Zero-cycles on rational surfaces over number fields

Abstract

Let X be a smooth, projective, geometrically integral surface defined over a perfect field k. Let kbe an algebraic closure of k and F be the function field of X : = k • X. Then X is said to be rational if F is a purely transcendental extension of k-. The tame symbols define a map from KE(F ) to the sum of function fields H k-(7)* of all irreducible divisors 7 on X. Let M be the cokernel of this map. If k is a number field and kv the completion at a place v of k, let kv be an algebraic closure of kv and M~ the corresponding cokernel for k-~ x X. Further, let I I I l (k ,M) be the kernel of the natural map from Hl(Gal(k /k) ,M) to [-I H1 (Gal(k-v/kv), My). In [5, w Colliot-Th616ne and Sansuc conjectured that