Question on effective Mordell conjecture Suppose $F(x,y,z)$ is a homogeneous polynomial over $\mathbb{Q}$, where $C:F(x,y,z)=0$ is a curve of genus $g\geq 2$.
Question: Faltings proved that $C$ has finite many rational points. Suppose that there is no rational points on $C$. Is there any effective version of Mordell conjecture, i.e., $$\vert F(x,y,z)\vert\gg_{F,g} g(H(x,y,z)),(x,y,z)\in\mathbb{Z}^3$$
where $H$ is naive height and $g(x)$ tends to infinity as $x$ tends to infinity?
 A: Have a look at the work of Gael Remond; see e.g.
http://plms.oxfordjournals.org/content/101/3/759.abstract
In that paper Remond proves an explicit upper bound on the number of rational points on a curve. You won't be able to do better than that I think with the current state-of-affairs. 
Added in edit: The above is not relevant to your question, but it does address but people usually refer to as Effective Mordell. As Felipe Voloch points out, what is usually meant by "Effective Mordell" is an explicit/effective bound on the height of a rational point on a smooth projective curve $X$ over a number field $K$ in terms of data associated to $X$ and $K$. The work of Gael Remond (cited above) is concerned with the number of points on $X$, and gives upper bounds for this number. If you are willing to believe that the Shafarevich conjecture can be made effective (see another paper of Gael Remond) then you can also prove an effective version of the Mordell conjecture. Anyway, this doesn't answer your question, but it might be good to know.
