Density of rational points on isotrivial rational elliptic surfaces

TL;DR

This paper demonstrates the density of rational points on a broad class of isotrivial rational elliptic surfaces and establishes a variant of weak-weak approximation, contingent on Tate-Shafarevich group finiteness.

Contribution

It provides new results on the density of rational points and weak-weak approximation for isotrivial rational elliptic surfaces, linking these properties to root number variations.

Findings

Rational points are dense in these surfaces for the Zariski topology.

The surfaces satisfy a variant of weak-weak approximation.

Results depend on the finiteness of Tate-Shafarevich groups.

Abstract

For a large class of isotrivial rational elliptic surfaces (with section), we show that the set of rational points is dense for the Zariski topology, by carefully studying variations of root numbers among the fibers of these surfaces. We also prove that these surfaces satisfy a variant of weak-weak approximation. Our results are conditional on the finiteness of Tate-Shafarevich groups for elliptic curves over the field of rational numbers.