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 $Br(X)\times C{H}_0(X)\to \mathbb{Q}/\mathbb{Z}$. 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 $s>0$.

Improved version