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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.

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Great question, and I have no idea (nor have I ever heard Barry M. ask this...) I don't even know the answer to the following subquestion: is there a bound on the number of rational points on a curve whose Jacobian has Mordell-Weil rank 1? Or for that matter: given integers (a,b,c) summing to 0, are there curves over Q with rational points P,Q,R satisfying aP + bQ + cR = 0? Maybe it's easy to construct examples, I haven't thought about it. But that would be one way to start thinking about this problem. –  JSE Jun 18 '12 at 15:30
If I remember correctly, I heard Mazur mention this at a big AMS meeting in Louisville. I was still a grad student, so >25 years ago. –  David Feldman Jun 18 '12 at 15:39
I suspect that the low-hanging fruit would take the form of relative results: if such-and-such a configuration occurs, then by taking a suitable covering of the curve and extension of the number field so must one in some-or-other set. I'm not expert enough to be more precise. I also wonder about Hilbert-irreducibility-theorem type specialization: if a configuration exists in a suitable function field setting, then one can set the parameter and have it over a number field? –  David Feldman Jun 18 '12 at 18:38

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