  Over function fields of p-adic curves, we construct stably rational varieties
in the form of homogeneous spaces of SL_n with semisimple simply connected
stabilizers and we show that strong approximation away from a non-empty set of
places fails for such varieties. The construction combines the Lichtenbaum
duality and the degree 3 cohomological invariants of the stabilizers. We then
establish a reciprocity obstruction which accounts for this failure of strong
approximation. We show that this reciprocity obstruction to strong
approximation is the only one for counterexamples we constructed, and also for
classifying varieties of tori. We also show that this reciprocity obstruction
to strong approximation is compatible with known results for tori. At the end,
we explain how a similar point of view shows that the reciprocity obstruction
to weak approximation is the only one for classifying varieties of tori over
p-adic function fields.
