On the rank of the fibers of elliptic K3 surfaces

Abstract

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 ⊂ 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 ≠ i) with rank at least the generic rank of π i plus one.