# 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*.

## Use[edit]

They give conditions for the group *G*(*k*) to be dense in a restricted direct product of groups of the form *G*(*k*_{s}) for *k*_{s} 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.

## History[edit]

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.

## Formal definitions and properties[edit]

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 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 *E*_{8} 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.

## See also[edit]

## References[edit]

- 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,**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