The genus $g$ of a nonsingular curve $C$ of degree $n$ is defined as $g = \frac{1}{2}(n-1)(n-2)$. Let $C(Q)$ denote the set of rational points on $C$. By Faltings, we know that $C (Q) < \infty$ for all $g\geq 2$.
I'm curious whether there exists another result (apart from Faltings') that relates the size of $C(Q)$ and $g$ ? More strongly, does the size of $C(Q)$ vary directly as $g$ ?
It is still an open question whether, for every $g \geq 2$ and any number field $K$, there exists a uniform bound $B_g(K)$ such that every curve $C$ of genus $g$ over $K$ has at most $B_g(K)$ rational points. This is known as the uniform Mordell conjecture.
Thus, not only do we not know the size $C(K)$ in terms of $g$, we don't even know whether there is a uniform bound.