Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the kernel of the Brauer-Manin pairing   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 $s>0$.

17
The categories of this paper are: q-fin
5
The categories of this paper are: gr-qc
24
The categories of this paper are: dg-ga
22
The categories of this paper are: chao-dyn
27
The categories of this paper are: comp-gas
26
The categories of this paper are: bayes-an
36
The categories of this paper are: cmp-lg
20
The categories of this paper are: acc-phys
34
The categories of this paper are: chem-ph
21
The categories of this paper are: adap-org
31
The categories of this paper are: ao-sci
29
The categories of this paper are: funct-an
4
The categories of this paper are: cond-mat
35
The categories of this paper are: mtrl-th
3
The categories of this paper are: physics
32
The categories of this paper are: atom-ph