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A paper of Heath-Brown gives an heuristic argument for the density of rational points on two cubic surfaces: $x^3+y^3+z^3=kw^3,k=2,3$, say, the number of rational points of height less than $N$ on these cubic surfaces is about $cN$, where $c$ has an interpretation from circle method.

Definition: Let $$\Sigma=\{P|P=(x,y,z,w)\in\mathbb{Z}^4,x^3+y^3+z^3=kw^3,gcd(x,y,z,w)=1\}$$

and $$\Sigma'=\{P'|P'\in\Sigma\},$$

where $P'$ lies on a plane rational curve over $\mathbb{Q}$ and $P'$ cannot be a non-singular point on plane rational curves over $\mathbb{Q}$.

Let $h(P)=\max\{|x|,|y|,|z|,|w|\},P=(x,y,z,w)$.

Question: A numerical computation seems to suggests that $\#\{P'|P'\in\Sigma',h(P')<N\}\sim c'N$ for these two surfaces, but I have no idea about an heuristic interpretation for the constant $c'$ ($c'\approx 0.8 c$ in both cases,but not exactly the same). Is there any heuristic interpretation for such constants?

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  • $\begingroup$ I don't quite understand your definition of $\Sigma'$, can you elaborate a bit more? Namely the condition that "P′ cannot be a non-singular point on plane rational curves over $\mathbb{Q}$". Do you mean that for every hyperplane $H \subset \mathbb{P}^3$ which contains $P'$ and for which $S \cap H$ is a (possible union of) rational curves, the point $P'$ is always a singular point of $S \cap H$? Can you give some examples of the points which you are interested in? $\endgroup$
    – Daniel Loughran
    Commented Nov 17, 2013 at 18:53
  • $\begingroup$ @DanielLoughran: Yes,that is what I try to express(of course, all the rational curves are defined over $\mathbb{Q}$). In fact, $(1,1,1)$ is such a point on $x^3+y^3+z^3=3$. As for other points on the surface $x^3+y^3+z^3=3$, I cannot prove they are elements of $\Sigma'$(just because I cannot list all the rational points on some genus 3 curves), but there is some heuristic arguments suggests that rational points on genus 3 curves are of "small" height, so a naive search might give all the rational points on a genus 3 curve, which is key to ascertain whether a point is in $\Sigma'$ or not. $\endgroup$
    – Y. Zhao
    Commented Nov 18, 2013 at 7:53
  • $\begingroup$ I think you mean "degree $3$" rather than "genus $3$" here; an irreducible singular plane cubic curve has arithmetic genus one and its normalisation has genus zero. I don't buy your claim that the rational points on such curves have "small height". For example the curve $y^2 = x^3$ may be parametrised by the map $t \mapsto (t^2,t^3)$, so there are many rational points of large height. $\endgroup$
    – Daniel Loughran
    Commented Nov 18, 2013 at 9:54
  • $\begingroup$ Also I should note that your surfaces have Picard number one, in particular they contain no lines or conics. So your question is really about singular plane cubic curves. $\endgroup$
    – Daniel Loughran
    Commented Nov 18, 2013 at 9:55
  • $\begingroup$ Do you have a proof that the point $(1,1,1)$ satisfies your conditions? $\endgroup$
    – Daniel Loughran
    Commented Nov 18, 2013 at 9:58

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