  We shall show that the Picard number of the generic fiber, in the sense of
scheme, of an abelian fibered hyperk\"ahler manifold over the projective space
is always one. We then give a few applications for the Mordell-Weil group. In
particular, by deforming O'Grady's 10-dimensional manifold, we construct an
abelian fibered hyperk\"ahler manifold of Mordell-Weil rank 20, which is the
maximum possible among all known ones.
