Approximation in algebraic groups

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 embeds in G and G.
The question asked in weak approximation is whether the embedding of G in G 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 in G has dense image, or equivalently whether the set
is a dense subset in G. 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.
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.