Counting rational points on quadric surfaces

TL;DR

This paper establishes an explicit upper bound on the number of rational points of bounded height on quadric surfaces, which is optimal in terms of the height parameter and for typical quadratic forms.

Contribution

It provides a new, explicit upper bound for rational points on quadric surfaces that depends on the defining quadratic form, improving previous bounds and achieving optimality.

Findings

Bound is explicit and depends on the quadratic form Q

Bound is optimal with respect to the height B

Bound is also optimal for typical forms Q

Abstract

We give an upper bound for the number of rational points of height at most $B$, lying on a surface defined by a quadratic form $Q$. The bound shows an explicit dependence on $Q$. It is optimal with respect to $B$, and is also optimal for typical forms $Q$.