Approximation in algebraic groups
Encyclopedia
In mathematics
, strong approximation in linear algebraic group
s is an important arithmetic property of matrix group
s. In rough terms, it explains to what extent there can be an extension of the Chinese remainder theorem
to various kinds of matrices
. For example, for orthogonal matrices, there cannot be such an extension and there is a theory explaining why there is, and where the problem lies (it is in the spin groups).
Strong approximation was established in the 1960s and 1970s, for algebraic groups that are semisimple groups and simply-connected
, for global field
s. The results for number fields are due to Martin Kneser
and Vladimir Platonov
; the function field
case, over finite field
s, is due to Grigory Margulis
and Gopal Prasad
. In the number field case a result over local field
s, the Kneser–Tits conjecture of Kneser and Jacques Tits
, was proved along the way.
This article will consider only the rational number field case, to simplify notation somewhat. It will assume the concept of adelic algebraic group
, which makes statement of the result quick. There is a property, weak approximation, that can be stated in more elementary terms because it requires only the product topology
, rather than the restricted product used in the adelic theory.
Let G then be a linear algebraic group over Q. Its adelic group GA contains G(Q) embedded on the diagonal. The question asked in strong approximation is whether
is a dense subset in GA, for a certain class of subgroups H. Here H should run over the subgroups where the component for the real number
s and a certain finite set S of prime numbers p is set to the identity element e of G. That is, we look at all 'small enough' such subgroups, as S is a larger and larger finite set: the condition becomes harder to meet as S grows. If the answer is affirmative, then strong approximation holds.
Then it is known that strong approximation holds for G, if it is assumed semisimple and simply-connected, and for each simple factor of G it is true that the real points are not compact (this is comparable to asking a quadratic form
to be indefinite). These sufficient conditions are also necessary.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, strong approximation in linear algebraic group
Linear algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices that is defined by polynomial equations...
s is an important arithmetic property of matrix group
Matrix group
In mathematics, a matrix group is a group G consisting of invertible matrices over some field K, usually fixed in advance, with operations of matrix multiplication and inversion. More generally, one can consider n × n matrices over a commutative ring R...
s. In rough terms, it explains to what extent there can be an extension of the Chinese remainder theorem
Chinese remainder theorem
The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra.In its most basic form it concerned with determining n, given the remainders generated by division of n by several numbers...
to various kinds of matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
. For example, for orthogonal matrices, there cannot be such an extension and there is a theory explaining why there is, and where the problem lies (it is in the spin groups).
Strong approximation was established in the 1960s and 1970s, for algebraic groups that are semisimple groups and simply-connected
Simply connected space
In topology, a topological space is called simply connected if it is path-connected and every path between two points can be continuously transformed, staying within the space, into any other path while preserving the two endpoints in question .If a space is not simply connected, it is convenient...
, for global field
Global field
In mathematics, the term global field refers to either of the following:*an algebraic number field, i.e., a finite extension of Q, or*a global function field, i.e., the function field of an algebraic curve over a finite field, equivalently, a finite extension of Fq, the field of rational functions...
s. The results for number fields are due to Martin Kneser
Martin Kneser
Martin Kneser was a German mathematician. His father Hellmuth Kneser and grandfather Adolf Kneser were also mathematicians....
and Vladimir Platonov
Vladimir Platonov
- Academic career :In 1961 Platonov graduated with highest distinction from Belarus State University. Two years later he received his Ph.D. from Academy of Sciences of Belarus. Platonov received his Doctor of Science degree from the Academy of Sciences of USSR in 1967. Since 1972 he has been an...
; the function field
Function field
Function field may refer to:*Function field of an algebraic variety*Function field...
case, over finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
s, is due to Grigory Margulis
Grigory Margulis
Gregori Aleksandrovich Margulis is a Russian mathematician known for his far-reaching work on lattices in Lie groups, and the introduction of methods from ergodic theory into diophantine approximation. He was awarded a Fields Medal in 1978 and a Wolf Prize in Mathematics in 2005, becoming the...
and Gopal Prasad
Gopal Prasad
Gopal Prasad is an Indian mathematician. His research interests span the fields of Lie groups, their discrete subgroups, algebraic groups, arithmetic groups, geometry of locally symmetric spaces, and representation theory of reductive p-adic groups.He is the Raoul Bott Professor of Mathematics at...
. In the number field case a result over local field
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
s, the Kneser–Tits conjecture of Kneser and Jacques Tits
Jacques Tits
Jacques Tits is a Belgian and French mathematician who works on group theory and geometry and who introduced Tits buildings, the Tits alternative, and the Tits group.- Career :Tits received his doctorate in mathematics at the age of 20...
, was proved along the way.
This article will consider only the rational number field case, to simplify notation somewhat. It will assume the concept of adelic algebraic group
Adelic algebraic group
In abstract algebra, an adelic algebraic group is a topological group defined by an algebraic group G over a number field K, and the adele ring A = A of K. It consists of the points of G having values in A; the definition of the appropriate topology is straightforward only in case G is a linear...
, which makes statement of the result quick. There is a property, weak approximation, that can be stated in more elementary terms because it requires only the product topology
Product topology
In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology...
, rather than the restricted product used in the adelic theory.
Let G then be a linear algebraic group over Q. Its adelic group GA contains G(Q) embedded on the diagonal. The question asked in strong approximation is whether
- G(Q)H
is a dense subset in GA, for a certain class of subgroups H. Here H should run over the subgroups where the component for the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s and a certain finite set S of prime numbers p is set to the identity element e of G. That is, we look at all 'small enough' such subgroups, as S is a larger and larger finite set: the condition becomes harder to meet as S grows. If the answer is affirmative, then strong approximation holds.
Then it is known that strong approximation holds for G, if it is assumed semisimple and simply-connected, and for each simple factor of G it is true that the real points are not compact (this is comparable to asking a quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
to be indefinite). These sufficient conditions are also necessary.
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.