Suppose that F/Q is a number field.

Using automorphic forms, Borel computed the (R-) stable cohomology of SL_n(O_F), and as a result, computed K_i(O_F) tensor Q. Nowadays, using work of Suslin, Voevodksy, Rost, etc, one has an almost "complete" understanding of the actual integral groups K_i(Z), say, modulo Vandiver's conjecture. This does not directly give the stable cohomology of SL_n, because one set of groups is computing homotopy and the other is computing cohomology, but let us not worry about that distinction for the moment.

Borel also computed the (R-) stable cohomology of Sp_2n(O_F). My question, loosely speaking, is whether one can describe the stable integral cohomology of Sp_2n(Z) in as detailed away as the algebraic K-groups of Z "describe" the integral cohomology of SL_n(Z).

Edit: Andy has given references for the stable computation of H_2 (and hence, since one knows the rational cohomology groups, of H^2 and H^3). Next Challenge: H_3?