Approximation in algebraic groups

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Approximation in algebraic groups

In mathematics, strong approximation in linear algebraic groups is an important arithmetic property of matrix groups. In rough terms, it explains to what extent there can be an extension of the Chinese remainder theorem to various kinds of matrices. For example, for orthogonal matrices, there cannot be such an extension and there is a theory explaining why there is, and where the problem lies (it is in the spin groups).

Strong approximation was established in the 1960s and 1970s, for algebraic groups that are semisimple groups and simply-connected, for global fields. The results for number fields are due to Martin Kneser and Vladimir Platonov; the function field case, over finite fields, is due to Grigory Margulis and Gopal Prasad. In the number field case a result over local fields, the Kneser-Tits conjecture of Kneser and Jacques 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 of adelic 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 the product topology, rather than the restricted 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 the real numbers 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 groups and inner forms of Chevalley groups, 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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