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 $C\subset X$ 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 $\pi_i$ ($j\neq i$) with rank at least the generic rank of $\pi_i$ plus one.