Sur la théorie des corps de classes pour les variétés sur les corps p-adiques

Let k be a p-adic field. Consider a smooth, proper, geometrically integral k-variety X. In this paper, we study the reciprocity map φX: SK1(X) → πab1(X) introduced by S. Saito and prove that, assuming the Bloch-Kato conjecture in degree 3 for a prime l ≠ p (which is known for l = 2), its kernel is uniquely l-divisible for surfaces for which the l-adic cohomology group H2(X, ℚl) vanishes (so in particular for those with potentially good reduction). In higher dimension, we derive the same conclusion from a special case of a conjecture by Kato for varieties with good reduction. We also obtain finiteness results for the torsion part of the group SK1(X). The proofs exploit Voevodsky's motivic cohomology theory to which we furnish some complements in an appendix. In a second appendix, J.-L. Colliot-Thélène shows that the kernel of φX does indeed contain a huge uniquely divisible subgroup already in the case of curves of genus at least one.