cohomological obstructions and rational points 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. 
 A: 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.
A: 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...)
