Approximation in algebraic groups

Strong approximation was established in the 1960s and 1970s, for semisimple simply-connected algebraic groups over global fields.

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 AS for the ring of S-adeles and AS for the product of the completions ks, for s in the finite set S. For any choice of S, G(k) embeds in G(AS) and G(AS).

If the group G is connected and k-rational, then it satisfies weak approximation with respect to any set S (Platonov & Rapinchuk 1994, p.402).

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 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).