Approximation in algebraic groups

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 result over local fields called the Kneser–Tits conjecture.

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 (Platonov & Rapinchuk 1994, p.402). 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 (Platonov & Rapinchuk 1994, p.415). 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 (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 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 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.

• Superstrong approximation

• Eichler, Martin (1938), "Allgemeine Kongruenzklasseneinteilungen der Ideale einfacher Algebren über algebraischen Zahlkörpern und ihre L-Reihen.", Journal für die Reine und Angewandte Mathematik (in German), 179: 227–251, doi:10.1515/crll.1938.179.227, ISSN 0075-4102
• Kneser, Martin (1966), "Strong approximation", Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Providence, R.I.: American Mathematical Society, pp. 187–196, MR 0213361
• Margulis, G. A. (1977), "Cobounded subgroups in algebraic groups over local fields", Akademija Nauk SSSR. Funkcional'nyi Analiz i ego Priloženija, 11 (2): 45–57, 95, ISSN 0374-1990, MR 0442107
• Platonov, V. P. (1969), "The problem of strong approximation and the Kneser–Tits hypothesis for algebraic groups", Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, 33: 1211–1219, ISSN 0373-2436, MR 0258839
• Platonov, Vladimir; Rapinchuk, Andrei (1994), Algebraic groups and number theory. (Translated from the 1991 Russian original by Rachel Rowen.), Pure and Applied Mathematics, vol. 139, Boston, MA: Academic Press, Inc., ISBN 0-12-558180-7, MR 1278263
• Prasad, Gopal (1977), "Strong approximation for semi-simple groups over function fields", Annals of Mathematics, Second Series, 105 (3): 553–572, doi:10.2307/1970924, ISSN 0003-486X, JSTOR 1970924, MR 0444571