On the kernel of the Brauer-Manin pairing

Abstract

Let X be a regular scheme, flat and proper over the ring of integers of a p-adic field, with generic fiber X and special fiber Xs. We study the left kernel Br(X) of the Brauer-Manin pairing Br(X)×CH0(X)→Q/Z. Our main result is that the kernel of the reduction map Br(X)→Br(Xs) is the direct sum of (Q/Z[1p])s⊕(Q/Z)t and a finite p-group, where s+t=ρXs−ρX−I+1, for ρXs and ρX the Picard numbers of Xs and X, and I the number of irreducible components of Xs. Moreover, we show that t>0 implies s>0.