Dear user02138, you may be interested in the Bombieri-Lang conjecture. It has a weak form and a strong form. The latter is:
If $X$ s a smooth variety over a number field $K$ which is of general type over a number field, then there are only finitely many curves of (geometric) genus $\leq 1$ contained in $X$, and outside those curves $X$ has only finitely many $K$-rational points.
The key word here is "general type" which is a geometric condition (defined for a proper smooth variety as the canonical divisor $K$ being big, that is the order magnitude of the dimensions of the space of global section of the bundle $L(nK)$ is $n^d$ as $n$ goes to infinity), which for proper smooth curves is equivalent to the Mordell's condition $g \geq 2$. Under this condition (which some experts in algebraic geometry may discuss in more detail that I can -- for example is it a condition that can be read on $X(\mathbb C)$ as topological spacelike for curves?), you get strong results on the number of rational points. Of course this conjecture is wide-open, except in the cases of curves or more generally subvarieties of abelian varieties, proved by Faltings.