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According to Silverman, the Bombieri-Lang conjecture implies that the rational points of surface on general type lie on finite set of curves, except for a finite set of points.

Let $f$ be univariate squarefree polynomial with integer coefficients of degree $n$.

Consider the surface:

$$ y^2=f(x)f(z) \qquad (1)$$

According to MSE comment (1) is of general type for $n \ge 6$ in general.

For rational $d$ consider the twist of the hyperelliptic curve:

$$ dy^2=f(x) \qquad (2)$$

If (2) has two rational points $(x_1,y_1),(x_2,y_2)$ infinitely often, this gives rational point on (1) and on the curve $z=x_2,y^2=f(x)f(z)$.

Since the number of curves must be finite, this means there must be finite set of relations between $x_1$ and $x_2$, the simplest beeing $x_1=x_2$ and similar automorphisms.

Q1 Does Bombieri-Lang imply such restriction on rational points on the twist?

Q2 Is there reason to believe there will be simple relation between $x_1$ and $x_2$ infinitely often?

Assuming $abc$,$abcd$ and significant restrictions on $f$ and certain other non-vanishing conditions,$n$ sufficiently large, Granville showed there are more than one non-trivial solutions to (2) finite number of times.

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  • $\begingroup$ @René thanks for fixing the spelling. Remains to fix the grammar and the math. $\endgroup$
    – joro
    Commented Aug 27, 2015 at 15:00
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    $\begingroup$ Yes, it does. In Caporaso-Harris-Mazur, one says that the points $(x_1,y_1)$ and $(x_2,y_2)$ are correlated. It is by a similar flavor of construction that they proved, in wider generality, that Bombieri-Lang places a uniform bound on the number of rational points on a curve of a fixed genus. $\endgroup$
    – Vesselin Dimitrov
    Commented Aug 27, 2015 at 17:15

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