# Arithmetic property of a surface of general type

In my previous post I asked about the hyperbolicity of the affine surface $S'=\{zw \neq o\}$ in the projective surface $z^2 = P(x) Q(y)$ in $\mathbb{P}^3$, where $P$ and $Q$ are two general polynomials of the same degree $d$. Jason wrote that this affine surface is a finite unbranched cover of the product of two affine hyperbolic curve $(\mathbb{C}\setminus Z(P(x)))\times (\mathbb{C}\setminus Z(Q(y)))$,$(x,y,z) \mapsto (x/z, y/z)$ and therefore $S'$ is also hyperbolic as well (what is the generic condition on polynomials $P$ and $Q$?).//

additionally I could prove that this surface is a surface of general type if $d\ge 5$ and the roots of $P$ and $Q$ are not the same.//

Now assume the equation $z^2 = P(x) Q(y)$ on a number field $K$. I want to ask if the number of $K$ - rational points in the affine part $zw \neq 0$ is finite? I want to deduce this result from Lang conjectures which asserts that:

on a surface of general type the Zariski closure of rational points is a proper sub variety not of general type, i.e. in this case it is a finite union of rational and elliptic curves and a finite singular points.

• The map is $(x,y,z)\mapsto (x,y)$, not $(x,y,z)\mapsto (x/z,y/z)$. – Jason Starr Apr 1 '14 at 12:12
• How much is your degree $d$? If too small, the variety is certainly not of general type. Why do you think that you dont have rational curves on the surface? – Jérémy Blanc Apr 1 '14 at 12:28
• thanks Jason, What about the generic condition on the polynomials $P,Q$?Is it always true for all polynomials $P,Q$. – Mehdi Apr 1 '14 at 12:29
• If you take $P$ and $Q$ of small degree (for example of degree $1$), your surface $z^2=P(x)Q(y)$ is of course rational. That's why I asked what you assume on $P$, $Q$ to say that your variety is of general type. – Jérémy Blanc Apr 1 '14 at 12:46
• $d$ can be chosen arbitrary large. – Mehdi Apr 1 '14 at 12:52

It depends on the polynomials $P$ and $Q$ if they admit a polynomial parametrization they can even have infinitely many integer solutions.
• My question is for general polynomials $P$ and $Q$, i.e. the coefficients of $P, Q$ satisfy in some generic condition. – Mehdi Apr 1 '14 at 16:13
• Note that in my previous post it was shown that the affine part $z\neq 0$ is hyperbolic so it is natural (regarding Lang Conjecture) to expect that there is finite rational points on this part. – Mehdi Apr 1 '14 at 16:18