Approximation in algebraic groups

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In algebraic group theory, approximation theorems are an extension of the Chinese remainder theorem to algebraic groups G over global fields k. They give conditions for the group G(k) to be dense in a restricted direct product of groups of the form G(ks) for ks a completion of k at the place s. In weak approximation theorems the product is over a finite set of places s, while in strong approximation theorems the product is over all but a finite set of places.

Eichler (1938) 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 Kneser (1966) and Platonov (1969); the function field case, over finite fields, is due to Margulis (1977) and Prasad (1977). In the number field case Platonov also proved a related a result over local fields called the Kneser–Tits conjecture.

Let G be a linear algebraic group over a global field k, and A its ring of adeles. Its adelic group G(A) contains G(k) embedded on the diagonal. The question asked in strong approximation is whether

G(k)GS

is a dense subset in G(A), for a subgroup GS given by the product of G(ks) for s in the finite set S. If the answer is affirmative, then strong approximation holds. The main theorem of strong approximation (Kneser 1966, p.188) 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 Hs 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 E8 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.

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