The density of rational points on the cubic surface X³₀=X₁X₂X₃

Abstract

In this paper we are concerned with the problem of counting rational points of bounded height on rational cubic surfaces. For most such surfaces this question appears much too hard for current methods. We shall therefore examine a particular example, namely the surface

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for which the problem is tractable, though by no means trivial. This surface contains the lines X₀=X_i=0, for i=1, 2, 3, which we shall exclude from consideration. We therefore define

formula here

where an integer vector is said to be primitive if its components have no common factor. The function [Nscr ](H) counts points with the most naive height available. This suffices, however, to bring out the key features of the problem. We may observe at once that the condition x₀[les ]H is redundant, and that it suffices to assume that (x₁, x₂, x₃) is primitive.

Our principal result is the following theorem.