# Harari 2: Weak approximation on algebraic varieties (cohomology)

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.

1. ;

(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.

2. Let be a Galois, geometrically connected, nontrivial étale covering with group . Then , where is considered as a constant group scheme. Then (via Hermite's Theorem). It is possible to find , which implies Minchev's result that does not satisfy weak approximation.

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

where . Assume that . Then:
1. We have

2. Assume further that is of finite type, set such that ; then there exists a torsor under (a universal torsor, i.e. "as nontrivial as possible") such that

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

Then

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:

1. Châtelet surface: , , . Colliot-Thélène, Sansuc, Swinnerton-Dyer showed that , so the Brauer-Manin obstruction is the only one. If is irreducible, then , so satisfies weak approximation.

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 .

2. Conic bundles over with at most degenerate fibres. Results of Colliot-Thélène, Salberger, Skorobogatov covered at most . In the case of , the existence of a global rational point is easy to show, so the only problem is weak approximation, and which is due to Salberger, Skorobogatov 1993 (using descent and -theory).

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:

1. is torsion-free, where , ; e.g. rational, or smooth complete intersection of dimension at least three.
2. is finite.

Two ideas:

1. is an isomorphism for many -fibres (many' in the sense of Hilbert's irreducibility theorem).
2. If are elements of , assume , an open subset. Use the formal lemma': Take , , a finite set of places; then there exists , , finite such that:
1. for ;
2. for , where is the local invariant.

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:

1. If is a linear connected -group, , then

2. If is any commutative -group, , then

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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