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.



"Purely local" proof of local Langlands
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 (Rob is correct that Langlands has unpublished notes on the subject, but these are so complicated as to be unsatisfactory, and in any case it is truly unclear what the right characterization is for groups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](http://www.math.uni-bonn.de/people/scholze/LocalLanglands.pdf) 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.)


