  In this paper, we study an analogue of the Tate conjecture for $K_2$ of U,
the complement of split multiplicative fibers in an elliptic surface. A main
result is to give an upper bound of the rank of the Galois fixed part of the
etale cohomology $H^2(\bar{U},Q_p(2))$. As an application, we give an elliptic
K3 surface $X$ over a p-adic field for which the torsion part of the Chow group
$CH_0(X)$ of 0-cycles is finite. This would be the first example of a surface
$X$ over a p-adic field whose geometric genus is non-zero and for which the
torsion part of $CH_0(X)$ is finite.
