Rational points on surfaces of general type The weak Lang conjecture asserts that rational points on a variety of general type defined over $\mathbb{Q}$ are not Zariski dense (same replacing $\mathbb{Q}$ with a number field). This one is proved in dimension $1$, and in dimension $2$ it would in particular mean that the rational points of surface of general type are contained in the union of a finite number of rational and elliptic curves, plus a finite set of "isolated points".
It seems natural for me that a surface of general type has "less points" because the polynomials that define it are of "higher" degree. However, is there some more precise evidence for this conjecture ? 
And in which particular cases is the conjecture known to be true?
I am mostly interested in the case of surfaces, but higher dimension case is also interesting.
 A: A Theorem of Faltings states that any proper subvariety of an abelian variety has finitely many rational points provided this subvariety does not contain a translate of any non-trivial proper abelian subvariety:
G. Faltings, The general case of S. Lang’s conjecture, pages 175–182 of Barsotti Symposium
in Algebraic Geometry (Abano Terme, 1991), Perspect. Math. 15, Academic Press, San Diego,
CA, 1994.
Harris and Silverman use this to show that if C is curve of genus $\ge 2$ that is neither hyperelliptic nor bielliptic, then the set of rational points on $C^{(2)}$ is finite. Here $C^{(2)}$ is the symmetric square of $C$. 
J. Harris and J. H. Silverman, Bielliptic curves and symmetric products, Proceedings of the
American Mathematical Society 112 (1991), no. 2, 347–356.
My understanding is that if $C$ has genus $3$ (say a plane quartic) then $C^{(2)}$ is a surface of general type. 
I admit however that symmetric powers of curves are really special and probably not what the OP is hoping for.
