Faltings' theorem says (at least) that the set $C(K)$ of points of a higher genus curve $C$ rational over a number field $K$ constitutes a *finite* subset of the finitely generated (by Mordell-Weil) abelian group $J(C)(K)$ (of $K$-rational points of the Jacobian $J(C)$).

Decades ago I heard Mazur point out in a talk something along the following lines: one should now wonder what subsets of ${\Bbb Z}^n$ can arise by pulling back some $C(K)$ along an isomorphism from ${\Bbb Z}^n$ to the non-torsion part of some $J(C)(K)$. (One really has many questions here if one restricts the genus and $K$ in various ways.)

Question: What progress on this problem has occurred since Faltings' theorem? Progress can mean theorems, or an emerging conjectural picture, or merely the framing of technical questions that it seems "right" to study first.