- Approximation in algebraic groups
In

mathematics ,**strong approximation in**is an important arithmetic property oflinear algebraic group smatrix group s. In rough terms, it explains to what extent there can be an extension of theChinese remainder theorem to various kinds of matrices. For example, fororthogonal matrices , there cannot be such an extension and there is a theory explaining why there is, and where the problem lies (it is in thespin group s).Strong approximation was established in the 1960s and 1970s, for algebraic groups that are

semisimple group s and simply-connected, forglobal field s. The results fornumber field s are due toMartin Kneser andVladimir Platonov ; thefunction field case, overfinite field s, is due toGrigory Margulis andGopal Prasad . In the number field case a result overlocal field s, the**Kneser-Tits conjecture**of Kneser andJacques Tits , was proved along the way.This article will consider only the

rational number field case, to simplify notation somewhat. It will assume the concept ofadelic algebraic group , which makes statement of the result quick. There is a property,**weak approximation**, that can be stated in more elementary terms because it requires only theproduct topology , rather than therestricted product used in the adelic theory.Let "G" then be a linear algebraic group over

**Q**. Its adelic group "G"_{"A"}contains "G"("Q") embedded on the diagonal. The question asked in "strong" approximation is whether:"G"("Q")"H"

is a

dense subset in "G"_{"A"}, for a certain class of subgroups "H". Here "H" should run over the subgroups where the component for thereal number s and a certain finite set "S" of prime numbers "p" is set to the identity element "e" of "G". That is, we look at all 'small enough' such subgroups, as "S" is a larger and larger finite set: the condition becomes harder to meet as "S" grows. If the answer is affirmative, then**strong approximation**holds.Then it is known that strong approximation holds for "G", if it is assumed semisimple and simply-connected, and for each simple factor of "G" it is true that the real points are not compact (this is comparable to asking a

quadratic form to be indefinite). These sufficient conditions are also necessary.Weak approximation holds for a broader class of groups, including

adjoint group s andinner form s ofChevalley group s, showing that the strong approximation property is restrictive.**References***citation|id=MR|1278263

last=Platonov|first= Vladimir|last2= Rapinchuk|first2= Andrei

title=Algebraic groups and number theory. (Translated from the 1991 Russian original by Rachel Rowen.)

series=Pure and Applied Mathematics|volume= 139|publisher= Academic Press, Inc.|publication-place= Boston, MA|year= 1994|ISBN= 0-12-558180-7

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