Large rank jumps on elliptic surfaces and the Hilbert property

TL;DR

This paper investigates the distribution of fibers with increased Mordell--Weil rank on rational elliptic surfaces over number fields, providing conditions that prevent the set of fibers with rank jumps of at least 3 from being thin.

Contribution

It establishes specific conditions on singular fibers ensuring the set of fibers with significant rank jumps is not thin, advancing understanding of rank variation in elliptic surfaces.

Findings

Fibers with rank jumps of at least 3 are not thin under certain conditions.

Provides criteria relating singular fibers to rank jump behavior.

Enhances knowledge of the distribution of rational points on elliptic surfaces.

Abstract

Given a rational elliptic surface over a number field, we study the collection of fibers whose Mordell--Weil rank is greater than the generic rank. We give conditions on the singular fibers to assure that the collection of fibers for which the rank jumps of at least 3 is not thin.