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:

- 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 .
- Châtelet surfaces:
, where ,
. If is irreducible, then (a smooth projective model)
satisfies weak approximation. (Uses descent method.)
- The circle method: a smooth cubic hypersurface, then weak approximation holds for (Skinner 1997).

There are also results for linear algebraic groups:

- If is a -torus, and , then satisfies weak approximation beacuse is -rational (Voskreseskii).
- 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
*

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

(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).

*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.
*

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