See Salch on arxiv, and his webpage
Various material (from the above seminar I think) in folder AG/Top Langlands
http://mathoverflow.net/questions/7283/topological-langlands
A talk abstract: Andrew Salch (Johns Hopkins University). A computational motivation for topological Langlands correspondences. Abstract: We describe the most effective known method for computing the stable homotopy groups of spheres at odd primes; this uses formal modules over p-adic number rings, together with base-change properties of the “chromatic filtration,” i.e., local cohomology on the moduli stacks of formal modules, to construct the Adams-Novikov E_2-term. A consequence of these methods is the appearance of denominators of special values of Artin L-functions in the orders of the stable homotopy groups of spheres. We describe the role this phenomenon plays in motivating the (still very speculative) study of topological Langlands correspondences.
http://www.math.jhu.edu/~asalch/toplang/tltalk.pdf Talk of Salch
http://www.math.jhu.edu/~asalch/toplang/dca1.pdf Derived Carayol attack
nLab page on Topological Langlands