Let be a smooth, geometrically integral variety over (a number field), and suppose that is projective. We denote by the closure of in .
Here our aim is to: (i) explain the counterexamples to weak approximation; (ii) find `intermediate' sets between and ; (iii) in some cases, prove that .
General setting
Let be an algebraic group (usually linear, but not necessarily connected, e.g. finite). If is commutative: define the étale cohomology groups (; the cohomological dimension of a number field forgetting real places makes the higher cohomology groups uninteresting). In general, we have only the pointed set (defined by Cech cocycles for the étale topology). If , , where . If is linear, corresponds to -torsors over up to isomorphism.
Take , define
Obviously . We will see that in many cases
Example.
(Indeed the Brauer group of the ring of integers of is zero). is the Brauer-Manin set of . Manin showed in 1970 that for a genus one curve with finite Tate-Shafarevich group, the condition implies the existence of a rational point.
Remark. If is rational, then is finite, where . Then is `computable'.
Theorem. [H, Skorobogatov] If is linear and , then (and is "computable").
Abelian descent theory
This was developed by Colliot-Thélène and Sansuc, and recently completed by Skorobogatov.
Theorem. Define
This Theorem is difficult, see Skorobogatov's book for a complete account on the subject. One of the ideas is to recover the Brauer group of (mod. ) making cup-products , where and is the class of in .
Now assume that is a rational variety, so (since ). Assume . Consider a universal torsor . If , can define where
If you can prove that the torsors satisfy weak approximation, then , so the Brauer-Manin obstruction is the only one.
Example. There are many examples of this:
If is reducible, we can have a counterexample to weak approximation, e.g. , where , , in some cases there is an obstruction given by the Hilbert symbol .
Theorem. [Sansuc 1981] Let be a linear connected algebraic group over , a smooth compactification of , then the Brauer-Manin obstruction is the only one:
Back to fibration methods
If is a fibration, we saw that if the base and the fibres satisfy weak approximation, under certain circumstances then satisfies weak approximation.
Here we consider , a projective, surjective morphism (and the generic fibre is smooth). Assume also that all fibres are geometrically integral (can do with all but one because of strong approximation on the affine line).
Theorem. [H 1993, 1996] Yes, if you assume that:
Two ideas:
Applications: (i) Recover Sansuc's result just knowing the case of a torus; (ii) If you know that for a smooth cubic surface, then by induction the same holds for hypersurfaces, so if , then satisfies weak approximation.
Nonabelian descent
If is a finite but not commutative -group, it is possible that for , .
Theorem. [Skorobogatov 1997] There exists a bi-elliptic surface such that , .
Actually: for some , .
There are similar statements for weak approximation (H 1998), e.g. take any bi-elliptic surface, , then .
Nevertheless the Brauer-Manin condition is quite strong, as shows the following result :
Theorem. [H 2001] We have:
Open question : is the first part of this theorem still true for a which is an extension of a finite abelian group by a connected linear group ? My guess is "no".
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