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In algebraic group theory, approximation theorems are an extension of the
Chinese remainder theorem In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
to
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...
s ''G'' over
global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: * Algebraic number field: A finite extension of \mathbb *Global function fi ...
s ''k''.


History

proved strong approximation for some classical groups. Strong approximation was established in the 1960s and 1970s, for semisimple simply-connected algebraic groups over
global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: * Algebraic number field: A finite extension of \mathbb *Global function fi ...
s. The results for
number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...
s are due to and ; the function field case, over
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
s, is due to and . In the number field case Platonov also proved a related result over
local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact t ...
s called the Kneser–Tits conjecture.


Formal definitions and properties

Let ''G'' be a linear algebraic group over a global field ''k'', and ''A'' the adele ring of ''k''. If ''S'' is a non-empty finite set of places of ''k'', then we write ''A''''S'' for the ring of ''S''-adeles and ''A''''S'' for the product of the completions ''k''''s'', for ''s'' in the finite set ''S''. For any choice of ''S'', ''G''(''k'') embeds in ''G''(''A''''S'') and ''G''(''A''''S''). The question asked in ''weak'' approximation is whether the embedding of ''G''(''k'') in ''G''(''A''''S'') has dense image. If the group ''G'' is connected and ''k''-rational, then it satisfies weak approximation with respect to any set ''S'' . More generally, for any connected group ''G'', there is a finite set ''T'' of finite places of ''k'' such that ''G'' satisfies weak approximation with respect to any set ''S'' that is disjoint with ''T'' . In particular, if ''k'' is an algebraic number field then any connected group ''G'' satisfies weak approximation with respect to the set ''S'' = ''S'' of infinite places. The question asked in ''strong'' approximation is whether the embedding of ''G''(''k'') in ''G''(''A''''S'') has dense image, or equivalently whether the set :''G''(''k'')''G''(''A''''S'') is a
dense subset In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ra ...
in ''G''(''A''). The main theorem of strong approximation states that a non-solvable linear algebraic group ''G'' over a global field ''k'' has strong approximation for the finite set ''S'' if and only if its
radical Radical may refer to: Politics and ideology Politics *Radical politics, the political intent of fundamental societal change *Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe and ...
''N'' is
unipotent In mathematics, a unipotent element ''r'' of a ring ''R'' is one such that ''r'' − 1 is a nilpotent element; in other words, (''r'' − 1)''n'' is zero for some ''n''. In particular, a square matrix ''M'' is a unipotent ...
, ''G''/''N'' is simply connected, and each almost simple component ''H'' of ''G''/''N'' has a non-compact component ''H''''s'' for some ''s'' in ''S'' (depending on ''H''). The proofs of strong approximation depended on the
Hasse principle In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of eac ...
for algebraic groups, which for groups of type ''E''8 was only proved several years later. Weak approximation holds for a broader class of groups, including
adjoint group In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL(n ...
s and
inner form In mathematics, an inner form of an algebraic group G over a field K is another algebraic group H such that there exists an isomorphism \phi between G and H defined over \overline K (this means that H is a ''K-form'' of G) and in addition, for every ...
s of
Chevalley group In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phras ...
s, showing that the strong approximation property is restrictive.


See also

*
Superstrong approximation Superstrong approximation is a generalisation of strong approximation in algebraic groups ''G'', to provide spectral gap results. The spectrum in question is that of the Laplacian matrix associated to a family of quotients of a discrete group Γ; a ...


References

* * * * * *{{Citation , last1=Prasad , first1=Gopal , title=Strong approximation for semi-simple groups over function fields , jstor=1970924 , mr=0444571 , year=1977 , journal=
Annals of Mathematics The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the ...
, series=Second Series , issn=0003-486X , volume=105 , issue=3 , pages=553–572 , doi=10.2307/1970924 Algebraic groups Diophantine geometry