Rational points on conic bundles over elliptic curves

TL;DR

This paper investigates the limitations of the etale Brauer-Manin obstruction in explaining failures of the Hasse principle for rational points on conic bundles over elliptic curves with positive rank, and explores distribution properties of these points.

Contribution

It demonstrates that the etale Brauer-Manin obstruction is insufficient for certain conic bundles over elliptic curves and analyzes the distribution of rational points relative to the elliptic curve's rational points.

Findings

Etale Brauer-Manin obstruction does not fully explain failures of the Hasse principle.

Results on local-to-global principles for torsion points on elliptic curves over .

Insights into the distribution of rational points on conic bundles over elliptic curves.

Abstract

We study rational points on conic bundles over elliptic curves with positive rank over a number field. We show that the etale Brauer-Manin obstruction is insufficient to explain failures of the Hasse principle for such varieties. We then further consider properties of the distribution of the set of rational points with respect to its image in the rational points of the elliptic curve. In the process, we prove results on a local-to-global principle for torsion points on elliptic curves over $\mathbb{Q}$.