- Approximation in algebraic groups
mathematics, strong approximation in linear algebraic groups is an important arithmetic property of matrix groups. In rough terms, it explains to what extent there can be an extension of the Chinese remainder theoremto 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 fields. The results for number fields are due to Martin Kneserand Vladimir Platonov; the function fieldcase, over finite fields, is due to Grigory Margulisand Gopal Prasad. In the number field case a result over local fields, the Kneser-Tits conjecture of Kneser and Jacques Tits, was proved along the way.
This article will consider only the
rational number fieldcase, 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 productused in the adelic theory.
Let "G" then be a linear algebraic group over Q. Its adelic group "G""A" contains "G"("Q") embedded on the diagonal. The question asked in "strong" approximation is whether
dense subsetin "G""A", for a certain class of subgroups "H". Here "H" should run over the subgroups where the component for the real numbers 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 formto 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.
last=Platonov|first= Vladimir|last2= Rapinchuk|first2= Andrei
title=Algebraic groups and number theory. (Translated from the 1991 Russian original by Rachel Rowen.)
series=Pure and Applied Mathematics|volume= 139|publisher= Academic Press, Inc.|publication-place= Boston, MA|year= 1994|ISBN= 0-12-558180-7
Wikimedia Foundation. 2010.
Look at other dictionaries:
Algebraic topology — is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism. In many situations this is too much to hope for… … Wikipedia
Diophantine approximation — In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers. The absolute value of the difference between the real number to be approximated and… … Wikipedia
List of algebraic topology topics — This is a list of algebraic topology topics, by Wikipedia page. See also: topology glossary List of topology topics List of general topology topics List of geometric topology topics Publications in topology Topological property Contents 1… … Wikipedia
List of mathematics articles (A) — NOTOC A A Beautiful Mind A Beautiful Mind (book) A Beautiful Mind (film) A Brief History of Time (film) A Course of Pure Mathematics A curious identity involving binomial coefficients A derivation of the discrete Fourier transform A equivalence A … Wikipedia
Weil conjecture on Tamagawa numbers — In mathematics, the Weil conjecture on Tamagawa numbers was formulated by André Weil in the late 1950s and proved in 1989. It states that the Tamagawa number tau;( G ), where G is any connected and simply connected semisimple algebraic group G ,… … Wikipedia
Thin set (Serre) — In mathematics, a thin set in the sense of Serre is a certain kind of subset constructed in algebraic geometry over a given field K , by allowed operations that are in a definite sense unlikely . The two fundamental ones are: solving a polynomial … Wikipedia
Martin Kneser — Martin Kneser, 1973 Born 21 January 1928(1928 … Wikipedia
List of theorems — This is a list of theorems, by Wikipedia page. See also *list of fundamental theorems *list of lemmas *list of conjectures *list of inequalities *list of mathematical proofs *list of misnamed theorems *Existence theorem *Classification of finite… … Wikipedia
Glossary of arithmetic and Diophantine geometry — This is a glossary of arithmetic and Diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of… … Wikipedia
Grigory Margulis — Infobox Scientist name = Grigory Margulis box width = image width = 200px caption = Grigory Margulis birth date = February 24, 1946 (age 62) birth place = death date = death place = residence = citizenship = nationality = Russia ethnicity = field … Wikipedia