Cohomology of tori over p -adic curves

The Brauer–Manin pairing, class field theory, and motivic homology

“…In Section 5, we consider the case where X is a curve. In this case, the map ψ X was already studied by Scheiderer and van Hamel [33] (see Theorem 5.1 below), and the map ρ X is closely related to work of Hiranouchi [16] (see Theorem 5.4 and Remark 5.5 below). When X is of dimension 2 or higher, the maps ψ X and ρ X are not very close to an isomorphism even if X is projective over k. We study several examples of surfaces in Section 6.…”

The Brauer–Manin pairing, class field theory, and motivic homology

“…In this case, the map ψ X was already studied by Scheiderer-van Hamel [34] (see Theorem 5.1 below), and the map ρ X is closely related to work of Hiranouchi [16] (see Theorem 5.4 and Remark 5.5 below). When X is of dimension two or higher, the maps ψ X and ρ X are not very close to an isomorphism even if X is projective over k. We study several examples of surfaces in §6.…”

“…Theorem 5.1 (Scheiderer/van Hamel [34]). The homomorphism ψ U : C 0 (U ) → Br(U ) * is an injection with dense image.…”

Weak Approximation for Tori Over p-adic Function Fields

“…Note first that this direct limit is H 3 (K, T ) because H 2 (K, T ) ⊗ Q/Z = 0. But H 3 (K, T ) is the direct limit of the groups H 3 (V, T ) for V ⊂ U 0 and the latter groups are trivial by ( [26], Corollary 4.10).…”