  We constructed several families of elliptic K3 surfaces with Mordell-Weil
groups of ranks from 1 to 4. We studied F-theory compactifications on these
elliptic K3 surfaces times a K3 surface. Gluing pairs of identical rational
elliptic surfaces with nonzero Mordell-Weil ranks yields elliptic K3 surfaces,
the Mordell-Weil groups of which have nonzero ranks. The sum of the ranks of
the singularity type and the Mordell-Weil group of any rational elliptic
surface with a global section is 8. By utilizing this property, families of
rational elliptic surfaces with various nonzero Mordell-Weil ranks can be
obtained by choosing appropriate singularity types. Gluing pairs of these
rational elliptic surfaces yields families of elliptic K3 surfaces with various
nonzero Mordell-Weil ranks. We also determined the global structures of the
gauge groups that arise in F-theory compactifications on the resulting K3
surfaces times a K3 surface. $U(1)$ gauge fields arise in these
compactifications.
