  We discuss some aspects of the behavior of specialization at a finite place
of N\'eron-Severi groups of K3 surfaces over number fields. We give optimal
lower bounds for the Picard number of such specializations, thus answering a
question of Elsenhans and Jahnel. As a consequence of these results, we show
that it is possible to explicitly compute the Picard number of any given K3
surface over a number field.
