Let $X/k$ be a smooth curve of genus $g>1$ over a number field $k$. By the Faltings theorem (nee Mordell's conjecture), the set of $k$ - rational points $X(k)$ is finite. Due to the Mordell-Weil theorem, Faltings's theorem is a consequence of the following statement: If $X$ is a complex smooth curve of genus $g>1$, and $\Gamma\subset J$ is a finitely generated subgroup of the Jacobian $J=Jac(X)$, then $X\cap \Gamma$ is finite.
The latter statement is purely geometrical (and known to be true). So it is natural to try to prove it using geometric methods, but as far as I know, nobody did this. Are there any promising approaches towards geometric proof of the Faltings theorem? (I have heard of Chabauty's method, but probably it doesn't work for $\mathbb{C}$.)
