Let be a number field, and let be the completion of at . Let be the set of all places of .
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.
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:
There are also results for linear algebraic groups:
The Fibration Method
Theorem. Let be a projective, flat surjective morphism (with smooth, to simplify). Assume that
(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).
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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