serre have proven the correspondence for weight $k \geq 1$ modular forms over $GL(2) / \mathbb{Q}$.

  • For Maass wave forms, one knows that most of the time there is no corresponding Galois representation. For those, where we expect them to exist (Laplace eigenvalue $1/4$ for weight zero for example), we have no clue how to single them out among the remaining Maass wave forms. The only tool one has is the Arthur-Selberg trace formula, which analyzes all Maass wave forms simultaneously. The Selberg eigenvalue conjecture indicates, how little we know what happens close to $1/4$. There is no "pseudo matrix coefficient" for continuous series representations. For GL(3), there are no discrete series, so here you always have the same problem.

  • Besides the archimedean issues, in the function field world, there is an additional feature that you can work with $\ell$-adic cohomologies with $\ell$ coprime to the residue characteristic of all places. This is impossible in the number field world. Afaik $\ell = p$ makes the analysis more difficult.

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