On the kernel of the Brauer-Manin pairing

Abstract

Let $\mathcal X$ be a regular scheme, flat and proper over the ring of integers of a $p$-adic field, with generic fiber $X$ and special fiber $\mathcal X_s$. We study the left kernel $Br(\mathcal X)$ of the Brauer-Manin pairing $Br(X)\times CH_0(X)\to \mathbb Q/\mathbb Z$. Our main result is that the kernel of the reduction map $Br(\mathcal X)\to Br(\mathcal X_s)$ is the direct sum of $(\mathbb Q/\mathbb Z[\frac{1}{p}])^s\oplus (\mathbb Q/\mathbb Z)^t$ and a finite $p$-group, where $s+t= \rho_{\mathcal X_s}-\rho_X-I+1$, for $\rho_{\mathcal X_s}$ and $\rho_X$ the Picard numbers of $\mathcal X_s$ and $X$, and $I$ the number of irreducible components of $\mathcal X_s$. Moreover, we show that $t>0$ implies $s>0$.