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 ${\pi }_i$, $i=1,2$, 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 ${\pi }_i$ ($j\ne i$) with rank at least the generic rank of ${\pi }_i$ plus one.

9 pages

References in corpus (1)

• On the behaviour of root numbers in families of elliptic curves