# 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.

• 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. – Daniel Loughran Jan 28 '15 at 20:42
• 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. – Daniel Loughran Jan 28 '15 at 21:42
• @DanielLoughran: the Diophantine undecidability of $\mathbb{Q}$ is not known, although generally believed. – Laurent Moret-Bailly Jan 29 '15 at 6:57
• @Laurent Moret-Bailly: Yes thanks you are quite right. Anyway, I hope my point still stands. – Daniel Loughran Jan 29 '15 at 9:06

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.

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...)

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