Counting rational points on quartic del Pezzo surfaces with a rational conic

TL;DR

This paper establishes bounds for the number of rational points of bounded height on quartic del Pezzo surfaces over the rationals that contain a rational conic, advancing understanding of rational point distribution.

Contribution

It provides new upper and lower bounds for rational points on quartic del Pezzo surfaces with a rational conic, a case previously less understood.

Findings

Bounds are of the expected order of magnitude.

Results apply to all quartic del Pezzo surfaces with a rational conic.

Enhances understanding of rational points on algebraic surfaces.

Abstract

Upper and lower bounds, of the expected order of magnitude, are obtained for the number of rational points of bounded height on any quartic del Pezzo surface over $\mathbb{Q}$ that contains a conic defined over $\mathbb{Q}$.