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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.

11G35 Varieties over global fields
14G05 Rational points
Full Text: DOI
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