On the rank of the fibers of elliptic K3 surfaces

TL;DR

This paper investigates the rank behavior of elliptic K3 surfaces with multiple fibrations, showing the existence of curves that increase the generic rank and infinitely many fibers with elevated rank.

Contribution

It establishes the existence of specific curves that raise the generic rank and demonstrates infinitely many fibers with increased rank in elliptic K3 surfaces with multiple fibrations.

Findings

Existence of a curve increasing the generic rank after base extension.

Infinitely many fibers with rank at least the generic rank plus one.

Results apply to elliptic K3 surfaces with two distinct Jacobian fibrations.

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