What is know about a generalisation of Siegel's theorem on integral points to higher dimensional varieties?
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3$\begingroup$ We still don't know if there is a cubic surface with a non-zero finite number of rational points and people don't know whether to believe Lang's conjectures, so it's not clear to me we know much about anything in this generality. Are you interested in weak statements which might hold for all varieties, or stronger statements that only apply to smaller classes? And do you want theorems or conjectures? $\endgroup$– Kevin BuzzardCommented Feb 5, 2017 at 9:44
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1$\begingroup$ @Kevin Buzzard: Did you mean K3 surface, rather than cubic surface? $\endgroup$– Daniel LoughranCommented Feb 5, 2017 at 10:02
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4$\begingroup$ Well the main conjecture is the Lang-Vojta conjecture. This has a few formulations, which are conjecturally equivalent. One version says that any smooth Brody hyperbolic variety has only finitely many integral points (a smooth variety $X$ is called Brody hyperbolic if any holomorphic map $\mathbb{C} \to X(\mathbb{C})$ is constant.) This is known is some special cases, but of course wide open in general. $\endgroup$– Daniel LoughranCommented Feb 5, 2017 at 10:08
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2$\begingroup$ I also agree with Kevin that this question lacks focus somewhat. Mathoverflow does not work very well for "what is known" type questions. $\endgroup$– Daniel LoughranCommented Feb 5, 2017 at 10:31
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3$\begingroup$ Faltings proved that an affine open subset of an abelian variety has finitely many integral points. There is also a number of papers of Corvaja, Zannier and others proving some finiteness statements in various cases. I also agree that this question lacks focus. $\endgroup$– Felipe VolochCommented Feb 5, 2017 at 18:00
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