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. 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
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.
- Eichler, Martin (1938), "Allgemeine Kongruenzklasseneinteilungen der Ideale einfacher Algebren über algebraischen Zahlkörpern und ihre L-Reihen.", Journal für 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 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, ISSN 0003-486X, JSTOR 1970924, MR 0444571