On the rank of the fibers of elliptic K3 surfaces

Let X be an elliptic K3 surface endowed with two distinct Jacobian elliptic fibrations pi(i), i = 1, 2, defined over a number field k. We prove that there is an elliptic curve C subset of X such that the generic rank over k of X after a base extension by C is strictly larger than the generic rank of X. Moreover, if the generic rank of pi(j) is positive then there are infinitely many fibers of pi(i) (j not equal i) with rank at least the generic rank of pi(i) plus one.