# Status of local Langlands conjecture over positive characteristic

I am interested to know what the status of Local Langlands Conjectures in positive characteristic is? By a positive characteristic local field, I mean a field of the form $\mathbb{F}_q((t))$.

A nice summary of what is known for characteristic zero local fields is given in the first paragraph of

http://arxiv.org/pdf/1204.0132v2.pdf

I'm looking for a similar description in the positive characteristic case.

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Why do you not consider ${\mathbf F}_q((t))$ to be a positive characteristic local field, where $q$ is a prime power but not necessarily prime? Do you consider finite extensions of ${\mathbf Q}_p$ to be local fields of characteristic 0? –  KConrad Feb 4 '14 at 23:06
Sorry, that was a typo. It is correct now. Thanks. –  Dr. Evil Feb 4 '14 at 23:39

I am not an expert, but what I think I know on the subject is: For $GL_n$, the local Langlands conjecture has long been known by work of Laumon, Rapoport and Stuhler (Inventiones Math, 1993). Of course, in this case, even the global correspondence is now known, due to the Fields-Medal-winning work of Laurent Lafforgue.