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On the rank of the fibers of elliptic K3 surfaces

Let $X$ be an elliptic K3 surface endowed with two distinct Jacobian elliptic fibrations $π_i$, $i=1,2$, 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 $π_j$ is positive then there are infinitely many fibers of $π_i$ ($j\neq i$) with rank at least the generic rank of $π_i$ plus one.