Rank jumps and Multisections of elliptic fibrations on K3 surfaces

TL;DR

This paper investigates how elliptic fibrations on K3 surfaces exhibit rank jumps, especially in very general families, revealing conditions under which the Mordell--Weil rank increases and exploring geometric interactions with multisections.

Contribution

It establishes the potential Mordell--Weil rank jump property for families of K3 surfaces with elliptic fibrations, depending on the parameter d, and connects this to the geometry of multisections.

Findings

Rank jumps occur for very general members when d ≠ 2,3.

Rank jumps also occur for d ≡ 3 mod 4, d ≠ 3.

Explicit examples illustrate the phenomena.

Abstract

We consider the countably many families $\mathcal{L}_d$, $d\in\mathbb{N}_{\geq 2}$, 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 $d\neq 2,3$ and moreover the Mordell--Weil rank jump property for $d\equiv 3\mod 4$, $d\neq 3$. 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.