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Let $X$ a (nice) scheme over $\mathbb{Q}$. Are there cohomolgical obstructions answering the following questions:

1) is $X(\mathbb{Q})$ an empty set ?

2) is $X(\mathbb{Q})$ a finite (non empty) set ?

3) is $X(\mathbb{Q})$ an infinite set ?

thanks.

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  • $\begingroup$ What do you mean by "answering the following questions"? There are many cohomological obstructions to the existence of rational points, most of which are variants and refinements of the Brauer-Manin obstruction. In special cases it is known to be the only obstruction, but not in general; so I don't know if this can be used to "answer" your questions. It would be helpful if you could clarify more what you want, and what you know already. $\endgroup$ – Daniel Loughran Jan 28 '15 at 20:42
  • $\begingroup$ I guess my point is that these problems are very difficult in general; moreover 1) is even known to be undecidable (Hilbert's 10th problem). So the answer to all your questions in the stated generality is "no" with current technology. $\endgroup$ – Daniel Loughran Jan 28 '15 at 21:42
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    $\begingroup$ @DanielLoughran: the Diophantine undecidability of $\mathbb{Q}$ is not known, although generally believed. $\endgroup$ – Laurent Moret-Bailly Jan 29 '15 at 6:57
  • $\begingroup$ @Laurent Moret-Bailly: Yes thanks you are quite right. Anyway, I hope my point still stands. $\endgroup$ – Daniel Loughran Jan 29 '15 at 9:06
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Let me just give some pointers to further literature.

In the curve case, there is a cohomological obstruction to a curve over $\mathbb{Q}$ having infinitely many points, it is given by Faltings' solution of the Mordell conjecture (I know that this is almost certainly a deliberate misinterpretation of the question). In another direction, Grothendieck's section conjecture sets up a conjectural relation between rational points on curves and homotopical data, namely the etale fundamental group resp. the fundamental exact sequence.

In higher dimensions, as mentioned, the Brauer-Manin obstruction is the best known cohomological obstruction. There are refinements using non-abelian cohomology, a textbook treatment of which can be found in the book

  • A. Skorobogatov. Torsors and rational points. Cambridge Tracts in Mathematics, 144. Cambridge University Press, Cambridge, 2001.

Again, in a more homotopical direction (in a way generalizing both the Brauer-Manin obstruction and the above-mentioned section conjecture stuff), there is the work of Harpaz and Schlank on homotopy obstructions to rational points.

Finally, in case the variety actually has infinitely many points, I think the Manin conjecture should also be mentioned. It provides an asymptotic of the number of points of bounded height, but (as far as I remember) the numbers influencing the growth asymptotic are essentially cohomological in nature.

If you use the keywords provided to search, you will likely end up with a number of relevant interesting papers relating rational points on varieties to cohomological of homotopical information.

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The most well known (only?) cohomological obstruction to the existence of rational points is the Brauer-Manin obstruction (cf Manin's "Cubic Forms", 2nd ed; Lang's "Number Theory III" is a good short survey). It is primarily for varieties that are essentially rational, but there are known counterexamples to the Hasse principle for which the Brauer-Manin obstruction is not sufficient (Skorobogatov, Invent. Math. 135 (1999)). Poonen et al have done additional & important work on this topic.

(of course there is the good-old Tate-Shafarevich group, but I believe (?) Swinnerton-Dyer shows that it essentially reduces to Brauer-Manin if you assume some standard Schinzel-ish conjecture...)

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  • $\begingroup$ There are other obstructions than the Brauer-Manin obstruction, such as the étale Brauer-Manin obstruction, which is in general a finer obstruction. $\endgroup$ – Daniel Loughran Jan 29 '15 at 9:07

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