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As from this website http://math.uchicago.edu/~lxiao/workshop_site/

My question is: What does it mean by "purely local"? Also, I heard about this phrase "purely local" in other problems as well, mostly with the phrase "a purely local proof".

The other question is, for GL(1) and GL(2), are there already a "purely local proof"?

Thanks.

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    $\begingroup$ They just mean a proof that doesn't invoke the cohomology of Shimura varieties or the theory of automorphic representations. And, yes, local proofs are known for GL(1) and GL(2): In the first case, it's just local class field theory, and, for the second, a purely local proof can be found, for example, in the book of Bushnell-Henniart. $\endgroup$
    – Keerthi Madapusi
    Commented May 8, 2012 at 22:41
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    $\begingroup$ I'm not even sure that there is a published purely local statement of local Langlands, let alone a proof! If you just want "a bijection" then someone will come along and give you a counting argument bijecting the two sets, and that's no good. if you want "a natural bijection" then the question is "what do you mean by natural?". And if you go with the usual rigorous formulation, involving matching epsilon factors of pairs, then you have to define epsilon factors of pairs, and the usual method for doing this (and in particular proving well-definedness of the definition) is global. $\endgroup$
    – Kevin Buzzard
    Commented May 8, 2012 at 23:47
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    $\begingroup$ What is the history of local class field theory? My understanding is that it was derived from global class field theory originally. But once there was a statement, did people find a local proof? or was there a delay? $\endgroup$
    – Ben Wieland
    Commented May 9, 2012 at 0:31
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    $\begingroup$ @Alexander Chervov: Kevin isn't saying that you can't characterize the LLC by local properties, he's saying that you can't characterize the LLC by local properties whose existence has a (published) local proof. The epsilon factors are local (and they are measuring ramification), but the only published proof of their existence uses a global argument. In the n=1 case, things are simpler since the ramification of characters isn't as complicated. $\endgroup$
    – Rob Harron
    Commented May 10, 2012 at 14:51
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    $\begingroup$ Alexander: here are some other remarks that perhaps put things into some sort of context. There is a "local Langlands conjecture" for any connected reductive group over a local field, but as far as I know, if the local field isn't R or C, one cannot as yet write down a list of nice properties for which one can prove that there is at most one bijection with these properties: there is a list of properties, but currently they probably don't tie down the correspondence uniquely. Historically the same was true for GL(n) a long time ago, but the epsilon factor of pairs trick sufficed in this case. $\endgroup$
    – Kevin Buzzard
    Commented May 10, 2012 at 18:47

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The short answer to the question is that all currently known proofs of the local Langlands correspondence (and I'm just referring to GL(n) here) are "global" in the sense that they involve embedding the local problem into a global one. That is, the local field in question is realized as the completion of a global field at one of its places. Then the theory of automorphic forms over the global field may be applied. In particular, under certain circumstances, we know that Galois representations may be attached to automorphic representations. A purely local proof would not make reference to global fields at all.

Kevin commented that a purely local characterization (he uses the word statement) of the correspondences is a prerequisite for a purely local proof. The established characterization for GL(n) (and indeed, the one used in the proofs of Henniart and Harris-Taylor) is, as Kevin points out, through epsilon factors of pairs, and the existence of these is only defined through global means. (Rob is correct that Langlands has unpublished notes on the subject, but these are so complicated as to be unsatisfactory, and in any case it is truly unclear what the right characterization is for groups other than GL(n).)

Now to Alexander Chervov's important comment: what is the right characterization in the case of $n = 1$? Sure, you can make some quantitative conditions involving ramification. But let's recall that the most elegant path to local CFT is unquestionably through Lubin-Tate theory: the maximal totally ramified abelian extensions of a nonarchimedean local field are obtained by adjoining the torsion of a one-dimensional formal module of height one. Let us declare that Lubin-Tate theory itself provides the correct characterization of the local Langlands correspondence in the $n=1$ case (and to hell with conductors, Gauss sums, etc.).

This point of view suggests that variations on the theme of formal modules ought to provide the right purely local characterization of local Langlands (and also a hope for a purely local proof). Now already by 1990, Carayol conjectured ("Nonabelian Lubin-Tate theory") that certain deformation spaces of formal modules ("Lubin-Tate spaces") exhibit the local Langlands correspondence in their cohomology, at least for some classes of representations of GL(n). Harris and Taylor prove Carayol's conjecture for supercuspidal representations, which is enough to prove the existence of the correspondence in general. Here the characterization is still through epsilon factors of pairs, and therefore still global in nature.

The next big development along these lines is Peter Scholze's new proof of the correspondences for GL(n). While still global in nature, Scholze gives a purely local characterization of the correspondences, which satisfies Kevin's requirements for a "natural bijection", and which is compatible with the global theory. Suppose $\pi$ is a smooth irreducible representation of $\text{GL}_n(F)$ ($F$ a $p$-adic field). Scholze characterizes the corresponding (semisimplified) Weil representation $\sigma$ by giving an actual formula for the trace of $\sigma(\tau)$, for any element $\tau$ in the Weil group of $F$! Alas, the other side of Scholze's formula is too involved to describe here, but it involves deformation spaces of $p$-divisible groups in an ingenious way. When $n=1$, the formula reduces to the statement that local class field theory is realized in the torsion of Lubin-Tate formal modules. In my mind, purely local attacks on the local Langlands correspondence ought to start here.

(Not that any of the preceding is going to be mentioned in my talks tomorrow. My own meager contributions to this story don't yet connect to Scholze's work, but only to the theory of types, which figure prominently in the Bushnell-Henniart book mentioned by Keerthi.)

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  • $\begingroup$ Ah, I didn't know Scholze had a local characterization. It's very satisfying that it's in terms of something that pins down the representation of the Weil group using traces, rather than $L$- and $\epsilon$-factors. $\endgroup$
    – Rob Harron
    Commented May 12, 2012 at 12:12
  • $\begingroup$ @Jared : on the "other side of Scholze's formula", there is the trace of a certain "test function", say $f$. The latter function is obtained by "base change transfer" from another function $\phi$. While $\phi$ is undoubtedly purely local (involves cohomology of some deformation space for p-divisible groups), and while $f$ is characterized by purely local conditions (matching of orbital integrals), I wonder if the very existence of $f$ has been proved by local means ??? $\endgroup$
    – Jef
    Commented May 13, 2012 at 20:00
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    $\begingroup$ @Jef: Yes, the existence of $f$ is proved by purely local means in Chapter 1 of the Arthur-Clozel book on base change. That result is used to characterize automorphic base change. On the other hand, [AC] uses global means to establish the existence of automorphic base change. $\endgroup$
    – Jared Weinstein
    Commented May 14, 2012 at 3:27
  • $\begingroup$ @Jared, Thanks! $\endgroup$
    – natura
    Commented May 14, 2012 at 4:41
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This paper on a series of lectures have what your looking for with regards to your first question.

http://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/wedhorn/publications/LocalLanglands.pdf

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