# Harari 1: Weak approximation on algebraic varieties (introduction)

Let be a number field, and let be the completion of at . Let be the set of all places of .

Basic Facts

Theorem. [Weak Approximation] Let be a finite set of places of . Let for . Then there is an which is arbitrarily close to for .

This is a refinement of the Chinese remainder theorem. One reformulation of it is as follows: the diagonal embedding is dense, the product equipped with the product of the -adic topologies.

We have the slight refinement: is dense in .

Definition. Let be a geometrically integral algebraic variety. Then satisfies weak approximation if given a finite set of places and for , there exists a -rational point which is arbitrarily close to for .

Care must be taken if is empty; by convention, we will say that in this case satisfies weak approximation even if is empty.

We see weak approximation is equivalent to the statement that is dense in .

Remark. If is projective, and weak approximation is equivalent to strong approximation, namely, is dense in for the adelic topology. (Here, , a flat and proper model of .)

Let be smooth. Assume that is -birational to . Then satisfies weak approximation if and only if satisfies weak approximation (a consequence of the implicit function theorem for ).

We can speak about weak approximation for a function field : this means that weak approximation holds for any smooth (projective) model of .

Example. The spaces , and more generally, , satisfy weak approximation, as does any -rational variety, e.g. a smooth quadric with a -point.

More Examples

Theorem. Let a (smooth) projective quadric. Then satisfies weak approximation.

Here, we do not assume that there is a -rational point. This is the difficult part, the Hasse-Minkowski theorem: if for all , then .

There are several results for complete intersections:

1. A smooth intersection of quadrics (Colliot-Thélène, Sansuc, Swinnerton-Dyer 1987) satisfies weak approximation if or if and there exists a pair of skew-conjugate lines on .

2. Châtelet surfaces: , where , . If is irreducible, then (a smooth projective model) satisfies weak approximation. (Uses descent method.)

3. The circle method: a smooth cubic hypersurface, then weak approximation holds for (Skinner 1997).

There are also results for linear algebraic groups:

1. If is a -torus, and , then satisfies weak approximation beacuse is -rational (Voskreseskii).
2. If is a semi-simple, simply connected linear -group, then satisfies weak approximation (Kneser-Platonov, around 1969).

The Fibration Method

Theorem. Let be a projective, flat surjective morphism (with smooth, to simplify). Assume that

1. is projective and satisfies weak approximation;
2. Almost all -fibers of satisfy weak approximation; and
3. All fibers of are geometrically integral.
Then satisfies weak approximation.

(Here almost all means on a Zariski-dense open subset).

There are refinements when is the projective space : you can accept degenerate fibers on one hyperplane (using the strong approximation theorem for the affine space).

Applications: (i) Hasse-Minkowski theorem, from four variables to five; (ii) intersection of quadrics in for (here one uses a fibration in Châtelet surfaces) and with a pair of skew conjugate lines (to go from to by induction); (iii) cubic hypersurfaces of dimension with 3 conjugate singular points (Colliot-Thélène, Salberger).

Proof. Start with a smooth -point for any on . Project . Use weak approximation on , so can approximate by for . Consider the fiber ; has a -point close to for by the implicit function theorem. To apply weak approximation on , we check that for ; this is OK if is sufficiently large by the Weil estimates : here we use that all -fibers are geometrically irreducible, which implies that the reduction mod. of also is for a sufficiently large (independent of ).

Some Counterexamples

Cubic surfaces: the surface fails the Hasse Principle (Cassels, Guy).

Certain intersections of two quadrics in (see above).

Looking (over the rationals) at , , , it is possible to construct counterexamples to weak approximation. The idea: , ; there exists a finite set such that if and , then is a norm of (use a computation with valuations). If you find and such that there exists such that is not a local norm and there exists such that is a local norm, then there is no weak approximation. (Think: global reciprocity of class field theory.)

For tori, let be a biquadratic extension, then there are counterexamples like , where is a basis of ; this holds e.g. for , .

Theorem. [Minchev] Let be a projective, smooth -variety, assume that , where , an algebraic closure. Assume , then does not satisfy weak approximation.

Proof. [Sketch of proof] Enlarge the situation over where is a finite set of places. By assumption, there is a nontrivial geometrically connected covering , which for models gives . Take an arbitrary , the fibre where is an étale algebra ; each is unramified outside . Only finitely many are possible (by Hermite's Theorem). Find such that is totally split for each (by Cebotarev's Theorem); find such that the fiber of at for is not (this is possible because is geometrically connected, via a "geometric" Cebotarev-like Theorem). Then cannot be approximated by a rational point .

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