Rank jumps and Multisections of elliptic fibrations on K3 surfaces

We consider the countably many families Ld, d≥ 2, of K3 surfaces admitting an elliptic fibration with positive Mordell--Weil rank. We prove that the elliptic fibrations on the very general member of these families have the potential Mordell--Weil rank jump property for d≠ 2,3 and moreover the Mordell--Weil rank jump property for d≡ 3 4, d≠ 3. We provide explicit examples and discuss some extensions to subfamilies. The result is based on the geometric interaction between the (potential) Mordell--Weil rank jump property and the presence of special multisections of the fibration.