Weak Approximation Theorem

In algebraic group theory, approximation theorems are an extension of the Chinese remainder theorem to algebraic groups ''G'' over global fields ''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 fields. The results for number fields are due to and ; the function field case, over finite fields, is due to and . In the number field case Platonov also proved a related result over local fields 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 ''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 ''N'' is 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 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 groups and inner forms of Chevalley groups, showing that the strong approximation property is restrictive.

See also

*Superstrong approximation

References

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*{{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 , series=Second Series , issn=0003-486X , volume=105 , issue=3 , pages=553–572 , doi=10.2307/1970924

Algebraic groups
Diophantine geometry