  Andr\'e used Hodge-theoretic methods to show that in a smooth proper family X
to B of varieties over an algebraically closed field k of characteristic 0,
there exists a closed fiber having the same Picard number as the geometric
generic fiber, even if k is countable. We give a completely different approach
to Andr\'e's theorem, which also proves the following refinement: in a family
of varieties with good reduction at p, the locus on the base where the Picard
number jumps is nowhere p-adically dense. Our proof uses the ``p-adic Lefschetz
(1,1) theorem'' of Berthelot and Ogus, combined with an analysis of p-adic
power series. We prove analogous statements for cycles of higher codimension,
assuming a p-adic analogue of the variational Hodge conjecture, and prove that
this analogue implies the usual variational Hodge conjecture. Applications are
given to abelian schemes and to proper families of projective varieties.
