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|If G is defined over an algebraic number field or over a field of algebraic functions |
in one variable, then there arise problems about the arithmetic properties of G,
the study of which is the subject of the arithmetic theory of linear algebraic groups
|Therefore T(a, b) is not stably rational over Q. This is the simplest example of a |
linear algebraic group which is not rational over Q. Consider two cases. ... any
valuation v of Q. 2) There is a decomposition group of order 4, for example, L I Q(
§), is I 1, L = Q(\/:I, Class field theory tells ... Another arithmetic characteristic of G
arises when studying the set H l (k, G) of principal homogeneous spaces (PHS) of
|Introduction The arithmetic theory of linear algebraic groups was established as |
an independent field of mathematics within the last decade. One of its
characteristic features is a deep connection with many areas of mathematics, in
|Proceedings of the International Conference on the Algebraic and Arithmetic |
Theory of Quadratic Forms, December ... research network “Algebraic K-Theory,
Linear Algebraic Groups and Related Structures” HPRN-CT-2002-00287, and by
|For studying arithmetic groups, theory of linear algebraic groups is essential. |
Linear algebraic groups also provide a natural class of Lie groups and put real
Lie groups and p-adic Lie groups on the same footing. The reduction theory of ...
|C. Moore,  Group extensions of p-adic and adelic linear groups. Inst. Hautes ... |
 Theory of algebraic linear groups and periodic groups. ... V. P. Platonov, M. V.
Milovanov, [l] Determination of algebraic groups by arithmetic subgroups. Dokl ...
|... Algebraic group, Algebraic torus, Anisotropic group; Arithmetic group; Borel |
subgroup; Bruhat decomposition; Congruence problem; Diagonal group, Lie
group; Linear algebraic groups, arithmetic theory of; Linear group, Mumford
|The arithmetic subgroups defined in this more general case are also |
commensurable with “integral” elements. ... Alternatively, suppose that G is a Q-
simple linear algebraic group such that its real locus G can be written as a
product G = G1 X ...
|On Conformal Field Theories, Discrete Groups and Renormalization Pierre E. |
Cartier, Bernard Julia, Pierre Moussa, ... general theory of algebraic groups, one
could speak of the arithmetic subgroups of any linear algebraic group over Q.