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

Reciprocity obstruction to strong approximation over p-adic function fields