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This is a speculation and perhaps naive. The theorem of Siegel that

There exist only finitely many integral points on a curve of genus $\geq 1$ over a number ring $\mathcal O_{K, S}$ where $S$ is a finite set of places in a number field $K$

has proofs from arithmetic geometry, for instance by Parshin and Faltings, or by Kim. These proofs are interesting in that they use the modern developments in algebraic geometry and arithmetic to prove a theorem in number theory. But it is dissatisfying that as far as I am aware these proofs do not give an effective bound as opposed to the proofs using Baker's theorem. For instance, Silverman's book feels that the Parshin-Faltings proof is indirect.

But I do not know about the more recent developments and philosophy. I am wondering whether there is any hope that the more sophisticated proofs can be made effective, as it would give the best of both worlds. I hope the experts here can answer this.

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  • $\begingroup$ Bjorn Poonen is the person to ask about such questions, I think. Hopefully he takes a look. $\endgroup$
    – Qiaochu Yuan
    Commented Jul 25, 2010 at 15:07
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    $\begingroup$ Let's see! Kim himself is also a member of MO. $\endgroup$
    – Anweshi
    Commented Jul 25, 2010 at 15:08
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    $\begingroup$ Anweshi, I have a personal philosophy about this question which you could probably find discouraging. The point is that finding (a structure of) solutions to diophantine equations (showing that numbers are algebraic/algebraically or linearly dependent) stay on the "algebraic" side, while proving that there are no other solutions/relations belongs to analysis. There are several clever ways to hide analysis and claim that proofs are purely algebraic. All effective versions of Siegel's and Faltings's theorems are nevertheless analytic in nature... $\endgroup$
    – Wadim Zudilin
    Commented Jul 25, 2010 at 23:53
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    $\begingroup$ I would like to point out that Kim's proof is only for the projective line minus three points. Also, there is no known effective proof of the Mordell conjecture. This is a big open problem - see for instance "Séminaire sur Les Pinceaux de Courbes Elliptiques." À la recherche de "Mordell effectif''. Astérisque No. 183 (1990). Société Mathématique de France, Paris, 1990. pp. 1–135. The questions raised in this seminar are far from being solved. On the other hand, there are ways to bound the number of solutions (but not their size). See L. Szpiro, "Un peu d'effectivité" in Astérisque 127. $\endgroup$
    – Damian Rössler
    Commented Oct 9, 2011 at 21:04
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    $\begingroup$ Damn, there's already a "Siegel's Theorem"? I knew I shouldn't bother to think about zeros, disks, or upper half spaces, but there's no way around this one... $\endgroup$
    – Paul Siegel
    Commented Jan 20, 2012 at 15:23

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