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The Brauer-Manin obstruction for integral points on curves. (English) Zbl 1280.11038
The paper under review discusses the Brauer-Manin obstruction for integral points on hyperbolic smooth curves $$X$$ defined over number fields $$k$$. Let $$S$$ be a finite subset of places of $$k$$. By definition, Brauer-Manin suffices for $$X$$ if the diagonal image of $$S$$-integral points of $$X$$ (which is finite by Siegel’s theorem) coincides with the subset of the product of local integral points outside $$S$$ that are orthogonal to $$S$$-locally constant elements of the Brauer group.
The authors conjecture that Brauer-Manin suffices for any hyperbolic curves $$X$$ contained in $$\mathbb{P}^{1}$$. The main result of the paper gives several equivalent formulations of the conjecture, which concern about the adelic intersection in the generalized Jacobian of $$X$$. In particular, if $$X$$ is $$\mathbb{P}^{1}$$ minus three points, the conjecture implies an old conjecture on “exponential Diophantine equation” raised by T. Skolem [Avh. Norske Vid. Akad. Oslo 1937, No. 12, 1–16 (1937; Zbl 0017.24606, JFM 63.0889.03)].
Besides the main result, the authors also consider higher genus cases, they show that:
- If $$X$$ is an open subset of a projective curve of genus at least $$2$$, then the fact that Brauer-Manin suffices for $$X$$ follows from the conjecture raised by V. Scharaschkin [Local-global problems and the Brauer-Manin obstruction. Ph.D. thesis. University of Michigan (1999)] and A. Skorobogatov [Torsors and rational points. Cambridge: Cambridge University Press (2001; Zbl 0972.14015)] saying that the Brauer-Manin obstruction is the only obstruction to weak approximation for rational points on smooth projective curves.
If $$X$$ is an open subset of an elliptic curve with finite Mordell-Weil group and finite Tate-Shafarevich group, then Brauer-Manin suffices for $$X$$. Here the authors give an example (an elliptic curve over $$\mathbb{Q}$$ minus the infinity point) to show that the condition on finiteness of the Mordell-Weil group is necessary.

##### MSC:
 11G35 Varieties over global fields 14G05 Rational points
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