Rank jumps for Jacobians of Hyperelliptic curves on K3 surfaces

TL;DR

This paper investigates the phenomenon of Mordell-Weil rank jumps in families of Jacobians of genus-2 curves on K3 surfaces over number fields, identifying conditions for infinite rank jumps and non-thin sets.

Contribution

It demonstrates the existence of infinite rank jumps over finite extensions and describes geometric conditions for rank jumps on non-thin sets of fibers.

Findings

Rank jumps occur infinitely often over some finite extension.

Certain geometric conditions on K3 surfaces lead to rank jumps on non-thin sets.

The study advances understanding of Mordell-Weil groups in algebraic geometry.

Abstract

We study Mordell-Weil rank jumps on families of jacobians of a pencil of genus-2 curves on a K3 surface defined over a number field k. We exhibit a finite extension l/k over which the subset of fibers for which the rank jumps is infinite. Moreover, we describe further geometric conditions on the K3 surface under which the rank jumps on a non-thin set of fibers.